Research ArticlePricing FX Options in the HestonCIR Jump-Diffusion Modelwith Log-Normal and Log-Uniform Jump Amplitudes

Rehez Ahlip1 and Ante Prodan2

1School of Computing and Mathematics University of Western Sydney South Penrith NSW 1797 Australia2School of Computing Engineering and Mathematics University of Western Sydney Sydney NSW 1797 Australia

Correspondence should be addressed to Rehez Ahlip rahlipuwseduau

Received 26 November 2014 Accepted 7 May 2015

Academic Editor Enzo Orsingher

Copyright copy 2015 R Ahlip and A Prodan This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

We examine foreign exchange options in the jump-diffusion version of the Heston stochastic volatility model for the exchange ratewith log-normal jump amplitudes and the volatility model with log-uniformly distributed jump amplitudes We assume that thedomestic and foreign stochastic interest rates are governed by the CIR dynamics The instantaneous volatility is correlated withthe dynamics of the exchange rate return whereas the domestic and foreign short-term rates are assumed to be independent ofthe dynamics of the exchange rate and its volatility The main result furnishes a semianalytical formula for the price of the foreignexchange European call option

1 Introduction

We extend the results fromAhlip and Rutkowski [1] by deriv-ing a closed-form pricing formula for the foreign exchange(FX) options in a model where the spot exchange rate andits volatility are jump-diffusions with log-normal and log-uniform jump amplitudes respectively whereas the domesticand foreign interest rates are governed by the CoxndashIngersollndashRoss (CIR) dynamics postulated in [2] In particular ourmodel allows for correlation between the exchange rateprocess and its instantaneous volatility The interest rateprocesses are independent of one another and they are alsoindependent of the foreign exchange rate and its volatility

In the seminal paper by Heston [3] the author notedthat increasing the volatility of volatility only increases thekurtosis of spot returns and does not capture skewness Inorder to capture the skewness it is crucial to include alsothe properly specified correlation between the volatility andthe spot exchange rate returns In papers by Bakshi et al[4] Bates [5] and Duffie et al [6] the authors showed thatstochastic volatility models do not offer reliable prices forclose to expiration derivatives This motivated Bates [5] and

Bakshi et al [4] to introduce jumps to the dynamics ofthe exchange rate However as observed by Andersen andAndreasen [7] andAlizadeh et al [8] the addition of jumps tothe dynamics of the exchange rate is not sufficient to capturethe sudden increase in volatility due to market turbulenceSince the overall volatility in financial markets consists of ahighly persistent slowmoving and rapidmoving componentsEraker et al [9] proposed to introduce jump process to thedynamics of the volatility process in order to enhance thecross-sectional impact on option prices

More recently DrsquoIppoliti et al [10] obtained closed-formsolutions in the spirit of Heston in a model with jumps inboth spot returns of the underlying asset and its volatilityYan and Hanson [11] consider a model in which the stockprices follow a jump-diffusion process with log-uniformlydistributed jump amplitudes under the Heston volatilitymodel In the above-mentioned papers the authors assumeconstant interest rates Although the assumption of constantinterest rates leads to highly tractable FX models empiricalresults have confirmed that such models do not reflect themarket reality especially for long-dated hybrid FX productsFor these products the fluctuations of both the exchange rate

Hindawi Publishing CorporationInternational Journal of Stochastic AnalysisVolume 2015 Article ID 258217 15 pageshttpdxdoiorg1011552015258217

2 International Journal of Stochastic Analysis

and the interest rates are critical so that the constant interestrates assumption is clearly inappropriate for reliable valuationand hedging

Let us comment briefly on the existing literature inthe same vein van Haastrecht et al [12] have extendedthe stochastic volatility model of Schobel and Zhu [13] toequitycurrency derivatives by including stochastic interestrates and assuming all driving model factors to be instanta-neously correlated Since theirmodel is based on theGaussianprocesses it enjoys analytical tractability even in the mostgeneral case of a full correlation structure By contrast whenthe squared volatility is driven by the CIR process and theinterest rate is driven either by the Vasicek [14] or the Cox etal [2] process a full correlation structure leads to intractabil-ity of equity options even under a partial correlation of thedriving factors as have been documented by among othersvan Haastrecht and Pelsser [15] and Grzelak and Oosterlee[16 17] who examined in particular the HestonVasicekand HestonCIR hybrid models (see also Grzelak et al[18] where the SchobelndashZhuHullndashWhite and HestonHullndashWhite models for equity derivatives are studied)

Our goal is to derive semianalytical solutions for pricesof plain-vanilla FX options in a model in which the instanta-neous volatility component is specified by the extended Hes-ton model with log-normally and log-uniformly distributedjump amplitudes for the exchange rate and the volatilityprocess respectively whereas the short-term interest ratesfor the domestic and foreign economies are governed by theindependent CIR processes The model thus incorporatesimportant empirical characteristics of exchange rate returnvariability (a) the correlation between the exchange rateand its stochastic volatility (b) the presence of jumps in theexchange rate and volatility processes and (c) the randomcharacter of interest rates The practical importance of thisfeature of newly developed FX models is rather clear in viewof the existence of complex FX products that have a longlifetime and are sensitive to smiles or skews in the marketThe results obtained in this paper extend results obtained byGuoqing et al In their model only the stock price process issubject to jumps but the volatility of volatility is modeled bythe Heston dynamics

The paper is organized as follows In Section 2 we set theforeign exchange model examined in this work The optionspricing problem is introduced in Section 3 The main resultTheorem 3 of Section 4 furnishes the pricing formula forFX options It is worth stressing that the independence ofvolatility and interest rates appears to be a crucial assumptionfrom the point of view of analytical tractability and thus itcannot be relaxed Numerical illustrations of our method areprovided in Section 5 where the diffusion and jump-diffusionmodels are compared

2 The HestonCIR Jump-Diffusion ForeignExchange Model

Let (ΩFP) be an underlying probability space Let theexchange rate 119876 = (119876

119905)119905isin[0119879] its instantaneous squared

volatility V = (V119905)119905isin[0119879] the domestic short-term interest

rates 119903 = (119903119905)119905isin[0119879] and the foreign short-term interest rate

119903 = (119903119905)119905isin[0119879] be governed by the following system of SDEs

119889119876119905

= (119903119905

minus 119903119905

minus 120582120583119876

) 119876119905119889119905 + 119876

119905radicV119905

119889119882119876

119905+ 119876

119905minus119889119885

119876

119905

119889V119905

= (120579 minus 120581V119905) 119889119905 + 120590VradicV

119905119889119882

V119905

+ 119889119885V119905

119889119903119905

= (119886119889

minus 119887119889119903119905) 119889119905 + 120590

119889radic119903119905

119889119882119889

119905

119889119903119905

= (119886119891

minus 119887119891

119903119905) 119889119905 + 120590

119891radic119903

119905119889119882

119891

119905

(1)

We work under the following standing assumptions

(A1) Processes 119882119876 = (119882119876

119905)119905isin[0119879] and 119882V = (119882V

119905)119905isin[0119879]

are correlated Brownian motions with a constantcorrelation coefficient so that the quadratic covari-ation between the processes 119882

119876 and 119882V satisfies119889[119882119876 119882V]

119905= 120588 119889119905 for some constant 120588 isin [minus1 1]

(A2) Processes 119882119889 = (119882119889

119905)119905isin[0119879] and 119882119891 = (119882

119891

119905)119905isin[0119879]

are independent Brownian motions and they are alsoindependent of the Brownian motions 119882119876 and 119882Vhence the processes 119876 119903 and 119903 are independent

(A3) The process 119885119876

119905= sum

119873119876

119905

119896=1 119869119876

119896is the compound Poisson

process specifically the Poisson process 119873119876 has theintensity 120582

119876gt 0 and the random variables ln(1+ 119869

119876

119896)

119896 = 1 2 have the probability distribution119873(ln[1+

120583119876

] minus (12)1205902119876

1205902119876

) hence the jump sizes (119869119876

119896)infin

119896=1 arelog-normally distributed on (minus1 infin) with mean 120583

119876gt

minus1(A4) The process 119885V

119905= sum

119873V119905

119896=1 119869V119896is the compound Poisson

process specifically the Poisson process 119873V has theintensity 120582V gt 0 and the jump sizes 119869V

119896are uniformly

distributed(A5) The Poisson processes 119873119876 119873V and sequences of

random variables (119869119876

119896)infin

119896=1 and (119869V119896)infin

119896=1 are mutuallyindependent as well as independent of the Brownianmotions 119882

119876 119882V 119882119889 119882119891(A6) The modelrsquos parameters satisfy the stability condi-

tions 2120579 gt 1205902V gt 0 2119886

119889gt 1205902

119889gt 0 and 2119886

119891gt 1205902

119891gt 0

(see eg Wong and Heyde [19])

Note that we postulate that the instantaneous squaredvolatility process V the domestic short-term interest rate 119903and the foreign interest rate 119903 are independent stochasticprocesses We will argue in what follows that this assumptionis indeed crucial for analytical tractability For brevity werefer to the foreign exchange model given by SDEs (1) underAssumptions (A1)ndash(A6) as the HestonCIR jump-diffusionFX model

3 Foreign Exchange Call Option

We will first establish the general representation for thevalue of the foreign exchange (ie currency) Europeancall option with maturity 119879 gt 0 and a constant strikelevel 119870 gt 0 The probability measure P is interpreted as

International Journal of Stochastic Analysis 3

the domestic spot martingale measure (ie the domesticrisk-neutral probability) We denote by F = (F

119905)119905isin[0119879] the

filtration generated by the Brownian motions 119882119876 119882V 119882119889119882119891 and the compound Poisson processes 119885119876 and 119885V Wewrite EP

119905(sdot) and P

119905(sdot) to denote the conditional expectation

and the conditional probability underPwith respect to the120590-fieldF

119905 respectively In our computations we will adopt the

ldquodomesticrdquo point of viewwhichwill frequently be representedby the subscript 119889 Similarly we will use the subscript 119891

when referring to a foreign denominated variable Hence thearbitrage price 119862

119905(119879 119870) of the foreign exchange call option at

time 119905 isin [0 119879] is given as the conditional expectation withrespect to the 120590-field F

119905of the optionrsquos payoff at expiration

discounted by the domestic money market account that is

119862119905

(119879 119870) = EP119905

exp(minus int119879

119905

119903119906119889119906) 119862

119879(119879 119870)

= EP119905

exp(minus int119879

119905

119903119906119889119906) (119876

119879minus 119870)

+

(2)

or equivalently

119862119905

(119879 119870) = EP119905

exp(minus int119879

119905

119903119906119889119906) 119876

1198791119876119879gt119870

minus 119870EP119905

exp(minus int119879

119905

119903119906119889119906) 1

119876119879gt119870

(3)

Similarly the arbitrage price of the domestic discount bondmaturing at time 119879 equals for every 119905 isin [0 119879]

119861119889

(119905 119879) = EP119905

exp(minus int119879

119905

119903119906119889119906) (4)

and an analogous formula holds for the price process 119861119891

(119905 119879)

of the foreign discount bond under the foreign spot martin-gale measure (see eg Chapter 14 in Musiela and Rutkowski[20])

As a preliminary step towards the general valuation resultpresented in Section 4 we state the following well-knownproposition (see eg Cox et al [2] or Chapter 10 in Musielaand Rutkowski [20]) It is worth stressing that we use here inparticular the postulated independence of the foreign interestrate 119903 and the exchange rate process 119876 Under this standingassumption the dynamics of the foreign bond price 119861

119891(119905 119879)

under the domestic spot martingalemeasureP can be seen asan immediate consequence of formula (143) in Musiela andRutkowski [20] The simple form of the dynamics of 119861

119891(119905 119879)

under P is a consequence of the postulated independenceof 119882

119891 and 119882119876 (see Assumption (A2)) This crucial feature

underpins our further calculations and thus it cannot beeasily relaxed

Proposition 1 The prices at date 119905 of the domestic and foreigndiscount bonds maturing at time 119879 gt 119905 in the CIR model aregiven by the following expressions

119861119889

(119905 119879) = exp (119898119889

(119905 119879) minus 119899119889

(119905 119879) 119903119905)

119861119891

(119905 119879) = exp (119898119891

(119905 119879) minus 119899119891

(119905 119879) 119903119905)

(5)

where for 119894 isin 119889 119891

119898119894(119905 119879) =

2119886119894

1205902119894

sdot log[120574119894119890(12)119887119894(119879minus119905)

120574119894cosh (120574

119894(119879 minus 119905)) + (12) 119887

119894sinh (120574

119894(119879 minus 119905))

]

119899119894(119905 119879) =

sinh (120574119894(119879 minus 119905))

120574119894cosh (120574

119894(119879 minus 119905)) + (12) 119887

119894sinh (120574

119894(119879 minus 119905))

120574119894

=12

radic1198872119894

+ 21205902119894

(6)

Thedynamics of the domestic and foreign bond prices under thedomestic spot martingale measure P are given by

119889119861119889

(119905 119879) = 119861119889

(119905 119879) (119903119905119889119905 minus 120590

119889119899119889

(119905 119879) radic119903119905

119889119882119889

119905)

119889119861119891

(119905 119879) = 119861119891

(119905 119879) (119903119905119889119905 minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(7)

The following result is also well known (see eg Section1411 in Musiela and Rutkowski [20])

Lemma 2 The forward exchange rate 119865(119905 119879) at time 119905 forsettlement date 119879 equals

119865 (119905 119879) =119861119891

(119905 119879)

119861119889

(119905 119879)119876

119905 (8)

Since manifestly 119876119879

= 119865(119879 119879) the optionrsquos payoff atexpiration can also be expressed as follows

119862119879

(119879 119870) = 119865 (119879 119879) 1119865(119879119879)gt119870

minus 1198701119865(119879119879)gt119870

(9)

Consequently the optionrsquos value at time 119905 isin [0 119879] admits thefollowing representation

119862119905

(119879 119870)

= EP119905

exp(minus int119879

119905

119903119906119889119906) 119865 (119879 119879) 1

119865(119879119879)gt119870

minus 119870EP119905

exp(minus int119879

119905

119903119906119889119906) 1

119865(119879119879)gt119870

(10)

In what follows we will frequently use the notation 119909119905

=

ln119865(119905 119879) where 119905 isin [0 119879]

4 Pricing Formula for the FX Call Option

We are in a position to state the main result of the paperwhich furnishes a semianalytical formula for the arbitrageprice of the FX call option of European style under theHestonstochastic volatility for the exchange rate combined withthe independent CIR models for the domestic and foreignshort-term rates Since the proof of Theorem 3 relies on thederivation of the conditional characteristic function of thelogarithm of the exchange rate any suitable version of theFourier inversion technique or simulation technique can be

4 International Journal of Stochastic Analysis

applied to obtain the option price The interested reader isreferred to for instance Carr andMadan [21 22] or Lord andKahl [23 24] and the references therein as well as the recentpapers by Bernard et al [25] and Levendorskii [26] whodeveloped and examined in detail methods with essentialimprovements in accuracy andor efficiency

Theorem 3 Let the foreign exchange model be given by SDEs(1) under Assumptions (A1)ndash(A6) Then the price of theEuropean FX call option equals for every 119905 isin [0 119879]

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(11)

where the bond prices 119861119889(119905 119879) and 119861

119891(119905 119879) are given in

Proposition 1 and the functions 1198751 and 1198752 are given by for119895 = 1 2

119875119895

(119905 119876119905 V

119905 119903

119905 119903

119905 119870)

=12

+1120587

intinfin

0Re(119891

119895(120601)

exp (minus119894120601 ln119870)

119894120601) 119889120601

(12)

where the F119905-conditional characteristic functions 119891

119895(120601) =

119891119895(120601 119905 119876

119905 V

119905 119903

119905 119903

119905) 119895 = 1 2 of the random variable 119909

119879=

ln(119876119879

)under the probabilitymeasure P119879(seeDefinition 8) and

P119879(see Definition 6) respectively are given by

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)119894120601

119890minus(12)(1206012

+119894120601)1205902119876 minus 1)]

sdot exp[minus (119894120601120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)minus 1)

+(1 + 119894120601) 120588

120590V(V

119905+ 120579120591))] exp [minus119894120601 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [(1+ 119894120601) (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)] exp [minus1198661 (120591 1199041 1199042) V119905

minus 1198662 (120591 1199043 1199044) 119903119905

minus 1198663 (120591 1199045 1199046) 119903119905]

sdot exp [minus1205791198671 (120591 1199041 1199042) minus 1198861198891198672 (120591 1199043 1199044)

minus 119886119891

1198673 (120591 1199045 1199046)]

(13)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)119894120601

119890minus(12)(1206012

+119894120601)1205902119876 minus 1)]

sdot exp[minus (119894120601120582119876

120583119876

120591

+ 120582V120591 (120590V (119890

minus(120588(119894120601)120590V)119886 minus 119890minus(120588(119894120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)minus 1)

+119894120601120588

120590V(V

119905+ 120579120591))] exp [(1minus 119894120601) (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [119894120601 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)] exp [minus1198661

(120591 1199021 119902

2) V

119905

minus 1198662

(120591 1199023 119902

4) 119903

119905minus 119866

3(120591 119902

5 119902

6)]

sdot exp [minus1205791198671

(120591 1199021 119902

2) minus 119886

119889119867

2(120591 119902

3 119902

4)

minus 119886119891

1198673

(120591 1199025 119902

6)]

(14)

where the functions 1198661 1198662 1198663 1198671 1198672 1198673 are given inLemma 5 and 119888

119905equals

119888119905

= exp (119894120601119909119905) = exp (119894120601 ln119865 (119905 119879)) (15)

Moreover the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by

1199041 = minus(1 + 119894120601) 120588

120590V

1199042 = minus(1 + 119894120601)

2(1 minus 1205882)

2minus

(1 + 119894120601) 120588120581

120590V+1 + 119894120601

2

1199043 = 0

1199044 = minus 119894120601

1199045 = 0

1199046 = 1+ 119894120601

(16)

and the constants 1199021 1199022 1199023 1199024 1199025 1199026 equal

1199021 = minus119894120601120588

120590V

1199022 = minus(119894120601)

2(1 minus 1205882)

2minus

119894120601120588120581

120590V+

119894120601

2

1199023 = 0

1199024 = 1minus 119894120601

1199025 = 0

1199026 = 119894120601

(17)

International Journal of Stochastic Analysis 5

41 Auxiliary Results The proof of Theorem 3 hinges on anumber of lemmasWe start by stating the well-known resultwhich can be easily obtained from Proposition 8634 inJeanblanc et al [27] Let us denote 120591 = 119879 minus 119905 and let us set forall 0 le 119905 lt 119879

119869119876

(119905 119879) =

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) (18)

Note that we use here Assumptions (A3)ndash(A5)The property(A3) (resp (A4)) implies that the random variable 119869119876(119905 119879)

(resp 119885V119879

minus 119885V119905) is independent of the 120590-fieldF

119905 Let ]1 stand

for the Gaussian distribution 119873(ln(1+120583119876

)minus(12)1205902119876

1205902119876

) andlet ]2 stand for the uniform distribution with density

]2 (119911) =1

119887 minus 119886

1 119886 lt 119911 lt 119887

0 else(19)

where 0 lt 119886 lt 119887

Lemma 4 (i) Under Assumptions (A3) and (A5) the follow-ing equalities are valid

EP119905

exp (119894120601119869119876

(119905 119879))

= EP119905

exp(119894120601

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896))

= exp [120582119876

120591 int+infin

minusinfin

(119890119894120601119911

minus 1) ]1 (119889119911)]

= exp [120582119876

120591 ((1+ 120583119876

)119894120601

119890minus(12)1205902

119876(1206012+119894120601)

minus 1)]

(20)

(ii) Under Assumptions (A4) and (A5) the following equali-ties are valid for 119888 = 119886 + 119887119894 with 119886 le 0

EP119905

exp (119888 (119885V119879

minus 119885V119905)) = E

P119905

exp(119888

119873V119879

sum119896=119873

V119905 +1

119869V119896)

= exp [120582V120591 int+infin

minusinfin

(119890119888119911

minus 1) ]2 (119889119911)]

= exp[120582V120591 (119890119888119887 minus 119890119888119886

119888 (119887 minus 119886)minus 1)]

(21)

The next result extends Lemma 61 in Ahlip andRutkowski [28] (see also Duffie et al [6]) where the modelwithout the jump component in the dynamics of V wasexamined

Lemma 5 Let the dynamics of processes V 119903 and 119903 be givenby SDEs (1) with independent Brownian motions 119882

V 119882119889 and119882119891 For any complex numbers 120583 120582 120583 120583 we set

119865 (120591 V119905 119903

119905 119903

119905) = E

P119905

exp(minus120582V119879

minus 120583 int119879

119905

V119906119889119906 minus 119903

119879

minus 120583 int119879

119905

119903119906119889119906 minus 119903

119879minus 120583 int

119879

119905

119903119906119889119906)

(22)

Then

119865 (120591 V119905 119903

119905 119903

119905) = exp [minus1198661 (120591 120582 120583) V

119905minus 1198662 (120591 120583) 119903

119905

minus 1198663 (120591 120583) 119903119905

minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(23)

where

1198661 (120591 120582 120583)

=120582 [(120574 + 120581) + 119890

120574120591(120574 minus 120581)] + 2120583 (119890

120574120591minus 1)

1205902V120582 (119890120574120591 minus 1) + 120574 minus 120581 + 119890120574120591 (120574 + 120581)

1198662 (120591 120583)

= [(120574 + 119887

119889) + 119890120574120591 (120574 minus 119887

119889)] + 2120583 (119890120574120591 minus 1)

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

1198663 (120591 120583)

= [(120574 + 119887

119891) + 119890120574120591 (120574 minus 119887

119891)] + 2120583 (119890120574120591 minus 1)

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

)

1198671 (120591 120582 120583) = int120591

0(1198661 (119905 120582 120583)

+120582V

120579(1+

119890minus1198871198661(119905120582120583) minus 119890minus1198861198661(119905120582120583)

1198661 (119905 120582 120583) (119887 minus 119886))) 119889119905

1198672 (120591 120583) = minus2

1205902119889

sdot ln(2120574119890(120574+119887119889)1205912

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

)

1198673 (120591 120583) = minus2

1205902119891

sdot ln(2120574119890(120574+119887119891)1205912

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

))

(24)

where one denotes 120574 = radic1205812 + 21205902V120583 120574 = radic1198872

119889+ 21205902

119889120583 and 120574 =

radic1198872119891

+ 21205902119891

120583

Proof For the readerrsquos convenience we sketch the proof ofthe lemma Let us set for 119905 isin [0 119879]

119872119905

= 119865 (120591 V119905 119903

119905 119903

119905)

sdot exp(minus120583 int119905

0V119906119889119906 minus 120583 int

119905

0119903119906119889119906 minus 120583 int

119905

0119903119906119889119906)

(25)

6 International Journal of Stochastic Analysis

Then the process 119872 = (119872119905)119905isin[0119879] satisfies

119872119905

= EP119905

exp(minus120582V119879

minus 120583 int119879

0V119906119889119906 minus 119903

119879

minus 120583 int119879

0119903119906119889119906 minus 119903

119879minus 120583 int

119879

0119903119906119889119906)

(26)

and thus it is an F-martingale under P By applying theIto formula to the right-hand side in (25) and by settingthe drift term in the dynamics of 119872 to be zero we deducethat the function 119865(120591 V 119903 119903) satisfies the following partialintegrodifferential equation (PIDE)

minus120597119865

120597120591+12

1205902V V

1205972119865

120597V2

+ 120582V intinfin

0(119865 (120591 V+ 119911 119903 119903) minus 119865 (120591 V 119903 119903)) ]2 (119889119911)

+12

1205902119889119903

1205972119865

1205971199032+12

1205902119891

1199031205972119865

1205971199032+ (120579 minus 120581V)

120597119865

120597V

+ (119886119889

minus 119887119889119903)

120597119865

120597119903+ (119886

119891minus 119887

119891119903)

120597119865

120597119903

minus (120583V+ 120583119903 + 120583119903) 119865 = 0

(27)

with the initial condition 119865(0 V 119903 119903) = exp(minus120582V minus 119903 minus 119903)We search for a solution to this PIDE in the form

119865 (120591 V 119903 119903) = exp [minus1198661 (120591 120582 120583) Vminus 1198662 (120591 120583) 119903

minus 1198663 (120591 120583) 119903 minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(28)

with

1198661 (0 120582 120583) = 120582

1198662 (0 120583) =

1198663 (0 120583) =

1198671 (0 120582 120583) = 1198672 (0 120583) = 1198673 (0 120583) = 0

(29)

By substituting this expression in the PIDE and using part (ii)in Lemma 4 we obtain the following system of ODEs for the

functions 1198661 1198662 1198663 1198671 1198672 1198673 (for brevity we suppress thelast three arguments)

1205971198661 (120591)

120597120591= minus

12

1205902V119866

21 (120591) minus 1205811198661 (120591) + 120583

1205971198671 (120591)

120597120591= 1198661 (120591) +

120582V

120579(1+

119890minus1198871198661(120591) minus 119890minus1198861198661(120591)

1198661 (120591) (119887 minus 119886))

1205971198662 (120591)

120597120591= minus

12

120590211988911986622 (120591) minus 119887

1198891198662 (120591) + 120583

1205971198672 (120591)

120597120591= 1198662 (120591)

1205971198663 (120591)

120597120591= minus

12

1205902119891

11986623 (120591) minus 119887

1198911198663 (120591) + 120583

1205971198673 (120591)

120597120591= 1198663 (120591)

(30)

By solving these equations we obtain the stated formu-lae

Under the assumptions of Lemma 5 it is possible tofactorize 119865 as a product of two conditional expectationsThismeans that the functions1198661 (1198671)1198662 (1198672) and1198663 (1198673) are ofthe same form except that they correspond to different setsof parameters 120579 120581 120590V for 1198661 1198671 119886

119889 119887

119889 120590

119903for 1198662 1198672 and 119886

119891

119887119891 120590

119891for 1198663 1198673 Note however that the roles played by the

processes V 119903 and 119903 in our model are clearly differentIt should also be stressed that no closed-form analytical

expression for 119865(120591 V119905 119903

119905 119903

119905) is available in the case of cor-

related Brownian motions 119882V 119882119903 119882119891 Brigo and Alfonsi[29] who deal with this issue in a different context proposeto use a simple Gaussian approximation instead of the exactsolution More recently Grzelak and Oosterlee [16] proposedmore sophisticated approximations in the framework of theHestonCIR hybrid model We do not follow this path herehowever and we focus instead on finding a semianalyticalsolution since this goal can be achieved under Assumptions(A1)ndash(A6)

Let us now introduce a convenient change of the underly-ing probability measure from the domestic spot martingalemeasure P to the domestic forward martingale measure P

119879

Definition 6 The domestic forward martingale measure P119879

equivalent to P on (ΩF119879

) is defined by the Radon-Nikodym derivative process 120578 = (120578

119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P

10038161003816100381610038161003816100381610038161003816F119905= exp(minus int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(31)

An application of the Girsanov theorem shows that theprocess 119882119879 = (119882119879

119905)119905isin[0119879] which is given by the equality

119882119879

119905= 119882

119889

119905+ int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119906 (32)

International Journal of Stochastic Analysis 7

is the Brownian motion under the domestic forward martin-gale measure P

119879 Using the standard change of a numeraire

technique one can check that the price of the European for-eign exchange call option admits the following representationunder the probability measure P

119879

119862119905

(119879 119870) = 119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

minus 119870119861119889

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(33)

The following auxiliary result is easy to establish and thusits proof is omitted Recall that 119869119876(119905 119879) is given by equality(18)

Lemma 7 Under Assumptions (A1)ndash(A6) the dynamics ofthe forward exchange rate 119865(119905 119879) under the domestic forwardmartingale measure P

119879are given by the SDE

119889119865 (119905 119879) = 119865 (119905 119879) (119889119885119876

119905minus 120582

119876120583119876

119889119905 + radicV119905

119889119882119876

119905

+ 120590119889119899119889

(119905 119879) radic119903119905

119889119882119879

119905minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(34)

or equivalently

119865 (119879 119879) = 119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus

1

2int

119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(35)

where the dot sdot denotes the inner product inR3 (119865

(119905 119879))119905isin[0119879]

is the R3-valued process (row vector) given by

119865

(119905 119879) = [radicV119905 120590

119889119899119889

(119905 119879) radic119903119905 minus 120590

119891119899119891

(119905 119879) radic119903119905] (36)

and 119879 = (119879

119905)119905isin[0119879] stands for the R3-valued process

(column vector) given by

119879

119905= [119882

119876

119905 119882

119879

119905 119882

119891

119905]lowast

(37)

It is easy to check that under Assumptions (A1)ndash(A6)the process 119879 is the three-dimensional standard Brownianmotion under P

119879 In view of Lemma 7 we have that

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

) = 119861119889

(119905 119879)

sdotEP119879119905

119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot 1119865(119879119879)gt119870

= 119876119905119861119891

(119905 119879)EP119879119905

exp(119869119876

(119905 119879)

minus 120582119876

120583119876

(119879 minus 119905) + int119879

119905

119865

(119906 119879) sdot 119889119879

119906

minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906) 1119865(119879119879)gt119870

(38)

To deal with the first term in the right-hand side of (33) weintroduce another auxiliary probability measure

Definition 8 Themodified domestic forward martingale mea-sure P

119879 equivalent to P

119879on (ΩF

119879) is defined by the

Radon-Nikodym derivative process 120578 = (120578119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(39)

Using Lemma 7 and (8) we obtain

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

120578119879

)

EP119879119905

(120578119879

)

(40)

and thus the Bayes formula and Definition 8 yield

119861119889

(119905 119879)EP119879t (119865 (119879 119879) 1

119865(119879119879)gt119870)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(41)

This shows that P119879is a martingale measure associated with

the choice of the price process119876119905119861119891

(119905 119879) as a numeraire assetWe are now in a position to state the following lemma

Lemma 9 The price of the FX call option satisfies

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119876119879

gt 119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119876119879

gt 119870 | F119905)

(42)

or equivalently

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119909119879

gt ln119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119909119879

gt ln119870 | F119905)

(43)

To complete the proof Theorem 3 it remains to evaluatethe conditional probabilities arising in formula (43) Byanother application of the Girsanov theorem one can checkthat the process (119876 V 119903 119903) has theMarkov property under theprobability measuresP

119879and P

119879 In view of Proposition 1 and

Lemma 2 the random variable 119909119879is a function of 119876

119879 119903

119879 and

119903119879 We thus conclude that

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(44)

8 International Journal of Stochastic Analysis

where we denote

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(45)

To obtain explicit formulae for the conditional probabili-ties above it suffices to derive the corresponding conditionalcharacteristic functions

1198911 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

(46)

The idea is to use the Radon-Nikodym derivatives in order toobtain convenient expressions for the characteristic functionsin terms of conditional expectations under the domestic spotmartingale measure P The following lemma will allow us toachieve this goal

Lemma 10 The following equality holds

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(47)

Proof Straightforward computations show that

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

119889P119879

119889P

10038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906) = exp(int

119905

0radicV

119906119889119882

119876

119906

+ int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906) exp(minus

12

sdot int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(48)

Using (32) we now obtain

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(49)

which is the desired expression

In view of the formula established in Lemma 10and the abstract Bayes formula to compute 1198911(120601) =

1198911(120601 119905 119876119905 V

119905 119903

119905 119903

119905) it suffices to focus on the following

conditional expectation under P

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp(int119879

119905

radicV119906

119889119882119876

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(50)

Similarly in view of formula (31) we obtain for 1198912(120601) =

1198912(120601 119905 119876119905 V

119905 119903

119905 119903

119905)

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(51)

To proceed we will need the following result which is animmediate consequence of Lemma 7

Corollary 11 Under Assumptions (A1)ndash(A4) the process119909119905

= ln119865(119905 119879) admits the following representation under thedomestic forward martingale measure P

119879

119909119879

= 119909119905

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906

+ 119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

(52)

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 International Journal of Stochastic Analysis

and the interest rates are critical so that the constant interestrates assumption is clearly inappropriate for reliable valuationand hedging

Let us comment briefly on the existing literature inthe same vein van Haastrecht et al [12] have extendedthe stochastic volatility model of Schobel and Zhu [13] toequitycurrency derivatives by including stochastic interestrates and assuming all driving model factors to be instanta-neously correlated Since theirmodel is based on theGaussianprocesses it enjoys analytical tractability even in the mostgeneral case of a full correlation structure By contrast whenthe squared volatility is driven by the CIR process and theinterest rate is driven either by the Vasicek [14] or the Cox etal [2] process a full correlation structure leads to intractabil-ity of equity options even under a partial correlation of thedriving factors as have been documented by among othersvan Haastrecht and Pelsser [15] and Grzelak and Oosterlee[16 17] who examined in particular the HestonVasicekand HestonCIR hybrid models (see also Grzelak et al[18] where the SchobelndashZhuHullndashWhite and HestonHullndashWhite models for equity derivatives are studied)

Our goal is to derive semianalytical solutions for pricesof plain-vanilla FX options in a model in which the instanta-neous volatility component is specified by the extended Hes-ton model with log-normally and log-uniformly distributedjump amplitudes for the exchange rate and the volatilityprocess respectively whereas the short-term interest ratesfor the domestic and foreign economies are governed by theindependent CIR processes The model thus incorporatesimportant empirical characteristics of exchange rate returnvariability (a) the correlation between the exchange rateand its stochastic volatility (b) the presence of jumps in theexchange rate and volatility processes and (c) the randomcharacter of interest rates The practical importance of thisfeature of newly developed FX models is rather clear in viewof the existence of complex FX products that have a longlifetime and are sensitive to smiles or skews in the marketThe results obtained in this paper extend results obtained byGuoqing et al In their model only the stock price process issubject to jumps but the volatility of volatility is modeled bythe Heston dynamics

The paper is organized as follows In Section 2 we set theforeign exchange model examined in this work The optionspricing problem is introduced in Section 3 The main resultTheorem 3 of Section 4 furnishes the pricing formula forFX options It is worth stressing that the independence ofvolatility and interest rates appears to be a crucial assumptionfrom the point of view of analytical tractability and thus itcannot be relaxed Numerical illustrations of our method areprovided in Section 5 where the diffusion and jump-diffusionmodels are compared

2 The HestonCIR Jump-Diffusion ForeignExchange Model

Let (ΩFP) be an underlying probability space Let theexchange rate 119876 = (119876

119905)119905isin[0119879] its instantaneous squared

volatility V = (V119905)119905isin[0119879] the domestic short-term interest

rates 119903 = (119903119905)119905isin[0119879] and the foreign short-term interest rate

119903 = (119903119905)119905isin[0119879] be governed by the following system of SDEs

119889119876119905

= (119903119905

minus 119903119905

minus 120582120583119876

) 119876119905119889119905 + 119876

119905radicV119905

119889119882119876

119905+ 119876

119905minus119889119885

119876

119905

119889V119905

= (120579 minus 120581V119905) 119889119905 + 120590VradicV

119905119889119882

V119905

+ 119889119885V119905

119889119903119905

= (119886119889

minus 119887119889119903119905) 119889119905 + 120590

119889radic119903119905

119889119882119889

119905

119889119903119905

= (119886119891

minus 119887119891

119903119905) 119889119905 + 120590

119891radic119903

119905119889119882

119891

119905

(1)

We work under the following standing assumptions

(A1) Processes 119882119876 = (119882119876

119905)119905isin[0119879] and 119882V = (119882V

119905)119905isin[0119879]

are correlated Brownian motions with a constantcorrelation coefficient so that the quadratic covari-ation between the processes 119882

119876 and 119882V satisfies119889[119882119876 119882V]

119905= 120588 119889119905 for some constant 120588 isin [minus1 1]

(A2) Processes 119882119889 = (119882119889

119905)119905isin[0119879] and 119882119891 = (119882

119891

119905)119905isin[0119879]

are independent Brownian motions and they are alsoindependent of the Brownian motions 119882119876 and 119882Vhence the processes 119876 119903 and 119903 are independent

(A3) The process 119885119876

119905= sum

119873119876

119905

119896=1 119869119876

119896is the compound Poisson

process specifically the Poisson process 119873119876 has theintensity 120582

119876gt 0 and the random variables ln(1+ 119869

119876

119896)

119896 = 1 2 have the probability distribution119873(ln[1+

120583119876

] minus (12)1205902119876

1205902119876

) hence the jump sizes (119869119876

119896)infin

119896=1 arelog-normally distributed on (minus1 infin) with mean 120583

119876gt

minus1(A4) The process 119885V

119905= sum

119873V119905

119896=1 119869V119896is the compound Poisson

process specifically the Poisson process 119873V has theintensity 120582V gt 0 and the jump sizes 119869V

119896are uniformly

distributed(A5) The Poisson processes 119873119876 119873V and sequences of

random variables (119869119876

119896)infin

119896=1 and (119869V119896)infin

119896=1 are mutuallyindependent as well as independent of the Brownianmotions 119882

119876 119882V 119882119889 119882119891(A6) The modelrsquos parameters satisfy the stability condi-

tions 2120579 gt 1205902V gt 0 2119886

119889gt 1205902

119889gt 0 and 2119886

119891gt 1205902

119891gt 0

(see eg Wong and Heyde [19])

Note that we postulate that the instantaneous squaredvolatility process V the domestic short-term interest rate 119903and the foreign interest rate 119903 are independent stochasticprocesses We will argue in what follows that this assumptionis indeed crucial for analytical tractability For brevity werefer to the foreign exchange model given by SDEs (1) underAssumptions (A1)ndash(A6) as the HestonCIR jump-diffusionFX model

3 Foreign Exchange Call Option

We will first establish the general representation for thevalue of the foreign exchange (ie currency) Europeancall option with maturity 119879 gt 0 and a constant strikelevel 119870 gt 0 The probability measure P is interpreted as

International Journal of Stochastic Analysis 3

the domestic spot martingale measure (ie the domesticrisk-neutral probability) We denote by F = (F

119905)119905isin[0119879] the

filtration generated by the Brownian motions 119882119876 119882V 119882119889119882119891 and the compound Poisson processes 119885119876 and 119885V Wewrite EP

119905(sdot) and P

119905(sdot) to denote the conditional expectation

and the conditional probability underPwith respect to the120590-fieldF

119905 respectively In our computations we will adopt the

ldquodomesticrdquo point of viewwhichwill frequently be representedby the subscript 119889 Similarly we will use the subscript 119891

when referring to a foreign denominated variable Hence thearbitrage price 119862

119905(119879 119870) of the foreign exchange call option at

time 119905 isin [0 119879] is given as the conditional expectation withrespect to the 120590-field F

119905of the optionrsquos payoff at expiration

discounted by the domestic money market account that is

119862119905

(119879 119870) = EP119905

exp(minus int119879

119905

119903119906119889119906) 119862

119879(119879 119870)

= EP119905

exp(minus int119879

119905

119903119906119889119906) (119876

119879minus 119870)

+

(2)

or equivalently

119862119905

(119879 119870) = EP119905

exp(minus int119879

119905

119903119906119889119906) 119876

1198791119876119879gt119870

minus 119870EP119905

exp(minus int119879

119905

119903119906119889119906) 1

119876119879gt119870

(3)

Similarly the arbitrage price of the domestic discount bondmaturing at time 119879 equals for every 119905 isin [0 119879]

119861119889

(119905 119879) = EP119905

exp(minus int119879

119905

119903119906119889119906) (4)

and an analogous formula holds for the price process 119861119891

(119905 119879)

of the foreign discount bond under the foreign spot martin-gale measure (see eg Chapter 14 in Musiela and Rutkowski[20])

As a preliminary step towards the general valuation resultpresented in Section 4 we state the following well-knownproposition (see eg Cox et al [2] or Chapter 10 in Musielaand Rutkowski [20]) It is worth stressing that we use here inparticular the postulated independence of the foreign interestrate 119903 and the exchange rate process 119876 Under this standingassumption the dynamics of the foreign bond price 119861

119891(119905 119879)

under the domestic spot martingalemeasureP can be seen asan immediate consequence of formula (143) in Musiela andRutkowski [20] The simple form of the dynamics of 119861

119891(119905 119879)

under P is a consequence of the postulated independenceof 119882

119891 and 119882119876 (see Assumption (A2)) This crucial feature

underpins our further calculations and thus it cannot beeasily relaxed

Proposition 1 The prices at date 119905 of the domestic and foreigndiscount bonds maturing at time 119879 gt 119905 in the CIR model aregiven by the following expressions

119861119889

(119905 119879) = exp (119898119889

(119905 119879) minus 119899119889

(119905 119879) 119903119905)

119861119891

(119905 119879) = exp (119898119891

(119905 119879) minus 119899119891

(119905 119879) 119903119905)

(5)

where for 119894 isin 119889 119891

119898119894(119905 119879) =

2119886119894

1205902119894

sdot log[120574119894119890(12)119887119894(119879minus119905)

120574119894cosh (120574

119894(119879 minus 119905)) + (12) 119887

119894sinh (120574

119894(119879 minus 119905))

]

119899119894(119905 119879) =

sinh (120574119894(119879 minus 119905))

120574119894cosh (120574

119894(119879 minus 119905)) + (12) 119887

119894sinh (120574

119894(119879 minus 119905))

120574119894

=12

radic1198872119894

+ 21205902119894

(6)

Thedynamics of the domestic and foreign bond prices under thedomestic spot martingale measure P are given by

119889119861119889

(119905 119879) = 119861119889

(119905 119879) (119903119905119889119905 minus 120590

119889119899119889

(119905 119879) radic119903119905

119889119882119889

119905)

119889119861119891

(119905 119879) = 119861119891

(119905 119879) (119903119905119889119905 minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(7)

The following result is also well known (see eg Section1411 in Musiela and Rutkowski [20])

Lemma 2 The forward exchange rate 119865(119905 119879) at time 119905 forsettlement date 119879 equals

119865 (119905 119879) =119861119891

(119905 119879)

119861119889

(119905 119879)119876

119905 (8)

Since manifestly 119876119879

= 119865(119879 119879) the optionrsquos payoff atexpiration can also be expressed as follows

119862119879

(119879 119870) = 119865 (119879 119879) 1119865(119879119879)gt119870

minus 1198701119865(119879119879)gt119870

(9)

Consequently the optionrsquos value at time 119905 isin [0 119879] admits thefollowing representation

119862119905

(119879 119870)

= EP119905

exp(minus int119879

119905

119903119906119889119906) 119865 (119879 119879) 1

119865(119879119879)gt119870

minus 119870EP119905

exp(minus int119879

119905

119903119906119889119906) 1

119865(119879119879)gt119870

(10)

In what follows we will frequently use the notation 119909119905

=

ln119865(119905 119879) where 119905 isin [0 119879]

4 Pricing Formula for the FX Call Option

We are in a position to state the main result of the paperwhich furnishes a semianalytical formula for the arbitrageprice of the FX call option of European style under theHestonstochastic volatility for the exchange rate combined withthe independent CIR models for the domestic and foreignshort-term rates Since the proof of Theorem 3 relies on thederivation of the conditional characteristic function of thelogarithm of the exchange rate any suitable version of theFourier inversion technique or simulation technique can be

4 International Journal of Stochastic Analysis

applied to obtain the option price The interested reader isreferred to for instance Carr andMadan [21 22] or Lord andKahl [23 24] and the references therein as well as the recentpapers by Bernard et al [25] and Levendorskii [26] whodeveloped and examined in detail methods with essentialimprovements in accuracy andor efficiency

Theorem 3 Let the foreign exchange model be given by SDEs(1) under Assumptions (A1)ndash(A6) Then the price of theEuropean FX call option equals for every 119905 isin [0 119879]

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(11)

where the bond prices 119861119889(119905 119879) and 119861

119891(119905 119879) are given in

Proposition 1 and the functions 1198751 and 1198752 are given by for119895 = 1 2

119875119895

(119905 119876119905 V

119905 119903

119905 119903

119905 119870)

=12

+1120587

intinfin

0Re(119891

119895(120601)

exp (minus119894120601 ln119870)

119894120601) 119889120601

(12)

where the F119905-conditional characteristic functions 119891

119895(120601) =

119891119895(120601 119905 119876

119905 V

119905 119903

119905 119903

119905) 119895 = 1 2 of the random variable 119909

119879=

ln(119876119879

)under the probabilitymeasure P119879(seeDefinition 8) and

P119879(see Definition 6) respectively are given by

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)119894120601

119890minus(12)(1206012

+119894120601)1205902119876 minus 1)]

sdot exp[minus (119894120601120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)minus 1)

+(1 + 119894120601) 120588

120590V(V

119905+ 120579120591))] exp [minus119894120601 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [(1+ 119894120601) (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)] exp [minus1198661 (120591 1199041 1199042) V119905

minus 1198662 (120591 1199043 1199044) 119903119905

minus 1198663 (120591 1199045 1199046) 119903119905]

sdot exp [minus1205791198671 (120591 1199041 1199042) minus 1198861198891198672 (120591 1199043 1199044)

minus 119886119891

1198673 (120591 1199045 1199046)]

(13)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)119894120601

119890minus(12)(1206012

+119894120601)1205902119876 minus 1)]

sdot exp[minus (119894120601120582119876

120583119876

120591

+ 120582V120591 (120590V (119890

minus(120588(119894120601)120590V)119886 minus 119890minus(120588(119894120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)minus 1)

+119894120601120588

120590V(V

119905+ 120579120591))] exp [(1minus 119894120601) (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [119894120601 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)] exp [minus1198661

(120591 1199021 119902

2) V

119905

minus 1198662

(120591 1199023 119902

4) 119903

119905minus 119866

3(120591 119902

5 119902

6)]

sdot exp [minus1205791198671

(120591 1199021 119902

2) minus 119886

119889119867

2(120591 119902

3 119902

4)

minus 119886119891

1198673

(120591 1199025 119902

6)]

(14)

where the functions 1198661 1198662 1198663 1198671 1198672 1198673 are given inLemma 5 and 119888

119905equals

119888119905

= exp (119894120601119909119905) = exp (119894120601 ln119865 (119905 119879)) (15)

Moreover the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by

1199041 = minus(1 + 119894120601) 120588

120590V

1199042 = minus(1 + 119894120601)

2(1 minus 1205882)

2minus

(1 + 119894120601) 120588120581

120590V+1 + 119894120601

2

1199043 = 0

1199044 = minus 119894120601

1199045 = 0

1199046 = 1+ 119894120601

(16)

and the constants 1199021 1199022 1199023 1199024 1199025 1199026 equal

1199021 = minus119894120601120588

120590V

1199022 = minus(119894120601)

2(1 minus 1205882)

2minus

119894120601120588120581

120590V+

119894120601

2

1199023 = 0

1199024 = 1minus 119894120601

1199025 = 0

1199026 = 119894120601

(17)

International Journal of Stochastic Analysis 5

41 Auxiliary Results The proof of Theorem 3 hinges on anumber of lemmasWe start by stating the well-known resultwhich can be easily obtained from Proposition 8634 inJeanblanc et al [27] Let us denote 120591 = 119879 minus 119905 and let us set forall 0 le 119905 lt 119879

119869119876

(119905 119879) =

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) (18)

Note that we use here Assumptions (A3)ndash(A5)The property(A3) (resp (A4)) implies that the random variable 119869119876(119905 119879)

(resp 119885V119879

minus 119885V119905) is independent of the 120590-fieldF

119905 Let ]1 stand

for the Gaussian distribution 119873(ln(1+120583119876

)minus(12)1205902119876

1205902119876

) andlet ]2 stand for the uniform distribution with density

]2 (119911) =1

119887 minus 119886

1 119886 lt 119911 lt 119887

0 else(19)

where 0 lt 119886 lt 119887

Lemma 4 (i) Under Assumptions (A3) and (A5) the follow-ing equalities are valid

EP119905

exp (119894120601119869119876

(119905 119879))

= EP119905

exp(119894120601

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896))

= exp [120582119876

120591 int+infin

minusinfin

(119890119894120601119911

minus 1) ]1 (119889119911)]

= exp [120582119876

120591 ((1+ 120583119876

)119894120601

119890minus(12)1205902

119876(1206012+119894120601)

minus 1)]

(20)

(ii) Under Assumptions (A4) and (A5) the following equali-ties are valid for 119888 = 119886 + 119887119894 with 119886 le 0

EP119905

exp (119888 (119885V119879

minus 119885V119905)) = E

P119905

exp(119888

119873V119879

sum119896=119873

V119905 +1

119869V119896)

= exp [120582V120591 int+infin

minusinfin

(119890119888119911

minus 1) ]2 (119889119911)]

= exp[120582V120591 (119890119888119887 minus 119890119888119886

119888 (119887 minus 119886)minus 1)]

(21)

The next result extends Lemma 61 in Ahlip andRutkowski [28] (see also Duffie et al [6]) where the modelwithout the jump component in the dynamics of V wasexamined

Lemma 5 Let the dynamics of processes V 119903 and 119903 be givenby SDEs (1) with independent Brownian motions 119882

V 119882119889 and119882119891 For any complex numbers 120583 120582 120583 120583 we set

119865 (120591 V119905 119903

119905 119903

119905) = E

P119905

exp(minus120582V119879

minus 120583 int119879

119905

V119906119889119906 minus 119903

119879

minus 120583 int119879

119905

119903119906119889119906 minus 119903

119879minus 120583 int

119879

119905

119903119906119889119906)

(22)

Then

119865 (120591 V119905 119903

119905 119903

119905) = exp [minus1198661 (120591 120582 120583) V

119905minus 1198662 (120591 120583) 119903

119905

minus 1198663 (120591 120583) 119903119905

minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(23)

where

1198661 (120591 120582 120583)

=120582 [(120574 + 120581) + 119890

120574120591(120574 minus 120581)] + 2120583 (119890

120574120591minus 1)

1205902V120582 (119890120574120591 minus 1) + 120574 minus 120581 + 119890120574120591 (120574 + 120581)

1198662 (120591 120583)

= [(120574 + 119887

119889) + 119890120574120591 (120574 minus 119887

119889)] + 2120583 (119890120574120591 minus 1)

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

1198663 (120591 120583)

= [(120574 + 119887

119891) + 119890120574120591 (120574 minus 119887

119891)] + 2120583 (119890120574120591 minus 1)

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

)

1198671 (120591 120582 120583) = int120591

0(1198661 (119905 120582 120583)

+120582V

120579(1+

119890minus1198871198661(119905120582120583) minus 119890minus1198861198661(119905120582120583)

1198661 (119905 120582 120583) (119887 minus 119886))) 119889119905

1198672 (120591 120583) = minus2

1205902119889

sdot ln(2120574119890(120574+119887119889)1205912

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

)

1198673 (120591 120583) = minus2

1205902119891

sdot ln(2120574119890(120574+119887119891)1205912

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

))

(24)

where one denotes 120574 = radic1205812 + 21205902V120583 120574 = radic1198872

119889+ 21205902

119889120583 and 120574 =

radic1198872119891

+ 21205902119891

120583

Proof For the readerrsquos convenience we sketch the proof ofthe lemma Let us set for 119905 isin [0 119879]

119872119905

= 119865 (120591 V119905 119903

119905 119903

119905)

sdot exp(minus120583 int119905

0V119906119889119906 minus 120583 int

119905

0119903119906119889119906 minus 120583 int

119905

0119903119906119889119906)

(25)

6 International Journal of Stochastic Analysis

Then the process 119872 = (119872119905)119905isin[0119879] satisfies

119872119905

= EP119905

exp(minus120582V119879

minus 120583 int119879

0V119906119889119906 minus 119903

119879

minus 120583 int119879

0119903119906119889119906 minus 119903

119879minus 120583 int

119879

0119903119906119889119906)

(26)

and thus it is an F-martingale under P By applying theIto formula to the right-hand side in (25) and by settingthe drift term in the dynamics of 119872 to be zero we deducethat the function 119865(120591 V 119903 119903) satisfies the following partialintegrodifferential equation (PIDE)

minus120597119865

120597120591+12

1205902V V

1205972119865

120597V2

+ 120582V intinfin

0(119865 (120591 V+ 119911 119903 119903) minus 119865 (120591 V 119903 119903)) ]2 (119889119911)

+12

1205902119889119903

1205972119865

1205971199032+12

1205902119891

1199031205972119865

1205971199032+ (120579 minus 120581V)

120597119865

120597V

+ (119886119889

minus 119887119889119903)

120597119865

120597119903+ (119886

119891minus 119887

119891119903)

120597119865

120597119903

minus (120583V+ 120583119903 + 120583119903) 119865 = 0

(27)

with the initial condition 119865(0 V 119903 119903) = exp(minus120582V minus 119903 minus 119903)We search for a solution to this PIDE in the form

119865 (120591 V 119903 119903) = exp [minus1198661 (120591 120582 120583) Vminus 1198662 (120591 120583) 119903

minus 1198663 (120591 120583) 119903 minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(28)

with

1198661 (0 120582 120583) = 120582

1198662 (0 120583) =

1198663 (0 120583) =

1198671 (0 120582 120583) = 1198672 (0 120583) = 1198673 (0 120583) = 0

(29)

By substituting this expression in the PIDE and using part (ii)in Lemma 4 we obtain the following system of ODEs for the

functions 1198661 1198662 1198663 1198671 1198672 1198673 (for brevity we suppress thelast three arguments)

1205971198661 (120591)

120597120591= minus

12

1205902V119866

21 (120591) minus 1205811198661 (120591) + 120583

1205971198671 (120591)

120597120591= 1198661 (120591) +

120582V

120579(1+

119890minus1198871198661(120591) minus 119890minus1198861198661(120591)

1198661 (120591) (119887 minus 119886))

1205971198662 (120591)

120597120591= minus

12

120590211988911986622 (120591) minus 119887

1198891198662 (120591) + 120583

1205971198672 (120591)

120597120591= 1198662 (120591)

1205971198663 (120591)

120597120591= minus

12

1205902119891

11986623 (120591) minus 119887

1198911198663 (120591) + 120583

1205971198673 (120591)

120597120591= 1198663 (120591)

(30)

By solving these equations we obtain the stated formu-lae

Under the assumptions of Lemma 5 it is possible tofactorize 119865 as a product of two conditional expectationsThismeans that the functions1198661 (1198671)1198662 (1198672) and1198663 (1198673) are ofthe same form except that they correspond to different setsof parameters 120579 120581 120590V for 1198661 1198671 119886

119889 119887

119889 120590

119903for 1198662 1198672 and 119886

119891

119887119891 120590

119891for 1198663 1198673 Note however that the roles played by the

processes V 119903 and 119903 in our model are clearly differentIt should also be stressed that no closed-form analytical

expression for 119865(120591 V119905 119903

119905 119903

119905) is available in the case of cor-

related Brownian motions 119882V 119882119903 119882119891 Brigo and Alfonsi[29] who deal with this issue in a different context proposeto use a simple Gaussian approximation instead of the exactsolution More recently Grzelak and Oosterlee [16] proposedmore sophisticated approximations in the framework of theHestonCIR hybrid model We do not follow this path herehowever and we focus instead on finding a semianalyticalsolution since this goal can be achieved under Assumptions(A1)ndash(A6)

Let us now introduce a convenient change of the underly-ing probability measure from the domestic spot martingalemeasure P to the domestic forward martingale measure P

119879

Definition 6 The domestic forward martingale measure P119879

equivalent to P on (ΩF119879

) is defined by the Radon-Nikodym derivative process 120578 = (120578

119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P

10038161003816100381610038161003816100381610038161003816F119905= exp(minus int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(31)

An application of the Girsanov theorem shows that theprocess 119882119879 = (119882119879

119905)119905isin[0119879] which is given by the equality

119882119879

119905= 119882

119889

119905+ int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119906 (32)

International Journal of Stochastic Analysis 7

is the Brownian motion under the domestic forward martin-gale measure P

119879 Using the standard change of a numeraire

technique one can check that the price of the European for-eign exchange call option admits the following representationunder the probability measure P

119879

119862119905

(119879 119870) = 119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

minus 119870119861119889

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(33)

The following auxiliary result is easy to establish and thusits proof is omitted Recall that 119869119876(119905 119879) is given by equality(18)

Lemma 7 Under Assumptions (A1)ndash(A6) the dynamics ofthe forward exchange rate 119865(119905 119879) under the domestic forwardmartingale measure P

119879are given by the SDE

119889119865 (119905 119879) = 119865 (119905 119879) (119889119885119876

119905minus 120582

119876120583119876

119889119905 + radicV119905

119889119882119876

119905

+ 120590119889119899119889

(119905 119879) radic119903119905

119889119882119879

119905minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(34)

or equivalently

119865 (119879 119879) = 119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus

1

2int

119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(35)

where the dot sdot denotes the inner product inR3 (119865

(119905 119879))119905isin[0119879]

is the R3-valued process (row vector) given by

119865

(119905 119879) = [radicV119905 120590

119889119899119889

(119905 119879) radic119903119905 minus 120590

119891119899119891

(119905 119879) radic119903119905] (36)

and 119879 = (119879

119905)119905isin[0119879] stands for the R3-valued process

(column vector) given by

119879

119905= [119882

119876

119905 119882

119879

119905 119882

119891

119905]lowast

(37)

It is easy to check that under Assumptions (A1)ndash(A6)the process 119879 is the three-dimensional standard Brownianmotion under P

119879 In view of Lemma 7 we have that

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

) = 119861119889

(119905 119879)

sdotEP119879119905

119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot 1119865(119879119879)gt119870

= 119876119905119861119891

(119905 119879)EP119879119905

exp(119869119876

(119905 119879)

minus 120582119876

120583119876

(119879 minus 119905) + int119879

119905

119865

(119906 119879) sdot 119889119879

119906

minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906) 1119865(119879119879)gt119870

(38)

To deal with the first term in the right-hand side of (33) weintroduce another auxiliary probability measure

Definition 8 Themodified domestic forward martingale mea-sure P

119879 equivalent to P

119879on (ΩF

119879) is defined by the

Radon-Nikodym derivative process 120578 = (120578119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(39)

Using Lemma 7 and (8) we obtain

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

120578119879

)

EP119879119905

(120578119879

)

(40)

and thus the Bayes formula and Definition 8 yield

119861119889

(119905 119879)EP119879t (119865 (119879 119879) 1

119865(119879119879)gt119870)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(41)

This shows that P119879is a martingale measure associated with

the choice of the price process119876119905119861119891

(119905 119879) as a numeraire assetWe are now in a position to state the following lemma

Lemma 9 The price of the FX call option satisfies

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119876119879

gt 119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119876119879

gt 119870 | F119905)

(42)

or equivalently

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119909119879

gt ln119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119909119879

gt ln119870 | F119905)

(43)

To complete the proof Theorem 3 it remains to evaluatethe conditional probabilities arising in formula (43) Byanother application of the Girsanov theorem one can checkthat the process (119876 V 119903 119903) has theMarkov property under theprobability measuresP

119879and P

119879 In view of Proposition 1 and

Lemma 2 the random variable 119909119879is a function of 119876

119879 119903

119879 and

119903119879 We thus conclude that

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(44)

8 International Journal of Stochastic Analysis

where we denote

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(45)

To obtain explicit formulae for the conditional probabili-ties above it suffices to derive the corresponding conditionalcharacteristic functions

1198911 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

(46)

The idea is to use the Radon-Nikodym derivatives in order toobtain convenient expressions for the characteristic functionsin terms of conditional expectations under the domestic spotmartingale measure P The following lemma will allow us toachieve this goal

Lemma 10 The following equality holds

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(47)

Proof Straightforward computations show that

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

119889P119879

119889P

10038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906) = exp(int

119905

0radicV

119906119889119882

119876

119906

+ int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906) exp(minus

12

sdot int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(48)

Using (32) we now obtain

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(49)

which is the desired expression

In view of the formula established in Lemma 10and the abstract Bayes formula to compute 1198911(120601) =

1198911(120601 119905 119876119905 V

119905 119903

119905 119903

119905) it suffices to focus on the following

conditional expectation under P

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp(int119879

119905

radicV119906

119889119882119876

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(50)

Similarly in view of formula (31) we obtain for 1198912(120601) =

1198912(120601 119905 119876119905 V

119905 119903

119905 119903

119905)

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(51)

To proceed we will need the following result which is animmediate consequence of Lemma 7

Corollary 11 Under Assumptions (A1)ndash(A4) the process119909119905

= ln119865(119905 119879) admits the following representation under thedomestic forward martingale measure P

119879

119909119879

= 119909119905

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906

+ 119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

(52)

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 3

the domestic spot martingale measure (ie the domesticrisk-neutral probability) We denote by F = (F

119905)119905isin[0119879] the

filtration generated by the Brownian motions 119882119876 119882V 119882119889119882119891 and the compound Poisson processes 119885119876 and 119885V Wewrite EP

119905(sdot) and P

119905(sdot) to denote the conditional expectation

and the conditional probability underPwith respect to the120590-fieldF

119905 respectively In our computations we will adopt the

ldquodomesticrdquo point of viewwhichwill frequently be representedby the subscript 119889 Similarly we will use the subscript 119891

when referring to a foreign denominated variable Hence thearbitrage price 119862

119905(119879 119870) of the foreign exchange call option at

time 119905 isin [0 119879] is given as the conditional expectation withrespect to the 120590-field F

119905of the optionrsquos payoff at expiration

discounted by the domestic money market account that is

119862119905

(119879 119870) = EP119905

exp(minus int119879

119905

119903119906119889119906) 119862

119879(119879 119870)

= EP119905

exp(minus int119879

119905

119903119906119889119906) (119876

119879minus 119870)

+

(2)

or equivalently

119862119905

(119879 119870) = EP119905

exp(minus int119879

119905

119903119906119889119906) 119876

1198791119876119879gt119870

minus 119870EP119905

exp(minus int119879

119905

119903119906119889119906) 1

119876119879gt119870

(3)

Similarly the arbitrage price of the domestic discount bondmaturing at time 119879 equals for every 119905 isin [0 119879]

119861119889

(119905 119879) = EP119905

exp(minus int119879

119905

119903119906119889119906) (4)

and an analogous formula holds for the price process 119861119891

(119905 119879)

of the foreign discount bond under the foreign spot martin-gale measure (see eg Chapter 14 in Musiela and Rutkowski[20])

As a preliminary step towards the general valuation resultpresented in Section 4 we state the following well-knownproposition (see eg Cox et al [2] or Chapter 10 in Musielaand Rutkowski [20]) It is worth stressing that we use here inparticular the postulated independence of the foreign interestrate 119903 and the exchange rate process 119876 Under this standingassumption the dynamics of the foreign bond price 119861

119891(119905 119879)

under the domestic spot martingalemeasureP can be seen asan immediate consequence of formula (143) in Musiela andRutkowski [20] The simple form of the dynamics of 119861

119891(119905 119879)

under P is a consequence of the postulated independenceof 119882

119891 and 119882119876 (see Assumption (A2)) This crucial feature

underpins our further calculations and thus it cannot beeasily relaxed

Proposition 1 The prices at date 119905 of the domestic and foreigndiscount bonds maturing at time 119879 gt 119905 in the CIR model aregiven by the following expressions

119861119889

(119905 119879) = exp (119898119889

(119905 119879) minus 119899119889

(119905 119879) 119903119905)

119861119891

(119905 119879) = exp (119898119891

(119905 119879) minus 119899119891

(119905 119879) 119903119905)

(5)

where for 119894 isin 119889 119891

119898119894(119905 119879) =

2119886119894

1205902119894

sdot log[120574119894119890(12)119887119894(119879minus119905)

120574119894cosh (120574

119894(119879 minus 119905)) + (12) 119887

119894sinh (120574

119894(119879 minus 119905))

]

119899119894(119905 119879) =

sinh (120574119894(119879 minus 119905))

120574119894cosh (120574

119894(119879 minus 119905)) + (12) 119887

119894sinh (120574

119894(119879 minus 119905))

120574119894

=12

radic1198872119894

+ 21205902119894

(6)

Thedynamics of the domestic and foreign bond prices under thedomestic spot martingale measure P are given by

119889119861119889

(119905 119879) = 119861119889

(119905 119879) (119903119905119889119905 minus 120590

119889119899119889

(119905 119879) radic119903119905

119889119882119889

119905)

119889119861119891

(119905 119879) = 119861119891

(119905 119879) (119903119905119889119905 minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(7)

The following result is also well known (see eg Section1411 in Musiela and Rutkowski [20])

Lemma 2 The forward exchange rate 119865(119905 119879) at time 119905 forsettlement date 119879 equals

119865 (119905 119879) =119861119891

(119905 119879)

119861119889

(119905 119879)119876

119905 (8)

Since manifestly 119876119879

= 119865(119879 119879) the optionrsquos payoff atexpiration can also be expressed as follows

119862119879

(119879 119870) = 119865 (119879 119879) 1119865(119879119879)gt119870

minus 1198701119865(119879119879)gt119870

(9)

Consequently the optionrsquos value at time 119905 isin [0 119879] admits thefollowing representation

119862119905

(119879 119870)

= EP119905

exp(minus int119879

119905

119903119906119889119906) 119865 (119879 119879) 1

119865(119879119879)gt119870

minus 119870EP119905

exp(minus int119879

119905

119903119906119889119906) 1

119865(119879119879)gt119870

(10)

In what follows we will frequently use the notation 119909119905

=

ln119865(119905 119879) where 119905 isin [0 119879]

4 Pricing Formula for the FX Call Option

We are in a position to state the main result of the paperwhich furnishes a semianalytical formula for the arbitrageprice of the FX call option of European style under theHestonstochastic volatility for the exchange rate combined withthe independent CIR models for the domestic and foreignshort-term rates Since the proof of Theorem 3 relies on thederivation of the conditional characteristic function of thelogarithm of the exchange rate any suitable version of theFourier inversion technique or simulation technique can be

4 International Journal of Stochastic Analysis

applied to obtain the option price The interested reader isreferred to for instance Carr andMadan [21 22] or Lord andKahl [23 24] and the references therein as well as the recentpapers by Bernard et al [25] and Levendorskii [26] whodeveloped and examined in detail methods with essentialimprovements in accuracy andor efficiency

Theorem 3 Let the foreign exchange model be given by SDEs(1) under Assumptions (A1)ndash(A6) Then the price of theEuropean FX call option equals for every 119905 isin [0 119879]

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(11)

where the bond prices 119861119889(119905 119879) and 119861

119891(119905 119879) are given in

Proposition 1 and the functions 1198751 and 1198752 are given by for119895 = 1 2

119875119895

(119905 119876119905 V

119905 119903

119905 119903

119905 119870)

=12

+1120587

intinfin

0Re(119891

119895(120601)

exp (minus119894120601 ln119870)

119894120601) 119889120601

(12)

where the F119905-conditional characteristic functions 119891

119895(120601) =

119891119895(120601 119905 119876

119905 V

119905 119903

119905 119903

119905) 119895 = 1 2 of the random variable 119909

119879=

ln(119876119879

)under the probabilitymeasure P119879(seeDefinition 8) and

P119879(see Definition 6) respectively are given by

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)119894120601

119890minus(12)(1206012

+119894120601)1205902119876 minus 1)]

sdot exp[minus (119894120601120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)minus 1)

+(1 + 119894120601) 120588

120590V(V

119905+ 120579120591))] exp [minus119894120601 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [(1+ 119894120601) (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)] exp [minus1198661 (120591 1199041 1199042) V119905

minus 1198662 (120591 1199043 1199044) 119903119905

minus 1198663 (120591 1199045 1199046) 119903119905]

sdot exp [minus1205791198671 (120591 1199041 1199042) minus 1198861198891198672 (120591 1199043 1199044)

minus 119886119891

1198673 (120591 1199045 1199046)]

(13)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)119894120601

119890minus(12)(1206012

+119894120601)1205902119876 minus 1)]

sdot exp[minus (119894120601120582119876

120583119876

120591

+ 120582V120591 (120590V (119890

minus(120588(119894120601)120590V)119886 minus 119890minus(120588(119894120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)minus 1)

+119894120601120588

120590V(V

119905+ 120579120591))] exp [(1minus 119894120601) (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [119894120601 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)] exp [minus1198661

(120591 1199021 119902

2) V

119905

minus 1198662

(120591 1199023 119902

4) 119903

119905minus 119866

3(120591 119902

5 119902

6)]

sdot exp [minus1205791198671

(120591 1199021 119902

2) minus 119886

119889119867

2(120591 119902

3 119902

4)

minus 119886119891

1198673

(120591 1199025 119902

6)]

(14)

where the functions 1198661 1198662 1198663 1198671 1198672 1198673 are given inLemma 5 and 119888

119905equals

119888119905

= exp (119894120601119909119905) = exp (119894120601 ln119865 (119905 119879)) (15)

Moreover the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by

1199041 = minus(1 + 119894120601) 120588

120590V

1199042 = minus(1 + 119894120601)

2(1 minus 1205882)

2minus

(1 + 119894120601) 120588120581

120590V+1 + 119894120601

2

1199043 = 0

1199044 = minus 119894120601

1199045 = 0

1199046 = 1+ 119894120601

(16)

and the constants 1199021 1199022 1199023 1199024 1199025 1199026 equal

1199021 = minus119894120601120588

120590V

1199022 = minus(119894120601)

2(1 minus 1205882)

2minus

119894120601120588120581

120590V+

119894120601

2

1199023 = 0

1199024 = 1minus 119894120601

1199025 = 0

1199026 = 119894120601

(17)

International Journal of Stochastic Analysis 5

41 Auxiliary Results The proof of Theorem 3 hinges on anumber of lemmasWe start by stating the well-known resultwhich can be easily obtained from Proposition 8634 inJeanblanc et al [27] Let us denote 120591 = 119879 minus 119905 and let us set forall 0 le 119905 lt 119879

119869119876

(119905 119879) =

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) (18)

Note that we use here Assumptions (A3)ndash(A5)The property(A3) (resp (A4)) implies that the random variable 119869119876(119905 119879)

(resp 119885V119879

minus 119885V119905) is independent of the 120590-fieldF

119905 Let ]1 stand

for the Gaussian distribution 119873(ln(1+120583119876

)minus(12)1205902119876

1205902119876

) andlet ]2 stand for the uniform distribution with density

]2 (119911) =1

119887 minus 119886

1 119886 lt 119911 lt 119887

0 else(19)

where 0 lt 119886 lt 119887

Lemma 4 (i) Under Assumptions (A3) and (A5) the follow-ing equalities are valid

EP119905

exp (119894120601119869119876

(119905 119879))

= EP119905

exp(119894120601

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896))

= exp [120582119876

120591 int+infin

minusinfin

(119890119894120601119911

minus 1) ]1 (119889119911)]

= exp [120582119876

120591 ((1+ 120583119876

)119894120601

119890minus(12)1205902

119876(1206012+119894120601)

minus 1)]

(20)

(ii) Under Assumptions (A4) and (A5) the following equali-ties are valid for 119888 = 119886 + 119887119894 with 119886 le 0

EP119905

exp (119888 (119885V119879

minus 119885V119905)) = E

P119905

exp(119888

119873V119879

sum119896=119873

V119905 +1

119869V119896)

= exp [120582V120591 int+infin

minusinfin

(119890119888119911

minus 1) ]2 (119889119911)]

= exp[120582V120591 (119890119888119887 minus 119890119888119886

119888 (119887 minus 119886)minus 1)]

(21)

The next result extends Lemma 61 in Ahlip andRutkowski [28] (see also Duffie et al [6]) where the modelwithout the jump component in the dynamics of V wasexamined

Lemma 5 Let the dynamics of processes V 119903 and 119903 be givenby SDEs (1) with independent Brownian motions 119882

V 119882119889 and119882119891 For any complex numbers 120583 120582 120583 120583 we set

119865 (120591 V119905 119903

119905 119903

119905) = E

P119905

exp(minus120582V119879

minus 120583 int119879

119905

V119906119889119906 minus 119903

119879

minus 120583 int119879

119905

119903119906119889119906 minus 119903

119879minus 120583 int

119879

119905

119903119906119889119906)

(22)

Then

119865 (120591 V119905 119903

119905 119903

119905) = exp [minus1198661 (120591 120582 120583) V

119905minus 1198662 (120591 120583) 119903

119905

minus 1198663 (120591 120583) 119903119905

minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(23)

where

1198661 (120591 120582 120583)

=120582 [(120574 + 120581) + 119890

120574120591(120574 minus 120581)] + 2120583 (119890

120574120591minus 1)

1205902V120582 (119890120574120591 minus 1) + 120574 minus 120581 + 119890120574120591 (120574 + 120581)

1198662 (120591 120583)

= [(120574 + 119887

119889) + 119890120574120591 (120574 minus 119887

119889)] + 2120583 (119890120574120591 minus 1)

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

1198663 (120591 120583)

= [(120574 + 119887

119891) + 119890120574120591 (120574 minus 119887

119891)] + 2120583 (119890120574120591 minus 1)

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

)

1198671 (120591 120582 120583) = int120591

0(1198661 (119905 120582 120583)

+120582V

120579(1+

119890minus1198871198661(119905120582120583) minus 119890minus1198861198661(119905120582120583)

1198661 (119905 120582 120583) (119887 minus 119886))) 119889119905

1198672 (120591 120583) = minus2

1205902119889

sdot ln(2120574119890(120574+119887119889)1205912

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

)

1198673 (120591 120583) = minus2

1205902119891

sdot ln(2120574119890(120574+119887119891)1205912

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

))

(24)

where one denotes 120574 = radic1205812 + 21205902V120583 120574 = radic1198872

119889+ 21205902

119889120583 and 120574 =

radic1198872119891

+ 21205902119891

120583

Proof For the readerrsquos convenience we sketch the proof ofthe lemma Let us set for 119905 isin [0 119879]

119872119905

= 119865 (120591 V119905 119903

119905 119903

119905)

sdot exp(minus120583 int119905

0V119906119889119906 minus 120583 int

119905

0119903119906119889119906 minus 120583 int

119905

0119903119906119889119906)

(25)

6 International Journal of Stochastic Analysis

Then the process 119872 = (119872119905)119905isin[0119879] satisfies

119872119905

= EP119905

exp(minus120582V119879

minus 120583 int119879

0V119906119889119906 minus 119903

119879

minus 120583 int119879

0119903119906119889119906 minus 119903

119879minus 120583 int

119879

0119903119906119889119906)

(26)

and thus it is an F-martingale under P By applying theIto formula to the right-hand side in (25) and by settingthe drift term in the dynamics of 119872 to be zero we deducethat the function 119865(120591 V 119903 119903) satisfies the following partialintegrodifferential equation (PIDE)

minus120597119865

120597120591+12

1205902V V

1205972119865

120597V2

+ 120582V intinfin

0(119865 (120591 V+ 119911 119903 119903) minus 119865 (120591 V 119903 119903)) ]2 (119889119911)

+12

1205902119889119903

1205972119865

1205971199032+12

1205902119891

1199031205972119865

1205971199032+ (120579 minus 120581V)

120597119865

120597V

+ (119886119889

minus 119887119889119903)

120597119865

120597119903+ (119886

119891minus 119887

119891119903)

120597119865

120597119903

minus (120583V+ 120583119903 + 120583119903) 119865 = 0

(27)

with the initial condition 119865(0 V 119903 119903) = exp(minus120582V minus 119903 minus 119903)We search for a solution to this PIDE in the form

119865 (120591 V 119903 119903) = exp [minus1198661 (120591 120582 120583) Vminus 1198662 (120591 120583) 119903

minus 1198663 (120591 120583) 119903 minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(28)

with

1198661 (0 120582 120583) = 120582

1198662 (0 120583) =

1198663 (0 120583) =

1198671 (0 120582 120583) = 1198672 (0 120583) = 1198673 (0 120583) = 0

(29)

By substituting this expression in the PIDE and using part (ii)in Lemma 4 we obtain the following system of ODEs for the

functions 1198661 1198662 1198663 1198671 1198672 1198673 (for brevity we suppress thelast three arguments)

1205971198661 (120591)

120597120591= minus

12

1205902V119866

21 (120591) minus 1205811198661 (120591) + 120583

1205971198671 (120591)

120597120591= 1198661 (120591) +

120582V

120579(1+

119890minus1198871198661(120591) minus 119890minus1198861198661(120591)

1198661 (120591) (119887 minus 119886))

1205971198662 (120591)

120597120591= minus

12

120590211988911986622 (120591) minus 119887

1198891198662 (120591) + 120583

1205971198672 (120591)

120597120591= 1198662 (120591)

1205971198663 (120591)

120597120591= minus

12

1205902119891

11986623 (120591) minus 119887

1198911198663 (120591) + 120583

1205971198673 (120591)

120597120591= 1198663 (120591)

(30)

By solving these equations we obtain the stated formu-lae

Under the assumptions of Lemma 5 it is possible tofactorize 119865 as a product of two conditional expectationsThismeans that the functions1198661 (1198671)1198662 (1198672) and1198663 (1198673) are ofthe same form except that they correspond to different setsof parameters 120579 120581 120590V for 1198661 1198671 119886

119889 119887

119889 120590

119903for 1198662 1198672 and 119886

119891

119887119891 120590

119891for 1198663 1198673 Note however that the roles played by the

processes V 119903 and 119903 in our model are clearly differentIt should also be stressed that no closed-form analytical

expression for 119865(120591 V119905 119903

119905 119903

119905) is available in the case of cor-

related Brownian motions 119882V 119882119903 119882119891 Brigo and Alfonsi[29] who deal with this issue in a different context proposeto use a simple Gaussian approximation instead of the exactsolution More recently Grzelak and Oosterlee [16] proposedmore sophisticated approximations in the framework of theHestonCIR hybrid model We do not follow this path herehowever and we focus instead on finding a semianalyticalsolution since this goal can be achieved under Assumptions(A1)ndash(A6)

Let us now introduce a convenient change of the underly-ing probability measure from the domestic spot martingalemeasure P to the domestic forward martingale measure P

119879

Definition 6 The domestic forward martingale measure P119879

equivalent to P on (ΩF119879

) is defined by the Radon-Nikodym derivative process 120578 = (120578

119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P

10038161003816100381610038161003816100381610038161003816F119905= exp(minus int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(31)

An application of the Girsanov theorem shows that theprocess 119882119879 = (119882119879

119905)119905isin[0119879] which is given by the equality

119882119879

119905= 119882

119889

119905+ int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119906 (32)

International Journal of Stochastic Analysis 7

is the Brownian motion under the domestic forward martin-gale measure P

119879 Using the standard change of a numeraire

technique one can check that the price of the European for-eign exchange call option admits the following representationunder the probability measure P

119879

119862119905

(119879 119870) = 119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

minus 119870119861119889

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(33)

The following auxiliary result is easy to establish and thusits proof is omitted Recall that 119869119876(119905 119879) is given by equality(18)

Lemma 7 Under Assumptions (A1)ndash(A6) the dynamics ofthe forward exchange rate 119865(119905 119879) under the domestic forwardmartingale measure P

119879are given by the SDE

119889119865 (119905 119879) = 119865 (119905 119879) (119889119885119876

119905minus 120582

119876120583119876

119889119905 + radicV119905

119889119882119876

119905

+ 120590119889119899119889

(119905 119879) radic119903119905

119889119882119879

119905minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(34)

or equivalently

119865 (119879 119879) = 119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus

1

2int

119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(35)

where the dot sdot denotes the inner product inR3 (119865

(119905 119879))119905isin[0119879]

is the R3-valued process (row vector) given by

119865

(119905 119879) = [radicV119905 120590

119889119899119889

(119905 119879) radic119903119905 minus 120590

119891119899119891

(119905 119879) radic119903119905] (36)

and 119879 = (119879

119905)119905isin[0119879] stands for the R3-valued process

(column vector) given by

119879

119905= [119882

119876

119905 119882

119879

119905 119882

119891

119905]lowast

(37)

It is easy to check that under Assumptions (A1)ndash(A6)the process 119879 is the three-dimensional standard Brownianmotion under P

119879 In view of Lemma 7 we have that

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

) = 119861119889

(119905 119879)

sdotEP119879119905

119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot 1119865(119879119879)gt119870

= 119876119905119861119891

(119905 119879)EP119879119905

exp(119869119876

(119905 119879)

minus 120582119876

120583119876

(119879 minus 119905) + int119879

119905

119865

(119906 119879) sdot 119889119879

119906

minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906) 1119865(119879119879)gt119870

(38)

To deal with the first term in the right-hand side of (33) weintroduce another auxiliary probability measure

Definition 8 Themodified domestic forward martingale mea-sure P

119879 equivalent to P

119879on (ΩF

119879) is defined by the

Radon-Nikodym derivative process 120578 = (120578119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(39)

Using Lemma 7 and (8) we obtain

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

120578119879

)

EP119879119905

(120578119879

)

(40)

and thus the Bayes formula and Definition 8 yield

119861119889

(119905 119879)EP119879t (119865 (119879 119879) 1

119865(119879119879)gt119870)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(41)

This shows that P119879is a martingale measure associated with

the choice of the price process119876119905119861119891

(119905 119879) as a numeraire assetWe are now in a position to state the following lemma

Lemma 9 The price of the FX call option satisfies

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119876119879

gt 119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119876119879

gt 119870 | F119905)

(42)

or equivalently

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119909119879

gt ln119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119909119879

gt ln119870 | F119905)

(43)

To complete the proof Theorem 3 it remains to evaluatethe conditional probabilities arising in formula (43) Byanother application of the Girsanov theorem one can checkthat the process (119876 V 119903 119903) has theMarkov property under theprobability measuresP

119879and P

119879 In view of Proposition 1 and

Lemma 2 the random variable 119909119879is a function of 119876

119879 119903

119879 and

119903119879 We thus conclude that

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(44)

8 International Journal of Stochastic Analysis

where we denote

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(45)

To obtain explicit formulae for the conditional probabili-ties above it suffices to derive the corresponding conditionalcharacteristic functions

1198911 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

(46)

The idea is to use the Radon-Nikodym derivatives in order toobtain convenient expressions for the characteristic functionsin terms of conditional expectations under the domestic spotmartingale measure P The following lemma will allow us toachieve this goal

Lemma 10 The following equality holds

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(47)

Proof Straightforward computations show that

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

119889P119879

119889P

10038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906) = exp(int

119905

0radicV

119906119889119882

119876

119906

+ int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906) exp(minus

12

sdot int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(48)

Using (32) we now obtain

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(49)

which is the desired expression

In view of the formula established in Lemma 10and the abstract Bayes formula to compute 1198911(120601) =

1198911(120601 119905 119876119905 V

119905 119903

119905 119903

119905) it suffices to focus on the following

conditional expectation under P

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp(int119879

119905

radicV119906

119889119882119876

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(50)

Similarly in view of formula (31) we obtain for 1198912(120601) =

1198912(120601 119905 119876119905 V

119905 119903

119905 119903

119905)

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(51)

To proceed we will need the following result which is animmediate consequence of Lemma 7

Corollary 11 Under Assumptions (A1)ndash(A4) the process119909119905

= ln119865(119905 119879) admits the following representation under thedomestic forward martingale measure P

119879

119909119879

= 119909119905

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906

+ 119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

(52)

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 International Journal of Stochastic Analysis

applied to obtain the option price The interested reader isreferred to for instance Carr andMadan [21 22] or Lord andKahl [23 24] and the references therein as well as the recentpapers by Bernard et al [25] and Levendorskii [26] whodeveloped and examined in detail methods with essentialimprovements in accuracy andor efficiency

Theorem 3 Let the foreign exchange model be given by SDEs(1) under Assumptions (A1)ndash(A6) Then the price of theEuropean FX call option equals for every 119905 isin [0 119879]

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(11)

where the bond prices 119861119889(119905 119879) and 119861

119891(119905 119879) are given in

Proposition 1 and the functions 1198751 and 1198752 are given by for119895 = 1 2

119875119895

(119905 119876119905 V

119905 119903

119905 119903

119905 119870)

=12

+1120587

intinfin

0Re(119891

119895(120601)

exp (minus119894120601 ln119870)

119894120601) 119889120601

(12)

where the F119905-conditional characteristic functions 119891

119895(120601) =

119891119895(120601 119905 119876

119905 V

119905 119903

119905 119903

119905) 119895 = 1 2 of the random variable 119909

119879=

ln(119876119879

)under the probabilitymeasure P119879(seeDefinition 8) and

P119879(see Definition 6) respectively are given by

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)119894120601

119890minus(12)(1206012

+119894120601)1205902119876 minus 1)]

sdot exp[minus (119894120601120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)minus 1)

+(1 + 119894120601) 120588

120590V(V

119905+ 120579120591))] exp [minus119894120601 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [(1+ 119894120601) (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)] exp [minus1198661 (120591 1199041 1199042) V119905

minus 1198662 (120591 1199043 1199044) 119903119905

minus 1198663 (120591 1199045 1199046) 119903119905]

sdot exp [minus1205791198671 (120591 1199041 1199042) minus 1198861198891198672 (120591 1199043 1199044)

minus 119886119891

1198673 (120591 1199045 1199046)]

(13)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)119894120601

119890minus(12)(1206012

+119894120601)1205902119876 minus 1)]

sdot exp[minus (119894120601120582119876

120583119876

120591

+ 120582V120591 (120590V (119890

minus(120588(119894120601)120590V)119886 minus 119890minus(120588(119894120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)minus 1)

+119894120601120588

120590V(V

119905+ 120579120591))] exp [(1minus 119894120601) (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [119894120601 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)] exp [minus1198661

(120591 1199021 119902

2) V

119905

minus 1198662

(120591 1199023 119902

4) 119903

119905minus 119866

3(120591 119902

5 119902

6)]

sdot exp [minus1205791198671

(120591 1199021 119902

2) minus 119886

119889119867

2(120591 119902

3 119902

4)

minus 119886119891

1198673

(120591 1199025 119902

6)]

(14)

where the functions 1198661 1198662 1198663 1198671 1198672 1198673 are given inLemma 5 and 119888

119905equals

119888119905

= exp (119894120601119909119905) = exp (119894120601 ln119865 (119905 119879)) (15)

Moreover the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by

1199041 = minus(1 + 119894120601) 120588

120590V

1199042 = minus(1 + 119894120601)

2(1 minus 1205882)

2minus

(1 + 119894120601) 120588120581

120590V+1 + 119894120601

2

1199043 = 0

1199044 = minus 119894120601

1199045 = 0

1199046 = 1+ 119894120601

(16)

and the constants 1199021 1199022 1199023 1199024 1199025 1199026 equal

1199021 = minus119894120601120588

120590V

1199022 = minus(119894120601)

2(1 minus 1205882)

2minus

119894120601120588120581

120590V+

119894120601

2

1199023 = 0

1199024 = 1minus 119894120601

1199025 = 0

1199026 = 119894120601

(17)

International Journal of Stochastic Analysis 5

41 Auxiliary Results The proof of Theorem 3 hinges on anumber of lemmasWe start by stating the well-known resultwhich can be easily obtained from Proposition 8634 inJeanblanc et al [27] Let us denote 120591 = 119879 minus 119905 and let us set forall 0 le 119905 lt 119879

119869119876

(119905 119879) =

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) (18)

Note that we use here Assumptions (A3)ndash(A5)The property(A3) (resp (A4)) implies that the random variable 119869119876(119905 119879)

(resp 119885V119879

minus 119885V119905) is independent of the 120590-fieldF

119905 Let ]1 stand

for the Gaussian distribution 119873(ln(1+120583119876

)minus(12)1205902119876

1205902119876

) andlet ]2 stand for the uniform distribution with density

]2 (119911) =1

119887 minus 119886

1 119886 lt 119911 lt 119887

0 else(19)

where 0 lt 119886 lt 119887

Lemma 4 (i) Under Assumptions (A3) and (A5) the follow-ing equalities are valid

EP119905

exp (119894120601119869119876

(119905 119879))

= EP119905

exp(119894120601

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896))

= exp [120582119876

120591 int+infin

minusinfin

(119890119894120601119911

minus 1) ]1 (119889119911)]

= exp [120582119876

120591 ((1+ 120583119876

)119894120601

119890minus(12)1205902

119876(1206012+119894120601)

minus 1)]

(20)

(ii) Under Assumptions (A4) and (A5) the following equali-ties are valid for 119888 = 119886 + 119887119894 with 119886 le 0

EP119905

exp (119888 (119885V119879

minus 119885V119905)) = E

P119905

exp(119888

119873V119879

sum119896=119873

V119905 +1

119869V119896)

= exp [120582V120591 int+infin

minusinfin

(119890119888119911

minus 1) ]2 (119889119911)]

= exp[120582V120591 (119890119888119887 minus 119890119888119886

119888 (119887 minus 119886)minus 1)]

(21)

The next result extends Lemma 61 in Ahlip andRutkowski [28] (see also Duffie et al [6]) where the modelwithout the jump component in the dynamics of V wasexamined

Lemma 5 Let the dynamics of processes V 119903 and 119903 be givenby SDEs (1) with independent Brownian motions 119882

V 119882119889 and119882119891 For any complex numbers 120583 120582 120583 120583 we set

119865 (120591 V119905 119903

119905 119903

119905) = E

P119905

exp(minus120582V119879

minus 120583 int119879

119905

V119906119889119906 minus 119903

119879

minus 120583 int119879

119905

119903119906119889119906 minus 119903

119879minus 120583 int

119879

119905

119903119906119889119906)

(22)

Then

119865 (120591 V119905 119903

119905 119903

119905) = exp [minus1198661 (120591 120582 120583) V

119905minus 1198662 (120591 120583) 119903

119905

minus 1198663 (120591 120583) 119903119905

minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(23)

where

1198661 (120591 120582 120583)

=120582 [(120574 + 120581) + 119890

120574120591(120574 minus 120581)] + 2120583 (119890

120574120591minus 1)

1205902V120582 (119890120574120591 minus 1) + 120574 minus 120581 + 119890120574120591 (120574 + 120581)

1198662 (120591 120583)

= [(120574 + 119887

119889) + 119890120574120591 (120574 minus 119887

119889)] + 2120583 (119890120574120591 minus 1)

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

1198663 (120591 120583)

= [(120574 + 119887

119891) + 119890120574120591 (120574 minus 119887

119891)] + 2120583 (119890120574120591 minus 1)

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

)

1198671 (120591 120582 120583) = int120591

0(1198661 (119905 120582 120583)

+120582V

120579(1+

119890minus1198871198661(119905120582120583) minus 119890minus1198861198661(119905120582120583)

1198661 (119905 120582 120583) (119887 minus 119886))) 119889119905

1198672 (120591 120583) = minus2

1205902119889

sdot ln(2120574119890(120574+119887119889)1205912

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

)

1198673 (120591 120583) = minus2

1205902119891

sdot ln(2120574119890(120574+119887119891)1205912

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

))

(24)

where one denotes 120574 = radic1205812 + 21205902V120583 120574 = radic1198872

119889+ 21205902

119889120583 and 120574 =

radic1198872119891

+ 21205902119891

120583

Proof For the readerrsquos convenience we sketch the proof ofthe lemma Let us set for 119905 isin [0 119879]

119872119905

= 119865 (120591 V119905 119903

119905 119903

119905)

sdot exp(minus120583 int119905

0V119906119889119906 minus 120583 int

119905

0119903119906119889119906 minus 120583 int

119905

0119903119906119889119906)

(25)

6 International Journal of Stochastic Analysis

Then the process 119872 = (119872119905)119905isin[0119879] satisfies

119872119905

= EP119905

exp(minus120582V119879

minus 120583 int119879

0V119906119889119906 minus 119903

119879

minus 120583 int119879

0119903119906119889119906 minus 119903

119879minus 120583 int

119879

0119903119906119889119906)

(26)

and thus it is an F-martingale under P By applying theIto formula to the right-hand side in (25) and by settingthe drift term in the dynamics of 119872 to be zero we deducethat the function 119865(120591 V 119903 119903) satisfies the following partialintegrodifferential equation (PIDE)

minus120597119865

120597120591+12

1205902V V

1205972119865

120597V2

+ 120582V intinfin

0(119865 (120591 V+ 119911 119903 119903) minus 119865 (120591 V 119903 119903)) ]2 (119889119911)

+12

1205902119889119903

1205972119865

1205971199032+12

1205902119891

1199031205972119865

1205971199032+ (120579 minus 120581V)

120597119865

120597V

+ (119886119889

minus 119887119889119903)

120597119865

120597119903+ (119886

119891minus 119887

119891119903)

120597119865

120597119903

minus (120583V+ 120583119903 + 120583119903) 119865 = 0

(27)

with the initial condition 119865(0 V 119903 119903) = exp(minus120582V minus 119903 minus 119903)We search for a solution to this PIDE in the form

119865 (120591 V 119903 119903) = exp [minus1198661 (120591 120582 120583) Vminus 1198662 (120591 120583) 119903

minus 1198663 (120591 120583) 119903 minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(28)

with

1198661 (0 120582 120583) = 120582

1198662 (0 120583) =

1198663 (0 120583) =

1198671 (0 120582 120583) = 1198672 (0 120583) = 1198673 (0 120583) = 0

(29)

By substituting this expression in the PIDE and using part (ii)in Lemma 4 we obtain the following system of ODEs for the

functions 1198661 1198662 1198663 1198671 1198672 1198673 (for brevity we suppress thelast three arguments)

1205971198661 (120591)

120597120591= minus

12

1205902V119866

21 (120591) minus 1205811198661 (120591) + 120583

1205971198671 (120591)

120597120591= 1198661 (120591) +

120582V

120579(1+

119890minus1198871198661(120591) minus 119890minus1198861198661(120591)

1198661 (120591) (119887 minus 119886))

1205971198662 (120591)

120597120591= minus

12

120590211988911986622 (120591) minus 119887

1198891198662 (120591) + 120583

1205971198672 (120591)

120597120591= 1198662 (120591)

1205971198663 (120591)

120597120591= minus

12

1205902119891

11986623 (120591) minus 119887

1198911198663 (120591) + 120583

1205971198673 (120591)

120597120591= 1198663 (120591)

(30)

By solving these equations we obtain the stated formu-lae

Under the assumptions of Lemma 5 it is possible tofactorize 119865 as a product of two conditional expectationsThismeans that the functions1198661 (1198671)1198662 (1198672) and1198663 (1198673) are ofthe same form except that they correspond to different setsof parameters 120579 120581 120590V for 1198661 1198671 119886

119889 119887

119889 120590

119903for 1198662 1198672 and 119886

119891

119887119891 120590

119891for 1198663 1198673 Note however that the roles played by the

processes V 119903 and 119903 in our model are clearly differentIt should also be stressed that no closed-form analytical

expression for 119865(120591 V119905 119903

119905 119903

119905) is available in the case of cor-

related Brownian motions 119882V 119882119903 119882119891 Brigo and Alfonsi[29] who deal with this issue in a different context proposeto use a simple Gaussian approximation instead of the exactsolution More recently Grzelak and Oosterlee [16] proposedmore sophisticated approximations in the framework of theHestonCIR hybrid model We do not follow this path herehowever and we focus instead on finding a semianalyticalsolution since this goal can be achieved under Assumptions(A1)ndash(A6)

Let us now introduce a convenient change of the underly-ing probability measure from the domestic spot martingalemeasure P to the domestic forward martingale measure P

119879

Definition 6 The domestic forward martingale measure P119879

equivalent to P on (ΩF119879

) is defined by the Radon-Nikodym derivative process 120578 = (120578

119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P

10038161003816100381610038161003816100381610038161003816F119905= exp(minus int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(31)

An application of the Girsanov theorem shows that theprocess 119882119879 = (119882119879

119905)119905isin[0119879] which is given by the equality

119882119879

119905= 119882

119889

119905+ int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119906 (32)

International Journal of Stochastic Analysis 7

is the Brownian motion under the domestic forward martin-gale measure P

119879 Using the standard change of a numeraire

technique one can check that the price of the European for-eign exchange call option admits the following representationunder the probability measure P

119879

119862119905

(119879 119870) = 119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

minus 119870119861119889

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(33)

The following auxiliary result is easy to establish and thusits proof is omitted Recall that 119869119876(119905 119879) is given by equality(18)

Lemma 7 Under Assumptions (A1)ndash(A6) the dynamics ofthe forward exchange rate 119865(119905 119879) under the domestic forwardmartingale measure P

119879are given by the SDE

119889119865 (119905 119879) = 119865 (119905 119879) (119889119885119876

119905minus 120582

119876120583119876

119889119905 + radicV119905

119889119882119876

119905

+ 120590119889119899119889

(119905 119879) radic119903119905

119889119882119879

119905minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(34)

or equivalently

119865 (119879 119879) = 119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus

1

2int

119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(35)

where the dot sdot denotes the inner product inR3 (119865

(119905 119879))119905isin[0119879]

is the R3-valued process (row vector) given by

119865

(119905 119879) = [radicV119905 120590

119889119899119889

(119905 119879) radic119903119905 minus 120590

119891119899119891

(119905 119879) radic119903119905] (36)

and 119879 = (119879

119905)119905isin[0119879] stands for the R3-valued process

(column vector) given by

119879

119905= [119882

119876

119905 119882

119879

119905 119882

119891

119905]lowast

(37)

It is easy to check that under Assumptions (A1)ndash(A6)the process 119879 is the three-dimensional standard Brownianmotion under P

119879 In view of Lemma 7 we have that

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

) = 119861119889

(119905 119879)

sdotEP119879119905

119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot 1119865(119879119879)gt119870

= 119876119905119861119891

(119905 119879)EP119879119905

exp(119869119876

(119905 119879)

minus 120582119876

120583119876

(119879 minus 119905) + int119879

119905

119865

(119906 119879) sdot 119889119879

119906

minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906) 1119865(119879119879)gt119870

(38)

To deal with the first term in the right-hand side of (33) weintroduce another auxiliary probability measure

Definition 8 Themodified domestic forward martingale mea-sure P

119879 equivalent to P

119879on (ΩF

119879) is defined by the

Radon-Nikodym derivative process 120578 = (120578119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(39)

Using Lemma 7 and (8) we obtain

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

120578119879

)

EP119879119905

(120578119879

)

(40)

and thus the Bayes formula and Definition 8 yield

119861119889

(119905 119879)EP119879t (119865 (119879 119879) 1

119865(119879119879)gt119870)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(41)

This shows that P119879is a martingale measure associated with

the choice of the price process119876119905119861119891

(119905 119879) as a numeraire assetWe are now in a position to state the following lemma

Lemma 9 The price of the FX call option satisfies

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119876119879

gt 119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119876119879

gt 119870 | F119905)

(42)

or equivalently

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119909119879

gt ln119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119909119879

gt ln119870 | F119905)

(43)

To complete the proof Theorem 3 it remains to evaluatethe conditional probabilities arising in formula (43) Byanother application of the Girsanov theorem one can checkthat the process (119876 V 119903 119903) has theMarkov property under theprobability measuresP

119879and P

119879 In view of Proposition 1 and

Lemma 2 the random variable 119909119879is a function of 119876

119879 119903

119879 and

119903119879 We thus conclude that

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(44)

8 International Journal of Stochastic Analysis

where we denote

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(45)

To obtain explicit formulae for the conditional probabili-ties above it suffices to derive the corresponding conditionalcharacteristic functions

1198911 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

(46)

The idea is to use the Radon-Nikodym derivatives in order toobtain convenient expressions for the characteristic functionsin terms of conditional expectations under the domestic spotmartingale measure P The following lemma will allow us toachieve this goal

Lemma 10 The following equality holds

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(47)

Proof Straightforward computations show that

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

119889P119879

119889P

10038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906) = exp(int

119905

0radicV

119906119889119882

119876

119906

+ int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906) exp(minus

12

sdot int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(48)

Using (32) we now obtain

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(49)

which is the desired expression

In view of the formula established in Lemma 10and the abstract Bayes formula to compute 1198911(120601) =

1198911(120601 119905 119876119905 V

119905 119903

119905 119903

119905) it suffices to focus on the following

conditional expectation under P

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp(int119879

119905

radicV119906

119889119882119876

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(50)

Similarly in view of formula (31) we obtain for 1198912(120601) =

1198912(120601 119905 119876119905 V

119905 119903

119905 119903

119905)

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(51)

To proceed we will need the following result which is animmediate consequence of Lemma 7

Corollary 11 Under Assumptions (A1)ndash(A4) the process119909119905

= ln119865(119905 119879) admits the following representation under thedomestic forward martingale measure P

119879

119909119879

= 119909119905

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906

+ 119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

(52)

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 5

41 Auxiliary Results The proof of Theorem 3 hinges on anumber of lemmasWe start by stating the well-known resultwhich can be easily obtained from Proposition 8634 inJeanblanc et al [27] Let us denote 120591 = 119879 minus 119905 and let us set forall 0 le 119905 lt 119879

119869119876

(119905 119879) =

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) (18)

Note that we use here Assumptions (A3)ndash(A5)The property(A3) (resp (A4)) implies that the random variable 119869119876(119905 119879)

(resp 119885V119879

minus 119885V119905) is independent of the 120590-fieldF

119905 Let ]1 stand

for the Gaussian distribution 119873(ln(1+120583119876

)minus(12)1205902119876

1205902119876

) andlet ]2 stand for the uniform distribution with density

]2 (119911) =1

119887 minus 119886

1 119886 lt 119911 lt 119887

0 else(19)

where 0 lt 119886 lt 119887

Lemma 4 (i) Under Assumptions (A3) and (A5) the follow-ing equalities are valid

EP119905

exp (119894120601119869119876

(119905 119879))

= EP119905

exp(119894120601

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896))

= exp [120582119876

120591 int+infin

minusinfin

(119890119894120601119911

minus 1) ]1 (119889119911)]

= exp [120582119876

120591 ((1+ 120583119876

)119894120601

119890minus(12)1205902

119876(1206012+119894120601)

minus 1)]

(20)

(ii) Under Assumptions (A4) and (A5) the following equali-ties are valid for 119888 = 119886 + 119887119894 with 119886 le 0

EP119905

exp (119888 (119885V119879

minus 119885V119905)) = E

P119905

exp(119888

119873V119879

sum119896=119873

V119905 +1

119869V119896)

= exp [120582V120591 int+infin

minusinfin

(119890119888119911

minus 1) ]2 (119889119911)]

= exp[120582V120591 (119890119888119887 minus 119890119888119886

119888 (119887 minus 119886)minus 1)]

(21)

The next result extends Lemma 61 in Ahlip andRutkowski [28] (see also Duffie et al [6]) where the modelwithout the jump component in the dynamics of V wasexamined

Lemma 5 Let the dynamics of processes V 119903 and 119903 be givenby SDEs (1) with independent Brownian motions 119882

V 119882119889 and119882119891 For any complex numbers 120583 120582 120583 120583 we set

119865 (120591 V119905 119903

119905 119903

119905) = E

P119905

exp(minus120582V119879

minus 120583 int119879

119905

V119906119889119906 minus 119903

119879

minus 120583 int119879

119905

119903119906119889119906 minus 119903

119879minus 120583 int

119879

119905

119903119906119889119906)

(22)

Then

119865 (120591 V119905 119903

119905 119903

119905) = exp [minus1198661 (120591 120582 120583) V

119905minus 1198662 (120591 120583) 119903

119905

minus 1198663 (120591 120583) 119903119905

minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(23)

where

1198661 (120591 120582 120583)

=120582 [(120574 + 120581) + 119890

120574120591(120574 minus 120581)] + 2120583 (119890

120574120591minus 1)

1205902V120582 (119890120574120591 minus 1) + 120574 minus 120581 + 119890120574120591 (120574 + 120581)

1198662 (120591 120583)

= [(120574 + 119887

119889) + 119890120574120591 (120574 minus 119887

119889)] + 2120583 (119890120574120591 minus 1)

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

1198663 (120591 120583)

= [(120574 + 119887

119891) + 119890120574120591 (120574 minus 119887

119891)] + 2120583 (119890120574120591 minus 1)

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

)

1198671 (120591 120582 120583) = int120591

0(1198661 (119905 120582 120583)

+120582V

120579(1+

119890minus1198871198661(119905120582120583) minus 119890minus1198861198661(119905120582120583)

1198661 (119905 120582 120583) (119887 minus 119886))) 119889119905

1198672 (120591 120583) = minus2

1205902119889

sdot ln(2120574119890(120574+119887119889)1205912

1205902119889 (119890120574120591 minus 1) + 120574 minus 119887

119889+ 119890120574120591 (120574 + 119887

119889)

)

1198673 (120591 120583) = minus2

1205902119891

sdot ln(2120574119890(120574+119887119891)1205912

1205902119891

(119890120574120591 minus 1) + 120574 minus 119887119891

+ 119890120574120591 (120574 + 119887119891

))

(24)

where one denotes 120574 = radic1205812 + 21205902V120583 120574 = radic1198872

119889+ 21205902

119889120583 and 120574 =

radic1198872119891

+ 21205902119891

120583

Proof For the readerrsquos convenience we sketch the proof ofthe lemma Let us set for 119905 isin [0 119879]

119872119905

= 119865 (120591 V119905 119903

119905 119903

119905)

sdot exp(minus120583 int119905

0V119906119889119906 minus 120583 int

119905

0119903119906119889119906 minus 120583 int

119905

0119903119906119889119906)

(25)

6 International Journal of Stochastic Analysis

Then the process 119872 = (119872119905)119905isin[0119879] satisfies

119872119905

= EP119905

exp(minus120582V119879

minus 120583 int119879

0V119906119889119906 minus 119903

119879

minus 120583 int119879

0119903119906119889119906 minus 119903

119879minus 120583 int

119879

0119903119906119889119906)

(26)

and thus it is an F-martingale under P By applying theIto formula to the right-hand side in (25) and by settingthe drift term in the dynamics of 119872 to be zero we deducethat the function 119865(120591 V 119903 119903) satisfies the following partialintegrodifferential equation (PIDE)

minus120597119865

120597120591+12

1205902V V

1205972119865

120597V2

+ 120582V intinfin

0(119865 (120591 V+ 119911 119903 119903) minus 119865 (120591 V 119903 119903)) ]2 (119889119911)

+12

1205902119889119903

1205972119865

1205971199032+12

1205902119891

1199031205972119865

1205971199032+ (120579 minus 120581V)

120597119865

120597V

+ (119886119889

minus 119887119889119903)

120597119865

120597119903+ (119886

119891minus 119887

119891119903)

120597119865

120597119903

minus (120583V+ 120583119903 + 120583119903) 119865 = 0

(27)

with the initial condition 119865(0 V 119903 119903) = exp(minus120582V minus 119903 minus 119903)We search for a solution to this PIDE in the form

119865 (120591 V 119903 119903) = exp [minus1198661 (120591 120582 120583) Vminus 1198662 (120591 120583) 119903

minus 1198663 (120591 120583) 119903 minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(28)

with

1198661 (0 120582 120583) = 120582

1198662 (0 120583) =

1198663 (0 120583) =

1198671 (0 120582 120583) = 1198672 (0 120583) = 1198673 (0 120583) = 0

(29)

By substituting this expression in the PIDE and using part (ii)in Lemma 4 we obtain the following system of ODEs for the

functions 1198661 1198662 1198663 1198671 1198672 1198673 (for brevity we suppress thelast three arguments)

1205971198661 (120591)

120597120591= minus

12

1205902V119866

21 (120591) minus 1205811198661 (120591) + 120583

1205971198671 (120591)

120597120591= 1198661 (120591) +

120582V

120579(1+

119890minus1198871198661(120591) minus 119890minus1198861198661(120591)

1198661 (120591) (119887 minus 119886))

1205971198662 (120591)

120597120591= minus

12

120590211988911986622 (120591) minus 119887

1198891198662 (120591) + 120583

1205971198672 (120591)

120597120591= 1198662 (120591)

1205971198663 (120591)

120597120591= minus

12

1205902119891

11986623 (120591) minus 119887

1198911198663 (120591) + 120583

1205971198673 (120591)

120597120591= 1198663 (120591)

(30)

By solving these equations we obtain the stated formu-lae

Under the assumptions of Lemma 5 it is possible tofactorize 119865 as a product of two conditional expectationsThismeans that the functions1198661 (1198671)1198662 (1198672) and1198663 (1198673) are ofthe same form except that they correspond to different setsof parameters 120579 120581 120590V for 1198661 1198671 119886

119889 119887

119889 120590

119903for 1198662 1198672 and 119886

119891

119887119891 120590

119891for 1198663 1198673 Note however that the roles played by the

processes V 119903 and 119903 in our model are clearly differentIt should also be stressed that no closed-form analytical

expression for 119865(120591 V119905 119903

119905 119903

119905) is available in the case of cor-

related Brownian motions 119882V 119882119903 119882119891 Brigo and Alfonsi[29] who deal with this issue in a different context proposeto use a simple Gaussian approximation instead of the exactsolution More recently Grzelak and Oosterlee [16] proposedmore sophisticated approximations in the framework of theHestonCIR hybrid model We do not follow this path herehowever and we focus instead on finding a semianalyticalsolution since this goal can be achieved under Assumptions(A1)ndash(A6)

Let us now introduce a convenient change of the underly-ing probability measure from the domestic spot martingalemeasure P to the domestic forward martingale measure P

119879

Definition 6 The domestic forward martingale measure P119879

equivalent to P on (ΩF119879

) is defined by the Radon-Nikodym derivative process 120578 = (120578

119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P

10038161003816100381610038161003816100381610038161003816F119905= exp(minus int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(31)

An application of the Girsanov theorem shows that theprocess 119882119879 = (119882119879

119905)119905isin[0119879] which is given by the equality

119882119879

119905= 119882

119889

119905+ int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119906 (32)

International Journal of Stochastic Analysis 7

is the Brownian motion under the domestic forward martin-gale measure P

119879 Using the standard change of a numeraire

technique one can check that the price of the European for-eign exchange call option admits the following representationunder the probability measure P

119879

119862119905

(119879 119870) = 119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

minus 119870119861119889

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(33)

The following auxiliary result is easy to establish and thusits proof is omitted Recall that 119869119876(119905 119879) is given by equality(18)

Lemma 7 Under Assumptions (A1)ndash(A6) the dynamics ofthe forward exchange rate 119865(119905 119879) under the domestic forwardmartingale measure P

119879are given by the SDE

119889119865 (119905 119879) = 119865 (119905 119879) (119889119885119876

119905minus 120582

119876120583119876

119889119905 + radicV119905

119889119882119876

119905

+ 120590119889119899119889

(119905 119879) radic119903119905

119889119882119879

119905minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(34)

or equivalently

119865 (119879 119879) = 119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus

1

2int

119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(35)

where the dot sdot denotes the inner product inR3 (119865

(119905 119879))119905isin[0119879]

is the R3-valued process (row vector) given by

119865

(119905 119879) = [radicV119905 120590

119889119899119889

(119905 119879) radic119903119905 minus 120590

119891119899119891

(119905 119879) radic119903119905] (36)

and 119879 = (119879

119905)119905isin[0119879] stands for the R3-valued process

(column vector) given by

119879

119905= [119882

119876

119905 119882

119879

119905 119882

119891

119905]lowast

(37)

It is easy to check that under Assumptions (A1)ndash(A6)the process 119879 is the three-dimensional standard Brownianmotion under P

119879 In view of Lemma 7 we have that

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

) = 119861119889

(119905 119879)

sdotEP119879119905

119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot 1119865(119879119879)gt119870

= 119876119905119861119891

(119905 119879)EP119879119905

exp(119869119876

(119905 119879)

minus 120582119876

120583119876

(119879 minus 119905) + int119879

119905

119865

(119906 119879) sdot 119889119879

119906

minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906) 1119865(119879119879)gt119870

(38)

To deal with the first term in the right-hand side of (33) weintroduce another auxiliary probability measure

Definition 8 Themodified domestic forward martingale mea-sure P

119879 equivalent to P

119879on (ΩF

119879) is defined by the

Radon-Nikodym derivative process 120578 = (120578119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(39)

Using Lemma 7 and (8) we obtain

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

120578119879

)

EP119879119905

(120578119879

)

(40)

and thus the Bayes formula and Definition 8 yield

119861119889

(119905 119879)EP119879t (119865 (119879 119879) 1

119865(119879119879)gt119870)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(41)

This shows that P119879is a martingale measure associated with

the choice of the price process119876119905119861119891

(119905 119879) as a numeraire assetWe are now in a position to state the following lemma

Lemma 9 The price of the FX call option satisfies

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119876119879

gt 119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119876119879

gt 119870 | F119905)

(42)

or equivalently

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119909119879

gt ln119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119909119879

gt ln119870 | F119905)

(43)

To complete the proof Theorem 3 it remains to evaluatethe conditional probabilities arising in formula (43) Byanother application of the Girsanov theorem one can checkthat the process (119876 V 119903 119903) has theMarkov property under theprobability measuresP

119879and P

119879 In view of Proposition 1 and

Lemma 2 the random variable 119909119879is a function of 119876

119879 119903

119879 and

119903119879 We thus conclude that

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(44)

8 International Journal of Stochastic Analysis

where we denote

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(45)

To obtain explicit formulae for the conditional probabili-ties above it suffices to derive the corresponding conditionalcharacteristic functions

1198911 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

(46)

The idea is to use the Radon-Nikodym derivatives in order toobtain convenient expressions for the characteristic functionsin terms of conditional expectations under the domestic spotmartingale measure P The following lemma will allow us toachieve this goal

Lemma 10 The following equality holds

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(47)

Proof Straightforward computations show that

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

119889P119879

119889P

10038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906) = exp(int

119905

0radicV

119906119889119882

119876

119906

+ int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906) exp(minus

12

sdot int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(48)

Using (32) we now obtain

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(49)

which is the desired expression

In view of the formula established in Lemma 10and the abstract Bayes formula to compute 1198911(120601) =

1198911(120601 119905 119876119905 V

119905 119903

119905 119903

119905) it suffices to focus on the following

conditional expectation under P

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp(int119879

119905

radicV119906

119889119882119876

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(50)

Similarly in view of formula (31) we obtain for 1198912(120601) =

1198912(120601 119905 119876119905 V

119905 119903

119905 119903

119905)

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(51)

To proceed we will need the following result which is animmediate consequence of Lemma 7

Corollary 11 Under Assumptions (A1)ndash(A4) the process119909119905

= ln119865(119905 119879) admits the following representation under thedomestic forward martingale measure P

119879

119909119879

= 119909119905

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906

+ 119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

(52)

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

6 International Journal of Stochastic Analysis

Then the process 119872 = (119872119905)119905isin[0119879] satisfies

119872119905

= EP119905

exp(minus120582V119879

minus 120583 int119879

0V119906119889119906 minus 119903

119879

minus 120583 int119879

0119903119906119889119906 minus 119903

119879minus 120583 int

119879

0119903119906119889119906)

(26)

and thus it is an F-martingale under P By applying theIto formula to the right-hand side in (25) and by settingthe drift term in the dynamics of 119872 to be zero we deducethat the function 119865(120591 V 119903 119903) satisfies the following partialintegrodifferential equation (PIDE)

minus120597119865

120597120591+12

1205902V V

1205972119865

120597V2

+ 120582V intinfin

0(119865 (120591 V+ 119911 119903 119903) minus 119865 (120591 V 119903 119903)) ]2 (119889119911)

+12

1205902119889119903

1205972119865

1205971199032+12

1205902119891

1199031205972119865

1205971199032+ (120579 minus 120581V)

120597119865

120597V

+ (119886119889

minus 119887119889119903)

120597119865

120597119903+ (119886

119891minus 119887

119891119903)

120597119865

120597119903

minus (120583V+ 120583119903 + 120583119903) 119865 = 0

(27)

with the initial condition 119865(0 V 119903 119903) = exp(minus120582V minus 119903 minus 119903)We search for a solution to this PIDE in the form

119865 (120591 V 119903 119903) = exp [minus1198661 (120591 120582 120583) Vminus 1198662 (120591 120583) 119903

minus 1198663 (120591 120583) 119903 minus 1205791198671 (120591 120582 120583) minus 1198861198891198672 (120591 120583)

minus 119886119891

1198673 (120591 120583)]

(28)

with

1198661 (0 120582 120583) = 120582

1198662 (0 120583) =

1198663 (0 120583) =

1198671 (0 120582 120583) = 1198672 (0 120583) = 1198673 (0 120583) = 0

(29)

By substituting this expression in the PIDE and using part (ii)in Lemma 4 we obtain the following system of ODEs for the

functions 1198661 1198662 1198663 1198671 1198672 1198673 (for brevity we suppress thelast three arguments)

1205971198661 (120591)

120597120591= minus

12

1205902V119866

21 (120591) minus 1205811198661 (120591) + 120583

1205971198671 (120591)

120597120591= 1198661 (120591) +

120582V

120579(1+

119890minus1198871198661(120591) minus 119890minus1198861198661(120591)

1198661 (120591) (119887 minus 119886))

1205971198662 (120591)

120597120591= minus

12

120590211988911986622 (120591) minus 119887

1198891198662 (120591) + 120583

1205971198672 (120591)

120597120591= 1198662 (120591)

1205971198663 (120591)

120597120591= minus

12

1205902119891

11986623 (120591) minus 119887

1198911198663 (120591) + 120583

1205971198673 (120591)

120597120591= 1198663 (120591)

(30)

By solving these equations we obtain the stated formu-lae

Under the assumptions of Lemma 5 it is possible tofactorize 119865 as a product of two conditional expectationsThismeans that the functions1198661 (1198671)1198662 (1198672) and1198663 (1198673) are ofthe same form except that they correspond to different setsof parameters 120579 120581 120590V for 1198661 1198671 119886

119889 119887

119889 120590

119903for 1198662 1198672 and 119886

119891

119887119891 120590

119891for 1198663 1198673 Note however that the roles played by the

processes V 119903 and 119903 in our model are clearly differentIt should also be stressed that no closed-form analytical

expression for 119865(120591 V119905 119903

119905 119903

119905) is available in the case of cor-

related Brownian motions 119882V 119882119903 119882119891 Brigo and Alfonsi[29] who deal with this issue in a different context proposeto use a simple Gaussian approximation instead of the exactsolution More recently Grzelak and Oosterlee [16] proposedmore sophisticated approximations in the framework of theHestonCIR hybrid model We do not follow this path herehowever and we focus instead on finding a semianalyticalsolution since this goal can be achieved under Assumptions(A1)ndash(A6)

Let us now introduce a convenient change of the underly-ing probability measure from the domestic spot martingalemeasure P to the domestic forward martingale measure P

119879

Definition 6 The domestic forward martingale measure P119879

equivalent to P on (ΩF119879

) is defined by the Radon-Nikodym derivative process 120578 = (120578

119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P

10038161003816100381610038161003816100381610038161003816F119905= exp(minus int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(31)

An application of the Girsanov theorem shows that theprocess 119882119879 = (119882119879

119905)119905isin[0119879] which is given by the equality

119882119879

119905= 119882

119889

119905+ int

119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119906 (32)

International Journal of Stochastic Analysis 7

is the Brownian motion under the domestic forward martin-gale measure P

119879 Using the standard change of a numeraire

technique one can check that the price of the European for-eign exchange call option admits the following representationunder the probability measure P

119879

119862119905

(119879 119870) = 119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

minus 119870119861119889

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(33)

The following auxiliary result is easy to establish and thusits proof is omitted Recall that 119869119876(119905 119879) is given by equality(18)

Lemma 7 Under Assumptions (A1)ndash(A6) the dynamics ofthe forward exchange rate 119865(119905 119879) under the domestic forwardmartingale measure P

119879are given by the SDE

119889119865 (119905 119879) = 119865 (119905 119879) (119889119885119876

119905minus 120582

119876120583119876

119889119905 + radicV119905

119889119882119876

119905

+ 120590119889119899119889

(119905 119879) radic119903119905

119889119882119879

119905minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(34)

or equivalently

119865 (119879 119879) = 119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus

1

2int

119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(35)

where the dot sdot denotes the inner product inR3 (119865

(119905 119879))119905isin[0119879]

is the R3-valued process (row vector) given by

119865

(119905 119879) = [radicV119905 120590

119889119899119889

(119905 119879) radic119903119905 minus 120590

119891119899119891

(119905 119879) radic119903119905] (36)

and 119879 = (119879

119905)119905isin[0119879] stands for the R3-valued process

(column vector) given by

119879

119905= [119882

119876

119905 119882

119879

119905 119882

119891

119905]lowast

(37)

It is easy to check that under Assumptions (A1)ndash(A6)the process 119879 is the three-dimensional standard Brownianmotion under P

119879 In view of Lemma 7 we have that

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

) = 119861119889

(119905 119879)

sdotEP119879119905

119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot 1119865(119879119879)gt119870

= 119876119905119861119891

(119905 119879)EP119879119905

exp(119869119876

(119905 119879)

minus 120582119876

120583119876

(119879 minus 119905) + int119879

119905

119865

(119906 119879) sdot 119889119879

119906

minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906) 1119865(119879119879)gt119870

(38)

To deal with the first term in the right-hand side of (33) weintroduce another auxiliary probability measure

Definition 8 Themodified domestic forward martingale mea-sure P

119879 equivalent to P

119879on (ΩF

119879) is defined by the

Radon-Nikodym derivative process 120578 = (120578119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(39)

Using Lemma 7 and (8) we obtain

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

120578119879

)

EP119879119905

(120578119879

)

(40)

and thus the Bayes formula and Definition 8 yield

119861119889

(119905 119879)EP119879t (119865 (119879 119879) 1

119865(119879119879)gt119870)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(41)

This shows that P119879is a martingale measure associated with

the choice of the price process119876119905119861119891

(119905 119879) as a numeraire assetWe are now in a position to state the following lemma

Lemma 9 The price of the FX call option satisfies

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119876119879

gt 119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119876119879

gt 119870 | F119905)

(42)

or equivalently

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119909119879

gt ln119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119909119879

gt ln119870 | F119905)

(43)

To complete the proof Theorem 3 it remains to evaluatethe conditional probabilities arising in formula (43) Byanother application of the Girsanov theorem one can checkthat the process (119876 V 119903 119903) has theMarkov property under theprobability measuresP

119879and P

119879 In view of Proposition 1 and

Lemma 2 the random variable 119909119879is a function of 119876

119879 119903

119879 and

119903119879 We thus conclude that

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(44)

8 International Journal of Stochastic Analysis

where we denote

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(45)

To obtain explicit formulae for the conditional probabili-ties above it suffices to derive the corresponding conditionalcharacteristic functions

1198911 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

(46)

The idea is to use the Radon-Nikodym derivatives in order toobtain convenient expressions for the characteristic functionsin terms of conditional expectations under the domestic spotmartingale measure P The following lemma will allow us toachieve this goal

Lemma 10 The following equality holds

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(47)

Proof Straightforward computations show that

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

119889P119879

119889P

10038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906) = exp(int

119905

0radicV

119906119889119882

119876

119906

+ int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906) exp(minus

12

sdot int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(48)

Using (32) we now obtain

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(49)

which is the desired expression

In view of the formula established in Lemma 10and the abstract Bayes formula to compute 1198911(120601) =

1198911(120601 119905 119876119905 V

119905 119903

119905 119903

119905) it suffices to focus on the following

conditional expectation under P

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp(int119879

119905

radicV119906

119889119882119876

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(50)

Similarly in view of formula (31) we obtain for 1198912(120601) =

1198912(120601 119905 119876119905 V

119905 119903

119905 119903

119905)

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(51)

To proceed we will need the following result which is animmediate consequence of Lemma 7

Corollary 11 Under Assumptions (A1)ndash(A4) the process119909119905

= ln119865(119905 119879) admits the following representation under thedomestic forward martingale measure P

119879

119909119879

= 119909119905

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906

+ 119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

(52)

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Algebra

Discrete Dynamics in Nature and Society

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 7

is the Brownian motion under the domestic forward martin-gale measure P

119879 Using the standard change of a numeraire

technique one can check that the price of the European for-eign exchange call option admits the following representationunder the probability measure P

119879

119862119905

(119879 119870) = 119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

minus 119870119861119889

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(33)

The following auxiliary result is easy to establish and thusits proof is omitted Recall that 119869119876(119905 119879) is given by equality(18)

Lemma 7 Under Assumptions (A1)ndash(A6) the dynamics ofthe forward exchange rate 119865(119905 119879) under the domestic forwardmartingale measure P

119879are given by the SDE

119889119865 (119905 119879) = 119865 (119905 119879) (119889119885119876

119905minus 120582

119876120583119876

119889119905 + radicV119905

119889119882119876

119905

+ 120590119889119899119889

(119905 119879) radic119903119905

119889119882119879

119905minus 120590

119891119899119891

(119905 119879) radic119903119905

119889119882119891

119905)

(34)

or equivalently

119865 (119879 119879) = 119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus

1

2int

119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(35)

where the dot sdot denotes the inner product inR3 (119865

(119905 119879))119905isin[0119879]

is the R3-valued process (row vector) given by

119865

(119905 119879) = [radicV119905 120590

119889119899119889

(119905 119879) radic119903119905 minus 120590

119891119899119891

(119905 119879) radic119903119905] (36)

and 119879 = (119879

119905)119905isin[0119879] stands for the R3-valued process

(column vector) given by

119879

119905= [119882

119876

119905 119882

119879

119905 119882

119891

119905]lowast

(37)

It is easy to check that under Assumptions (A1)ndash(A6)the process 119879 is the three-dimensional standard Brownianmotion under P

119879 In view of Lemma 7 we have that

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

) = 119861119889

(119905 119879)

sdotEP119879119905

119865 (119905 119879) exp(119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot 1119865(119879119879)gt119870

= 119876119905119861119891

(119905 119879)EP119879119905

exp(119869119876

(119905 119879)

minus 120582119876

120583119876

(119879 minus 119905) + int119879

119905

119865

(119906 119879) sdot 119889119879

119906

minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906) 1119865(119879119879)gt119870

(38)

To deal with the first term in the right-hand side of (33) weintroduce another auxiliary probability measure

Definition 8 Themodified domestic forward martingale mea-sure P

119879 equivalent to P

119879on (ΩF

119879) is defined by the

Radon-Nikodym derivative process 120578 = (120578119905)119905isin[0119879] where

120578119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

(39)

Using Lemma 7 and (8) we obtain

119861119889

(119905 119879)EP119879119905

(119865 (119879 119879) 1119865(119879119879)gt119870

)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

120578119879

)

EP119879119905

(120578119879

)

(40)

and thus the Bayes formula and Definition 8 yield

119861119889

(119905 119879)EP119879t (119865 (119879 119879) 1

119865(119879119879)gt119870)

= 119876119905119861119891

(119905 119879)EP119879119905

(1119865(119879119879)gt119870

)

(41)

This shows that P119879is a martingale measure associated with

the choice of the price process119876119905119861119891

(119905 119879) as a numeraire assetWe are now in a position to state the following lemma

Lemma 9 The price of the FX call option satisfies

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119876119879

gt 119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119876119879

gt 119870 | F119905)

(42)

or equivalently

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) P119879

(119909119879

gt ln119870 | F119905)

minus 119870119861119889

(119905 119879)P119879

(119909119879

gt ln119870 | F119905)

(43)

To complete the proof Theorem 3 it remains to evaluatethe conditional probabilities arising in formula (43) Byanother application of the Girsanov theorem one can checkthat the process (119876 V 119903 119903) has theMarkov property under theprobability measuresP

119879and P

119879 In view of Proposition 1 and

Lemma 2 the random variable 119909119879is a function of 119876

119879 119903

119879 and

119903119879 We thus conclude that

119862119905

(119879 119870) = 119876119905119861119891

(119905 119879) 1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

minus 119870119861119889

(119905 119879) 1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870)

(44)

8 International Journal of Stochastic Analysis

where we denote

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(45)

To obtain explicit formulae for the conditional probabili-ties above it suffices to derive the corresponding conditionalcharacteristic functions

1198911 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

(46)

The idea is to use the Radon-Nikodym derivatives in order toobtain convenient expressions for the characteristic functionsin terms of conditional expectations under the domestic spotmartingale measure P The following lemma will allow us toachieve this goal

Lemma 10 The following equality holds

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(47)

Proof Straightforward computations show that

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

119889P119879

119889P

10038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906) = exp(int

119905

0radicV

119906119889119882

119876

119906

+ int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906) exp(minus

12

sdot int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(48)

Using (32) we now obtain

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(49)

which is the desired expression

In view of the formula established in Lemma 10and the abstract Bayes formula to compute 1198911(120601) =

1198911(120601 119905 119876119905 V

119905 119903

119905 119903

119905) it suffices to focus on the following

conditional expectation under P

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp(int119879

119905

radicV119906

119889119882119876

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(50)

Similarly in view of formula (31) we obtain for 1198912(120601) =

1198912(120601 119905 119876119905 V

119905 119903

119905 119903

119905)

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(51)

To proceed we will need the following result which is animmediate consequence of Lemma 7

Corollary 11 Under Assumptions (A1)ndash(A4) the process119909119905

= ln119865(119905 119879) admits the following representation under thedomestic forward martingale measure P

119879

119909119879

= 119909119905

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906

+ 119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

(52)

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 International Journal of Stochastic Analysis

where we denote

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(45)

To obtain explicit formulae for the conditional probabili-ties above it suffices to derive the corresponding conditionalcharacteristic functions

1198911 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)]

(46)

The idea is to use the Radon-Nikodym derivatives in order toobtain convenient expressions for the characteristic functionsin terms of conditional expectations under the domestic spotmartingale measure P The following lemma will allow us toachieve this goal

Lemma 10 The following equality holds

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(47)

Proof Straightforward computations show that

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

=119889P

119879

119889P119879

100381610038161003816100381610038161003816100381610038161003816F119905

119889P119879

119889P

10038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0119865

(119906 119879) sdot 119889119879

119906minus12

int119905

0

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906) = exp(int

119905

0radicV

119906119889119882

119876

119906

+ int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906) exp(minus

12

sdot int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

sdot exp(minus int119905

0120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

sdot int119905

012059021198891198992119889

(119906 119879) 119903119906119889119906)

(48)

Using (32) we now obtain

119889P119879

119889P

100381610038161003816100381610038161003816100381610038161003816F119905

= exp(int119905

0radicV

119906119889119882

119876

119906

minus int119905

0120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119905

0(V

119906+ 120590

2119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(49)

which is the desired expression

In view of the formula established in Lemma 10and the abstract Bayes formula to compute 1198911(120601) =

1198911(120601 119905 119876119905 V

119905 119903

119905 119903

119905) it suffices to focus on the following

conditional expectation under P

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp(int119879

119905

radicV119906

119889119882119876

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)

(50)

Similarly in view of formula (31) we obtain for 1198912(120601) =

1198912(120601 119905 119876119905 V

119905 119903

119905 119903

119905)

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(51)

To proceed we will need the following result which is animmediate consequence of Lemma 7

Corollary 11 Under Assumptions (A1)ndash(A4) the process119909119905

= ln119865(119905 119879) admits the following representation under thedomestic forward martingale measure P

119879

119909119879

= 119909119905

+ int119879

119905

119865

(119906 119879) sdot 119889119879

119906minus12

int119879

119905

1003817100381710038171003817119865

(119906 119879)10038171003817100381710038172

119889119906

+ 119869119876

(119905 119879) minus 120582119876

120583119876

(119879 minus 119905)

(52)

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 9

or more explicitly

119909119879

= 119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

minus12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906

+

119873119876

119879

sum

119896=119873119876

119905 +1

ln (1+ 119869119876

119896) minus 120582

119876120583119876

(119879 minus 119905)

(53)

Using equality (50) and Corollary 11 we obtain

1198911 (120601) = EP119905

exp (119894120601119909119879

) exp [int119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906]

= EP119905

exp [119894120601 (119909119905

+ int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601

2int

119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [int

119879

119905

radicV119906

119889119882119876

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906]

sdot exp [minus12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906] exp [119894120601119869

119876(119905 119879) minus 119894120601120582

119876120583119876

(119879 minus 119905)]

(54)

For the sake of conciseness we denote 120572 = 1 + 119894120601 120573 = 119894120601and 119888

119905= exp(119894120601119909

119905) After simplifications and rearrangement

the formula above becomes

1198911 (120601) = 119888119905EP119905

exp [120572 (int119879

119905

radicV119906

119889119882119876

119906minus12

int119879

119905

V119906119889119906)]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

120583119876

(119879 minus 119905)]

(55)

In view of Assumptions (A1)ndash(A6) we may use thefollowing representation for the Brownian motion 119882119876

119882119876

119905= 120588119882

V119905

+ radic1 minus 1205882119882119905 (56)

where 119882 = (119882119905)119905isin[0119879] is a Brownian motion under P

independent of the Brownian motions 119882V 119882

119889 and 119882119891

Consequently the conditional characteristic function 1198911(120601)

can be represented in the following way

1198911 (120601) = 119888119905EP119905

exp [120572120588 int119879

119905

radicV119906

119889119882V119906

+ 120572radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus120572

2int

119879

119905

V119906119889119906]

sdot exp [120573 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120572 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

sdot exp [120573119869119876

(119905 119879) minus 120573120582119876

(119879 minus 119905) 120583119876

]

(57)

By combining Proposition 1 with Definition 6 we obtainthe following auxiliary result which will be helpful in theproof of Theorem 3

Lemma 12 Given the dynamics (1) of processes V 119903 and 119903 andformula (32) we obtain the following equalities

int119879

119905

radicV119906

119889119882V119906

=1120590V

(V119879

minus V119905

minus 120579120591 + 120581 int119879

119905

V119906119889119906 minus (119885

V119879

minus 119885V119905))

int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906

= minus 119899119889

(119905 119879) 119903119905

minus int119879

119905

119886119889119899119889

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906

= minus 119899119891

(119905 119879) 119903119905

minus int119879

119905

119886119891

119899119891

(119906 119879) 119889119906 + int119879

119905

119903119906119889119906

(58)

Proof The first asserted formula is an immediate conse-quence of (1) For the second we recall that the function119899119889(119905 119879) is known to satisfy the following differential equation

for any fixed 119879 gt 0

120597119899119889

(119905 119879)

120597119905minus12

12059021198891198992119889

(119905 119879) minus 119887119889119899119889

(119905 119879) + 1 = 0 (59)

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Operations ResearchAdvances in

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Discrete Dynamics in Nature and Society

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Decision SciencesAdvances in

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 International Journal of Stochastic Analysis

with the terminal condition 119899119889(119879 119879) = 0Therefore using the

Ito formula and equality (32) we obtain

119889 (119899119889

(119905 119879) 119903119905) = 119903

119905119889119899

119889(119905 119879) + 119899

119889(119905 119879) 119889119903

119905

= 119903119905

(12

12059021198891198992119889

(119905 119879) + 119887119889119899119889

(119905 119879) minus 1) 119889119905

+ 119899119889

(119905 119879) (119886119889

minus 119887119889119903119905) 119889119905 + 119899

119889(119905 119879) 120590

119889radic119903119905

119889119882119889

119905

=12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119889

119905

= minus12

12059021198891198992119889

(119905 119879) 119903119905119889119905 minus 119903

119905119889119905 + 119899

119889(119905 119879) 119886

119889119889119905

+ 119899119889

(119905 119879) 120590119889radic119903

119905119889119882

119879

119905

(60)

This yields the second asserted formula upon integrationbetween 119905 and 119879 The derivation of the last one is based onthe same arguments and thus it is omitted

42 Proof of Theorem 3 We split the proof ofTheorem 3 intotwo steps in which we deal with 1198911(120601) and 1198912(120601)

Step 1 We will first compute 1198911(120601) By combining (57) withthe equalities derived in Lemma 12 we obtain the followingrepresentation for 1198911(120601)

1198911 (120601) = 119888119905EP119905

exp [minus120572120588

120590V(V

119905+ 120579120591)

+ (120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906 + 120572radic1 minus 1205882 int

119879

119905

radicV119906

119889119882119906

+120572120588

120590VV119879

]

sdot exp [minus120573 (119899119889

(119905 119879) 119903119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)

+ 120573 int119879

119905

119903119906119889119906]

sdot exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)

minus 120572 int119879

119905

119903119906119889119906] exp [120573119869

119876(119905 119879) minus 120573120582

119876120583119876

(119879 minus 119905)

minus120572120588

120590V(119885

V119879

minus 119885V119905)]

(61)

Recall the well-known property that if 120577 has the standardnormal distribution then E(119890119911120577) = 119890119911

22 for any complex

number 119911 isin C

Consequently by conditioning first on the sample pathof the process (V 119903 119903) and using the independence of theprocesses (V 119903 119903) and 119882 under P and Lemma 4 we obtain

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp[120572120588

120590VV119879

+ (1205722 (1 minus 1205882)

2+

120572120588120581

120590Vminus

120572

2) int

119879

119905

V119906119889119906]

sdot exp [120573 int119879

119905

119903119906119889119906 minus 120572 int

119879

119905

119903119906119889119906]

(62)

where we denote 120574 = 1 minus 119894120601 This in turn implies that thefollowing equality holds

1198911 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 (120590V (119890minus(120588(1+119894120601)120590V)119886 minus 119890minus(120588(1+119894120601)120590V)119887)

120588 (1 + 119894120601) (119887 minus 119886)+ 1)

+120572120588

120590V(V

119905+ 120579120591))] exp [minus120573 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120572 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199041V119879

minus 1199042 int119879

119905

V119906119889119906 minus 1199043119903

119879minus 1199044 int

119879

119905

119903119906119889119906 minus 1199045119903

119879

minus 1199046 int119879

119905

119903119906119889119906]

(63)

where the constants 1199041 1199042 1199043 1199044 1199045 1199046 are given by (16) Adirect application of Lemma 5 furnishes an explicit formulafor 1198911(120601) as reported in the statement of Theorem 3

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 11

Step 2 In order to compute the conditional characteristicfunction

1198912 (120601) = 1198912 (120601 119905 119876119905 V

119905 119903

119905 119903

119905) = E

P119879119905

[exp (119894120601119909119879

)] (64)

we proceed in an analogous manner as for 1198911(120601) We firstrecall that (see (51))

1198912 (120601) = EP119905

exp (119894120601119909119879

)

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906

minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906]

(65)

Therefore using Corollary 11 we obtain

1198912 (120601) = 119888119905EP119905

exp [119894120601 (int119879

119905

radicV119906

119889119882119876

119906+ int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119879

119906minus int

119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus119894120601 (12

int119879

119905

(V119906

+ 12059021198891198992119889

(119906 119879) 119903119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)]

sdot exp [minus int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906minus12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906] exp [119894120601119869

119876(119905 119879)]

(66)

Consequently using formulae (32) and (56) and Lemma 4 weobtain the following expression for 1198912(120601)

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp [120573 (120588 int119879

119905

radicV119906

119889119882V119906

+ radic1 minus 1205882 int119879

119905

radicV119906

119889119882119906

minus int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906)]

sdot exp [minus120573 (12

int119879

119905

(V119906

+ 1205902119891

1198992119891

(119906 119879) 119903119906) 119889119906)] exp [minus120574 (int

119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

(67)

Similarly as in the case of 1198911(120601) we condition on thesample path of the process (V 119903 119903) and we use the postulatedindependence of the processes (V 119903 119903) and 119882 under P Byinvoking also Lemma 4 we obtain

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1) minus 120573120582119876

120583119876

120591]

sdotEP119905

exp[120573120588 int119879

119905

radicV119906

119889119882V119906

+1205732 (1 minus 1205882) minus 120573

2int

119879

119905

V119906119889119906]

sdot exp [minus120574 (int119879

119905

120590119889119899119889

(119906 119879) radic119903119906

119889119882119889

119906+12

int119879

119905

12059021198891198992119889

(119906 119879) 119903119906119889119906)]

sdot exp [minus120573 (int119879

119905

120590119891

119899119891

(119906 119879) radic119903119906

119889119882119891

119906

+12

int119879

119905

1205902119891

1198992119891

(119906 119879) 119903119906119889119906)]

(68)

Using Lemma 12 we conclude that

1198912 (120601) = 119888119905exp [120582

119876120591 ((1+ 120583

119876)120573

119890minus(12)1205731205741205902

119876 minus 1)]

sdot exp[minus (120573120582119876

120583119876

120591

+ 120582V120591 [120590V (119890(120588(119894120601)120590V)119886 minus 119890minus(120588(i120601)120590V)119887)

120588 (119894120601) (119887 minus 119886)+ 1]

+120573120588

120590V(V

119905+ 120579120591))] exp [120574 (119899

119889(119905 119879) 119903

119905

+ int119879

119905

119886119889119899119889

(119906 119879) 119889119906)] exp [120573 (119899119891

(119905 119879) 119903119905

+ int119879

119905

119886119891

119899119891

(119906 119879) 119889119906)]EP119905

exp [minus1199021V119879

minus 1199022 int119879

119905

V119906119889119906 minus 1199023119903

119879minus 1199024 int

119879

119905

119903119906119889119906 minus 1199025119903

119879

minus 1199026 int119879

119905

119903119906119889119906]

(69)

with the coefficients 1199021 1199022 1199023 1199024 1199025 1199026 reported in formula(17) Another straightforward application of Lemma 5 yieldsthe closed-form expression (14) for the conditional character-istic function 1198912(120601)

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 International Journal of Stochastic Analysis

To complete the proof ofTheorem 3 it suffices to combineformula (44)with the standard inversion formula (12) provid-ing integral representations for the conditional probabilities

1198751 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

1198752 (119905 119876119905 V

119905 119903

119905 119903

119905 119870) = P

119879(119909

119879gt ln119870 | 119876

119905 V

119905 119903

119905 119903

119905)

(70)

This ends the derivation of the pricing formula for theforeign exchange call option The price of the correspondingput option is readily available aswell due to the put-call parityrelationship for FX options (see formula (72) in Section 5)

5 Numerical Results

The goal of the final section is to illustrate our approachby means of numerical examples in which we apply our FXmarketmodel that is theHestonCIR jump-diffusionmodeland we compare this approach with other related models thatwere proposed inMoretto et al [30] andAhlip andRutkowski[1] to deal with the exchange rate derivatives

Let us start by noting that the foreign exchange marketdiffers from equity markets in that quotes for options are notmade in terms of strikes Indeed the FX option prices arequoted in terms of the associated implied volatilities for afixed forward delta Δ

119865and a fixed time to expiry 120591 = 119879 minus 119905

For more information about the market conventions theinterested reader is referred to for instanceMoretto et al [30]or Reiswich and Uwe [31]

For a quoted volatility 120590 the corresponding strike price119870 is obtained using the following conversion formula whichis based on the classic Garman-Kohlhagen lognormal modelfor the exchange rate

119870 = 119865 (119905 119879) exp(minus120575120590radic120591119873minus1

(120575Δ119865

) +12

1205902120591) (71)

where 119873minus1 is the inverse of the standard normal cumulativedistribution function and the auxiliary parameter 120575 satisfies120575 = 1 (120575 = minus1 resp) for the call (put resp) optionFormula (71) makes it clear that market quotations pricesbased on the implied volatility for fixed deltas are in factequivalent to quoting prices for fixed strikes Formore detailsthe interested reader is referred to Hakala andWystup [32] orReiswich and Uwe [31]

Another relevant feature is that currency derivativesare based on the notion of at-the-money forward (ATMF)rate that is the forward exchange rate 119865(119905 119879) obtained byexploiting the interest rate parity implicit in (8) Recall thatthe universal put-call parity formula for plain-vanilla foreignexchange options reads

119862119905

(119879 119870) minus 119875119905

(119879 119870) = 119876119905119861119891

(119905 119879) minus 119870119861119889

(119905 119879) (72)

where 119862119905(119879 119870) and 119875

119905(119879 119870) are prices of currency call and

put options respectively In particular the prices of ATMFcall and put options are equal in any arbitrage-free marketmodel

Table 1 Market volatility 120590MKT for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Delta minus10 minus15 minus25 ATMF(50) 25 15 10

1M 1036 1009 973 930 915 918 9252M 1028 1001 965 925 915 922 9313M 1022 995 962 925 919 928 9396M 1023 995 964 935 939 955 9749M 1022 996 996 940 949 968 9881 Y 1024 998 969 945 956 977 9992Y 1028 1002 974 955 972 998 1024

Table 2 Market strike prices for USDEUR derivative exchangerate on June 13 2005 (original source of data Banca Caboto SpAGruppo Intesa Milano)

Strike minus10 minus15 minus25 ATMF(50) 25 15 10

1M 11651 11745 11877 12101 12317 12435 125192M 11496 11626 11807 12116 12421 12591 127123M 11370 11529 11752 12134 12518 12735 128916M 11129 11350 11660 12189 12753 13081 133249M 10968 11233 11609 12246 12951 13369 136801 Y 10843 11147 11579 12307 13140 13638 140132 Y 10561 10984 11596 12562 13826 14606 15205

Table 3Market domestic (USD) and foreign (EUR) interest rates onJune 13 2005 (original source of data Banca Caboto SpA GruppoIntesa Milano)

Rates 119903119889

119903119891

1M 314 2092M 322 2093M 332 2106M 350 2099M 360 2091 Y 368 2092Y 402 219

51 Market Data In the numerical results presented inTables 1 2 and 3 we make use (with the kind permissionof the authors) of the data for the USDEUR exchange ratederivatives and interest rates from the paper by Moretto et al[30] (see page 469 therein)

52 Comparison of Model Prices The dynamics of theexchange rate and volatility as given by (1) involve theparameters 120582

119876 120583

119876 120590

119876 120582V 120581 120579 and 120590V In addition there

are three parameters for each of the interest rates In ournumerical examples the values of parameters 120581 120579 and 120590V areborrowed from Moretto et al [30] who proposed an exten-sion of the Heston model for the exchange rate under the

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 13

Table 4 Values of parameters of the HCIRLNLU model (1)

119886 119887 120579 120581 120582119876

120583119876

120582V 120588 120590119876

01 02 002606 0091 01000 000258 01000 09786 00644

Table 5 Prices of ATM USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00128496 00128912 001295122M 00190734 00192465 002330373M 00245511 00249469 003221936M 00390888 00407209 005855039M 00526127 00556335 0081688812M 00656178 00699566 0102082

Table 6 Prices of 25 USDEUR European exchange rate calloptions using data of June 13 2005

Maturity Heston price HCIR price HCIR-LN-LU price1M 00005469 00054274 000620942M 00088177 00089194 001401533M 00116882 00119536 002246286M 00205735 00216388 004570369M 00297421 00323061 0066326012M 00389762 00437765 00849682

assumption of constant interest rates as represented by themarket yield curve It should be acknowledged that the choiceof interest rate parameters in our model is rather artificialand it was made for illustrative purposes only We used thefollowing values of parameters for the HestonCIR (HCIR)model and the HestonCIRLog NormalLog Uniform Jump-Diffusion (HCIR-LN-LU) model 119886

119889= 00332 119887

119889= 003

119886119891

= 0021 119887119891

= 0024 120590119889

= 025 120590119891

= 024 1198760 = 12087and 120588 = 09786 For each maturity date the initial valueV0 = 00078The parameters given in Table 4 were taken fromDrsquoIppoliti et al [10] and were used for illustrative purposesonly The Heston model the HCIR model examined in Ahlipand Rutkowski [1] and the HCIR-LN-LUmodel put forwardin this paper were compared Although the numerical resultspresented here are only preliminary they neverthelessmake itclear that jumps in exchange rate and volatility dynamics andthe uncertain character of interest rates affect the valuation offoreign exchange derivatives

In Table 5 we report prices of ATM calls for expiriesranging from one month to one year We use here theATM volatilities for different maturities given in Table 1the corresponding ATM strike prices from Table 2 andthe interest rates from Table 3 As one can see the pricesobtained using our model (HCIR-LN-LU) are higher thanthe prices for HCIR model and substantially higher than theprices obtained for the Heston model In Table 6 we report25 USDEUR currency call option prices computed in theHeston model (refer to Figures 1 and 4) the HCIR modeland the present model using data of June 13 2005 In the nextthree examples (see Tables 7 8 and 9) we consider prices for

2 4 6 8 10 12Months

002

004

006

008

010

Opt

ion

valu

e

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Figure 1 Graphs for ATM options prices given in Table 5

Table 7 Prices for ATM USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00128496 00128546 00128697(121019) (121028) (121028)

2M 00190734 00190999 00222887(121184) (121217) (121217)

3M 0024422 002449323 00319434(121369) (121428) (121428)

6M 00386608 00390158 00573285(121992) (122289) (122289)

9M 00518264 00527189 00790228(122652) (123329) (123329)

12M 00644786 00681417 00988909(123356) (124071) (124071)

Table 8 Prices for 25 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00054139 00054181 00058999(123193) (123201) (123201)

2M 00086633 00086839 00138257(124274) (124308) (124308)

3M 00116882 00117434 00232394(125188) (125267) (125267)

6M 00204432 00207368 00480355(127581) (127892) (127892)

9M 00293928 00301153 00706473(129652) (130367) (130367)

12M 00385139 00399506 00909937(131587) (132884) (132884)

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

14 International Journal of Stochastic Analysis

Table 9 Prices for 15 USDEUR European exchange rate calloptions using data of June 13 2005 Values in brackets are strikelevels obtained using formula (71)

Maturity Heston price HCIR price HCIR-LN-LU price

1M 00031477 00031503 00032142(124388) (124397) (124397)

2M 00005302 00053178 00120455(126005) (127429) (127429)

3M 00074126 00074553 00208664(127349) (127429) (127429)

6M 00138051 00140366 00433107(130848) (1311672 (131167)

9M 00207329 00213553 00616101(133813) (134551) (134551)

1 Y 00280719 00293209 00437222(136544) (133788) (133788)

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 2 Graphs for 25 options prices given in Table 6

ATM and 25 and 15 volatilities (refer to Figures 3 2 and5) respectively For each maturity the corresponding strikelevel was obtained using (71) Prices for ATM 25 and 15USDEUR currency call options are computed in the HestonHCIR and HCIR-LN-LUmodels using data of June 13 2005and parameter values given in Table 4

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are grateful to Enrico Moretto for consentingto use data reported in [30] and Uwe Wystup who kindlyadvised them in regards to numerical examples presentedin Section 5 They also thank Marek Rutkowski Scott Joslin

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

002

004

006

008

010

Opt

ion

valu

e

Figure 3 Graphs for options prices given in Table 7

002

004

006

008

2 4 6 8 10 12Months

Heston-25HCIR-25HCIR-LN-LU-25

Opt

ion

valu

e

Figure 4 Graphs for options prices given in Table 8

002

004

006

2 4 6 8 10 12Months

Heston ATMHCIR ATMHCIR-LN-LU-ATM

Opt

ion

valu

e

Figure 5 Graph for option 15 prices given in Table 9

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

International Journal of Stochastic Analysis 15

Paul Glasserman and participants at the IMS-FPS-2014Workshop July 2ndash6 2014 UTS Sydney for valuable com-ments on an earlier version of the paper All remaining errorsare theirs

References

[1] R Ahlip and M Rutkowski ldquoPricing of foreign exchangeoptions under the Heston stochastic volatility model and CIRinterest ratesrdquo Quantitative Finance vol 13 no 6 pp 955ndash9662013

[2] J C Cox J E Ingersoll and S A Ross ldquoA theory of the termstructure of interest ratesrdquo Econometrica vol 53 no 2 pp 385ndash407 1985

[3] S L Heston ldquoA closed-form solution for options with stochasticvolatility with applications to bond and currency optionsrdquoReview of Financial Studies vol 6 no 2 pp 327ndash343 1993

[4] G Bakshi C Cao and Z Chen ldquoEmpirical performance ofalternative option pricing modelsrdquo Journal of Finance vol 52no 5 pp 2003ndash2049 1997

[5] D S Bates ldquoJumps and stochastic volatility exchange rate pro-cesses implicit in Deutsche Mark optionsrdquo Review of FinancialStudies vol 9 no 1 pp 69ndash107 1996

[6] DDuffie J Pan andK Singleton ldquoTransformanalysis and assetpricing for affine jump-diffusionsrdquo Econometrica vol 68 no 6pp 1343ndash1376 2000

[7] T Andersen and J Andreasen ldquoJump-diffusion processesvolatility smile fitting and numerical methodsrdquo The Journal ofFinancial Economics vol 4 pp 231ndash262 2000

[8] S AlizadehMW Brandt and F XDiebold ldquoRange-based esti-mation of stochastic volatility modelsrdquo The Journal of Financevol 57 no 3 pp 1047ndash1091 2002

[9] B Eraker M Johannes and N Polson ldquoThe impacts of jumpsin volatility and returnsrdquo Journal of Finance vol 58 no 3 pp1269ndash1300 2003

[10] F DrsquoIppoliti E Moretto S Pasquali and B Trivellato ldquoExactpricing with stochastic volatility and jumpsrdquo InternationalJournal ofTheoretical andApplied Finance vol 13 no 6 pp 901ndash929 2010

[11] G Yan and F BHanson ldquoOption pricing for stochastic volatilityjump diffusion model with log-uniform jump-amplitudesrdquoin Proceedings of the American Control Conference pp 1ndash7September 2006

[12] A van Haastrecht R Lord A Pelsser and D SchragerldquoPricing long-dated insurance contracts with stochastic interestrates and stochastic volatilityrdquo Insurance Mathematics andEconomics vol 45 no 3 pp 436ndash448 2009

[13] R Schobel and J Zhu ldquoStochastic volatility with an ornstein-uhlenbeck process an extensionrdquo Review of Finance vol 3 no1 pp 23ndash46 1999

[14] O Vasicek ldquoAn equilibrium characterization of the term struc-turerdquo Journal of Financial Economics vol 5 no 2 pp 177ndash1881977

[15] A van Haastrecht and A Pelsser ldquoGeneric pricing of FXinflation and stock options under stochastic interest rates andstochastic volatilityrdquoQuantitative Finance vol 11 no 5 pp 665ndash691 2011

[16] L A Grzelak and C W Oosterlee ldquoOn the Heston model withstochastic interest ratesrdquo SIAM Journal on Financial Mathemat-ics vol 2 no 1 pp 255ndash286 2011

[17] L A Grzelak and C W Oosterlee ldquoOn cross-currency modelswith stochastic volatility and correlated interest ratesrdquo AppliedMathematical Finance vol 19 no 1 pp 1ndash35 2012

[18] L A Grzelak CWOosterlee and S vanWeeren ldquoExtension ofstochastic volatility equity models with the Hull-White interestrate processrdquo Quantitative Finance vol 12 no 1 pp 89ndash1052012

[19] B Wong and C C Heyde ldquoOn the martingale property ofstochastic exponentialsrdquo Journal of Applied Probability vol 41no 3 pp 654ndash664 2004

[20] MMusiela andM RutkowskiMartingale Methods in FinancialModelling vol 36 of Stochastic Modelling and Applied Probabil-ity Springer New York NY USA 2nd edition 2005

[21] P Carr and D Madan ldquoOption valuation using the fast fouriertransformrdquo Journal of Computational Finance vol 2 pp 61ndash731999

[22] P Carr and D Madan ldquoSaddlepoint methods for optionpricingrdquo Journal of Computational Finance vol 13 no 1 pp 49ndash61 2009

[23] R Lord and C Kahl ldquoOptimal fourier inversion in semi-analytical option pricingrdquo Journal of Computational Financevol 10 pp 1ndash30 2007

[24] R Lord and C Kahl ldquoComplex logarithms in Heston-likemodelsrdquoMathematical Finance vol 20 no 4 pp 671ndash694 2010

[25] C Bernard Z Cui and D McLeish ldquoNearly exact option pricesimulation using characteristic functionsrdquo International Journalof Theoretical and Applied Finance vol 15 no 7 Article ID1250047 29 pages 2012

[26] S Levendorskii ldquoEfficient pricing and reliable calibration in theHeston modelrdquo International Journal of Theoretical and AppliedFinance vol 15 Article ID 1250050 44 pages 2012

[27] M Jeanblanc M Yor and M Chesney Mathematical methodsfor financial markets Springer Finance Springer London UK2009

[28] R Ahlip and M Rutkowski ldquoForward start options understochastic volatility and stochastic interest ratesrdquo InternationalJournal of Theoretical and Applied Finance vol 12 no 2 pp209ndash225 2009

[29] D Brigo and A Alfonsi ldquoCredit default swap calibration andderivatives pricing with the SSRD stochastic intensity modelrdquoFinance and Stochastics vol 9 no 1 pp 29ndash42 2005

[30] E Moretto S Pasquali and B Trivellato ldquoDerivative evaluationusing recombining trees under stochastic volatilityrdquo Advancesand Applications in Statistical Sciences vol 1 no 2 pp 453ndash4802010

[31] D Reiswich and W Uwe ldquoFX volatility smile constructionrdquoWilmott vol 2012 no 60 pp 58ndash69 2012

[32] J Hakala and U Wystup Foreign Exchange Risk ModelsInstruments and Strategies Risk Books London UK 2002

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

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OptimizationJournal of

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Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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Algebra

Discrete Dynamics in Nature and Society

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Discrete MathematicsJournal of

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Stochastic AnalysisInternational Journal of