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Research ArticleAsian Option Pricing with Transaction Costs and Dividendsunder the Fractional Brownian Motion Model
Yan Zhang Di Pan Sheng-Wu Zhou and Miao Han
College of Sciences China University of Mining and Technology Jiangsu Xuzhou 221116 China
Correspondence should be addressed to Sheng-Wu Zhou zswcumt163com
Received 25 December 2013 Accepted 12 February 2014 Published 26 March 2014
Academic Editor Nazim I Mahmudov
Copyright copy 2014 Yan Zhang et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
The pricing problem of geometric average Asian option under fractional Brownian motion is studied in this paper The partialdifferential equation satisfied by the optionrsquos value is presented on the basis of no-arbitrage principle and fractional formulaThen bysolving the partial differential equation the pricing formula and call-put parity of the geometric average Asian optionwith dividendpayment and transaction costs are obtained At last the influences of Hurst index and maturity on option value are discussed bynumerical examples
1 Introduction
Option pricing theory has been an unprecedented develop-ment since the classic Black-Scholes option pricing model [1]was proposed Asian options are a kind of common strongpath-dependent options whose value depends on the averageprice of the underlying asset during the life of the optionFusai and Meucci [2] have discretely studied Asian optionpricing problem under the Levy process Vecer [3] has gotthe unified algorithm of the Asian option value based on thebasic theory of stochastic analysis Vecer and Xu [4] extendedthe method of removing path correlation to the case in asemimartingale model and obtained the partial differentialequation of the option value under the standard Brownianmotion
However the empirical analysis shows that there is a long-term correlation between the underlying asset prices so thatthe geometric Brownian motion is not considered as an idealtool to describe the process of asset price Since fractionalBrownian motion has the properties of self-similarity thicktail and long-term correlation that fractional Brownianmotion has become a good tool to depict the process ofunderlying asset price According to the standard Brownianmotion Mandelbrot and Van Ness [5] obtained a stochasticintegral form of fractional Brownian motion Based on thewick product Duncan et al [6] introduced fractional Itointegral Elliott and van der Hoek and others [7] have studied
fractional Brownian motion with Hurst parameter belongingto the interval (12 1) and they obtained fractional Girsanovtheorem and fractional Ito formula by using wick productBecause an Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameter[8] was obtained by Christian Bender it brought greatconvenience to option pricing
In the reality of the securities market investors werefaced with considerable and nonignorable transaction costsand Leland [8] firstly examined the problems of optionpricing and hedging with transaction costs Due to infinitevariation of geometric Brownian motion transaction costswould become infinite in the continuous time completelyhedging strategy So Leland suggested that no-arbitrageassumption is replaced by Delta hedging strategy underthe condition of discrete time occasions and transactioncosts The model was then extended by Hoggard et al andothers [9] Guasoni [10] studied the standard option withtransaction costs under the fractional Brownian motion buthe did not obtain option pricing formula Then Liu andChang and others [11] extended the option pricing withtransaction costs under fractional Brownian motion andprovide an approximate solution of the nonlinear Hoggard-Whalley-Wilmott equation Wang et al and others [12ndash16]have systematically discussed the European option pricingproblems with transaction costs and long-range dependenceBut these studies are usually aimed at European standard
Hindawi Publishing CorporationJournal of Applied MathematicsVolume 2014 Article ID 652954 8 pageshttpdxdoiorg1011552014652954
2 Journal of Applied Mathematics
options To the authorsrsquo knowledge there does not existsystematic research about Asian options under time-varyingfractional Brownian motion
In this paper Asian option pricing problemswith transac-tion costs and dividends under fractional Brownian motionare studied Firstly the partial differential equation satisfiedby geometric average Asian option value is obtained on thebasis of no-arbitrage principle Then the analytic expressionsof option value and parity formula are presented by solvingthe partial differential equation At last the influences ofHurst exponent and maturity on option value are discussedby numerical examples
2 Geometric Average Asian Options PricingModel under Fractional Brownian Motion
Definition 1 (see [17]) Let (Ω 119865 119875) be a complete probabilityspace on which a standard fractional Brownian motion withHurst exponent 119867 (0 lt 119867 lt 1) is continuous centeredGaussian processes 119861
119867(119905) 119905 ge 0 with covariance functions
Cov(119861
119867(119905) 119861
119867(119904)) = (12)(|119905|
2119867+ |119904|
2119867minus |119905 minus 119904|
2119867) 119904 119905 gt 0
In 2003 Bender has obtained an Ito formula for gen-eralized functionals of a fractional Brownian motion witharbitrary Hurst parameter [18] The following lemma isobtained by using the integral Ito formula
Lemma 2 Suppose that stochastic process 119878
119905satisfied the
following equation
119889119878
119905= 120583
119905119878
119905119889119905 + 120590
119905119878
119905119889119861
119867 (119905) (1)
where 120583
119905and 120590
119905are respectively drift coefficient and diffusion
coefficient Suppose that stochastic process 119891 = 119891(119905 119869
119905 119878
119905)
then for any 119905 isin [0 119879] one has
119889119891 = (
120597119891
120597119905
+ 120583
119905119878
119905
120597119891
120597119878
119905
+
120597119891
120597119869
119905
119889119869
119905
119889119905
+ 119867(120590
119905119878
119905)
2119905
2119867minus1 1205972119891
120597119878
2
119905
) 119889119905
+ 120590
119905119878
119905
120597119891
120597119878
119905
119889119861
119867 (119905)
(2)
where 119869
119905= 119890
(1119905) int119905
0ln 119878120591119889120591 is geometric average of 119878
119905between the
time period of [0 119905]
In this paper the following basic assumptions wereneeded
(i) Underlying asset price 119878
119905 satisfied the stochastic
differential equations
119889119878
119905= (120583
119905minus 119902
119905) 119878
119905119889119905 + 120590119878
119905119889119861
119867 (119905) (3)
where 120583
119905is the expected return 119902
119905denotes dividend
yield120590 is volatility and119861
119867(119905) is a fractional Brownian
motion(ii) Risk-free interest rate 119903
119905is a certain function of time
119905
(iii) Transaction costs are proportional to the value ofthe transaction in the underlying Let 119896 denote thetransaction cost per unit dollar of transaction where119896 is a constant To buy or sell ]
119905shares of the
underlying asset need pay proportional transactioncosts (119896|]
119905|119878
119905) note that ]
119905gt 0 denotes buying the
underlying asset and ]119905
lt 0 denotes selling(iv) The expected return of the hedge portfolio equals the
risk-free rate 119903
119905
Let 119881 = 119881(119905 119869
119905 119878
119905) denote the value of the geometric
average Asian call at time 119905 where 119869
119905= 119890
(1119905) int119905
0ln 119878120591119889120591 is
geometric average of underlying asset in [0 119905] Construct aportfolioΠ long one position of the geometric average Asiancall and sell Δ shares of the underlying asset Then the valueof the portfolio at time 119905 is
Π
119905= 119881
119905minus Δ
119905119878
119905 (4)
After the time interval 120575119905 the change in the value of theportfolio Π is as follows
120575Π
119905= 120575119881
119905minus Δ
119905120575119878
119905minus Δ
119905119902
119905119878
119905120575119905 minus 119896
1003816
1003816
1003816
1003816
]119905
1003816
1003816
1003816
1003816
119878
119905+120575119905
= (
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
minus Δ
119905119902
119905119878
119905) 120575119905
+ (
120597119881
120597119878
119905
minus Δ
119905) 120575119878
119905+
120597119881
120597119869
119905
120575119869
119905minus 119896
1003816
1003816
1003816
1003816
]119905
1003816
1003816
1003816
1003816
119878
119905+120575119905
(5)
where 120575119878
119905denotes the change in the underlying asset price
and ]119905
= Δ
119905+120575119905minusΔ
119905is the change of the underlying asset share
in [119905 119905 + 120575119905] Choose Δ
119905= 120597119881120597119878
119905 then (5) becomes
120575Π
119905= (
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
minus
120597119881
120597119878
119905
119902
119905119878
119905) 120575119905
+
120597119881
120597119869
119905
120575119869
119905minus 119896
1003816
1003816
1003816
1003816
]119905
1003816
1003816
1003816
1003816
119878
119905+120575119905
(6)
where
]119905
= Δ
119905+120575119905minus Δ
119905=
120597119881
120597119878
119905+120575119905
minus
120597119881
120597119878
119905
=
120597
2119881
120597119878
2
119905
120575119878
119905+
120597
2119881
120597119878
119905120597119869
119905
120575119869
119905+ 119874 (120575119905)
=
120597
2119881
120597119878
2
119905
120590119878
119905120575119861
119867 (119905) + 119874 (120575119905)
(7)
Themathematical expectation of transaction costs is obtainedin the following form
119864 (119896
1003816
1003816
1003816
1003816
]119905
1003816
1003816
1003816
1003816
119878
119905+120575119905) = 119896
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120590119878
119905119864 (
1003816
1003816
1003816
1003816
120575119861
119867 (119905)
1003816
1003816
1003816
1003816
119878
119905+120575119889) + 119874 (120575119905)
=radic
2
120587
119896120590119878
2
119905(120575119905)
119867
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
+ 119874 (120575119905)
(8)
Journal of Applied Mathematics 3
By assumption (iv) the following relation holds
119864 (120575Π
119905) = 119903
119905Π
119905120575119905 (9)
By (6) and (8) one has
119864 (120575Π
119905) = (
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
minus
120597119881
120597119878
119905
119902
119905119878
119905) 120575119905
+
120597119881
120597119869
119905
120575119869
119905minus
radic
2
120587
119896120590119878
2
119905(120575119905)
119867
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
+ 119874 (120575119905)
(10)
Substituting (10) into (9) the following partial differentialequation is obtained
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119889119869
119905
119889119905
minusradic
2
120587
119896120590119878
2
119905(120575119905)
119867minus1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus 119903
119905119881 = 0
(11)
Substituting 119869
119905= 119890
(1119905) int119905
0ln 119878120591119889120591 and 119889119869
119905119889119905 = 119869
119905ln(119878
119905119869
119905)119905 into
(11) the following equation is obtained
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minusradic
2
120587
119896120590119878
2
119905(120575119905)
119867minus1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus 119903
119905119881 = 0
(12)
denoting
2= 2120590
2(119867119905
2119867minus1minus
radic
2
120587
119896
120590
(120575119905)
119867minus1 sign (119881
119878119878)) (13)
Substituting (13) into (12) the following result is obtained
Theorem3 Suppose that the underlying asset price 119878
119905satisfied
(3) then the value of the geometric average Asian call at time119905 (0 le 119905 le 119879) 119881(119905 119869
119905 119878
119905) satisfies the following mathematical
model
120597119881
120597119905
+
1
2
2119878
2
119905
120597
2119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minus 119903
119905119881 = 0
(14)
Remark 4 Theorem 3 is obtained for the long position of theoption If the short position of option is considered similarlywe can also get the mathematical model (14) and only thecorresponding modified volatility is given by the followingform
2= 2120590
2(119867119905
2119867minus1+
radic
2
120587
119896
120590
(120575119905)
119867minus1 sign (119881
119878119878)) (15)
Let Le(119867) =radic
2120587(119896120590)(120575119905)
119867minus1 which is called fractionalLeland number [8]
Remark 5 For the long position of a single European Asianoption its final payoff is (119869
119879minus 119870)
+ or (119870 minus 119869
119879)
+ and theyare both convex function so 119881
119869119869gt 0 and noticing 119869
119905=
119890
(1119905) int119905
0ln 119878120591119889120591 thus 119881
119878119878gt 0 However for the short position
of a single European Asian option its final payoff at maturityis minus(119869
119879minus 119870)
+ or minus(119870 minus 119869
119879)
+ and they are both concavefunction so that 119881
119869119869lt 0 119881
119878119878lt 0 So for a single European
Asian option (13) and (15) can be represented as
2= 2120590
2(119867119905
2119867minus1minus
radic
2
120587
119896
120590
(120575119905)
119867minus1) (16)
3 Option Pricing Formula
Theorem6 Suppose that the underlying asset price 119878
119905satisfied
(3) then the value 119881(119905 119869
119905 119878
119905) of the geometric average Asian
call with strike price 119870 maturity 119879 and transaction fee rate 119896
at time 119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(17)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + 119903
lowast(119879 minus 119905) + 120590
lowast2(119879
2119867minus 119905
2119867)
120590
lowastradic
119879
2119867minus 119905
2119867
119889
2= 119889
1minus 120590
lowastradic
119879
2119867minus 119905
2119867
119903
lowast=
int
119879
119905(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
Le (119867) 120590
2119879 minus 119905
119879
120590
lowast= 120590 times (1 minus
4119867 (119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879
2119867minus 119905
2119867)
+
119867 (119879
2119867+2minus 119905
2119867+2)
119879
2(119867 + 1) (119879
2119867minus 119905
2119867)
minus2 Le (119867)
(119879 minus 119905)
3
3119879
2(119879
2119867minus 119905
2119867)
)
12
119873 (119909) = int
119909
minusinfin
1
radic2120587
119890
minus11990522
119889119905
(18)
4 Journal of Applied Mathematics
Proof By Theorem 3 the value 119881(119905 119869
119905 119878
119905) of the geometric
average Asian call satisfies the following model
120597119881
120597119905
+
1
2
2119878
2
119905
120597
2119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minus 119903
119905119881 = 0
119881 (119879 119869
119879 119878
119879) = (119869
119879minus 119870)
+
(19)
Let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] 119881(119905 119869
119905 119878
119905) = 119880(119905 120585
119905)
then
120597119881
120597119905
=
ln (119869
119905119878
119905)
119879
120597119880
120597120585
119905
+
120597119880
120597119905
120597119881
120597119878
119905
=
119879 minus 119905
119879119878
119905
120597119880
120597120585
119905
120597
2119881
120597119878
2
119905
= (
119879 minus 119905
119879119878
119905
)
2120597
2119880
120597120585
2
119905
minus
119879 minus 119905
119879119878
2
119905
120597119880
120597120585
119905
120597119881
120597119869
119905
=
119905
119879119869
119905
120597119880
120597120585
119905
(20)
Combined with the boundary conditions of the call option119881(119879 119869
119879 119878
119879) = (119869
119879minus 119870)
+ the model (19) can be converted to
120597119880
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119880
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119880
120597120585
2
119905
minus 119903
119905119880 = 0
119880 (120585
119879 119879) = (119890
120585119879
minus 119870)
+
(21)
Let 120591 = 120574(119905) 120578
120591= 120585
119905+ 120572(119905) 119882(120591 120578
120591) = 119880(119905 120585
119905)119890
120573(119905) whichsatisfied the conditions
120572 (119879) = 120573 (119879) = 120574 (119879) = 0 (22)
and then we have
120597119880
120597119905
= 119890
minus120573(119905)(
120597119882
120597120591
120574
1015840(119905) minus 120573
1015840(119905) 119882 +
120597119882
120597120578
120591
120572
1015840(119905))
120597119880
120597120585
119905
= 119890
minus120573(119905) 120597119882
120597120578
120591
120597
2119880
120597120585
2
119905
= 119890
minus120573(119905) 1205972119882
120597120578
2
120591
(23)
Substituting (23) into (21) we can get
120574
1015840(119905)
120597119882
120597120591
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120578
2
120591
+ [(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
+ 120572
1015840(119905)]
120597119882
120597120578
120591
minus (119903
119905+ 120573
1015840(119905)) 119882 = 0
(24)
Set
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
+ 120572
1015840(119905) = 0 119903
119905+ 120573
1015840(119905) = 0
120574
1015840(119905) = minus
2
2
(
119879 minus 119905
119879
)
2
(25)
Combining with the terminal conditions 120572(119879) = 120573(119879) =
120574(119879) = 0 we have
120572 (119905) = int
119879
119905
(119903
120579minus 119902
120579)
119879 minus 120579
119879
119889120579 minus int
119879
119905
1
2
2(
119879 minus 120579
119879
) 119889120579
= 119903 lowast (119879 minus 119905)
120573 (119905) = int
119879
119905
119903
120579119889120579
120574 (119905) = int
119879
119905
1
2
2(
119879 minus 120579
119879
)
2
119889120579 =
120590
lowast2
2
(119879
2119867minus 119905
2119867)
(26)
where
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
Le (119867) 120590
2119879 minus 119905
119879
120590
lowast= 120590 times (1 minus
4119867 (119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879
2119867minus 119905
2119867)
+
119867 (119879
2119867+2minus 119905
2119867+2)
119879
2(119867 + 1) (119879
2119867minus 119905
2119867)
minus2Le (119867)
(119879 minus 119905)
3
3119879
2(119879
2119867minus 119905
2119867)
)
12
(27)
Thus themodel (21) is converted into the classic heat conduc-tion equation
120597119882
120597120591
=
120597
2119882
120597120578
2
120591
119882 (120578
0 0) = (119890
1205780
minus 119870)
+
(28)
Its solution is
119882 (120578
120591 120591) =
1
2radic120587120591
int
+infin
minusinfin
(119890
119910minus 119870)
+119890
minus(119910minus120578120591)24120591
119889119910
=
1
2radic120587120591
int
+infin
ln119870(119890
119910minus 119870) 119890
minus(119910minus120578120591)24120591
119889119910
= 119890
120578120591+120591
119873 (
2120591 + 120578
120591minus ln119870
radic2120591
) minus 119870119873 (
120578
120591minus ln119870
radic2120591
)
(29)
After variable reduction we have
119882 (120578
120591 120591) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119873 (119889
2)
(30)
Journal of Applied Mathematics 5
where
2120591 + 120578
120591minus ln119870
radic2120591
= (ln[
[
(119869
119905
119905119878
119879minus119905
119905)
1119879
119870
]
]
+ 119903
lowast(119879 minus 119905)
+ 120590
lowast2(119879
2119867minus 119905
2119867) )
times (120590
lowast(119879
2119867minus 119905
2119867)
12
)
minus1
≜ 119889
1
120578
120591minus ln119870
radic2120591
=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + 119903
lowast(119879 minus 119905)
120590
lowastradic
119879
2119867minus 119905
2119867
= 119889
1minus 120590
lowastradic
119879
2119867minus 119905
2119867≜ 119889
2
(31)
So the value of geometric average Asian call option at time 119905
is obtained
119881 (119905 119869
119905 119878
119905) = 119880 (120585
119905 119905) = 119882 (120578
120591 120591) 119890
minus120573(119905)
= (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(32)
Theorem 7 Suppose the underlying asset price 119878
119905satisfies
(3) then the relationship between 119881
119862(119905 119869
119905 119878
119905) the value of
geometric average Asian call option and 119881
119875(119905 119869
119905 119878
119905) the value
of put option with strike price 119870 maturity 119879 and transactionfee rate 119896 at time 119905 is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(33)
where 119903
lowast 120590
lowast are the same as above
Proof Let
119882 (119905 119869
119905 119878
119905) = 119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905) (34)
Then119882 is suitable for the following terminal question in 0 le
119878 lt infin 0 le 119869 lt infin 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119882
120597119878
119905
+
2
2
119878
2
119905
120597
2119882
120597119878
2
119905
+
120597119882
120597119869
119905
119869
119905
ln 119878
119905minus ln 119869
119905
119905
minus 119903
119905119882 = 0
119882|119905=119879= (119869
119879minus 119870)
+minus (119870 minus 119869
119879)
+= 119869
119879minus 119870
(35)
We let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] then 119882 is suitable for
the following in 120585
119905isin 119877 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119882
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120585
2
119905
minus 119903119882 = 0
119882|119905=119879= 119890
120585119879
minus 119870
(36)
Set the form solution of problem of (36) is
119882 (119905 119869
119905 119878
119905) = 119886 (119905) 119890
120585119905+ 119887 (119905)
(37)
Substituting (37) into (36) and comparing coefficients onehas
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
119886 (119905) +
2
2
(
119879 minus 119905
119879
)
2
119886 (119905)
+ 119886
1015840(119905) minus 119903
119905119886 (119905) = 0
119887
1015840(119905) minus 119903
119905119887 (119905) = 0
(38)
Taking 119886(119879) = 1 119887(119879) = minus119870 the solutions of (38) are
119886 (119905) = 119890
120572(119905)minusint119879
119905119903120591119889120591+120574(119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120591119889120591+(120590
lowast22)(1198792119867minus1199052119867)
119887 (119905) = minus 119870119890
minusint119879
119905119903120579119889120579
(39)
Thus
119882 = 119886 (119905) 119890
120585119905+ 119887 (119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879119878
(119879minus119905)119879
minus 119870119890
minusint119879
119905119903120579119889120579
(40)
The parity formula between call option and put option is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(41)
Pricing formula of the geometric average Asian putoptions can be obtained byTheorems 6 and 7
Theorem 8 Suppose 119878
119905satisfies (3) then the value 119881
119875(119905 119869
119905
119878
119905) of the geometric average Asian put option with strike price
119870 maturity 119879 and transaction fee rate 119896 at time 119905 is
119881
119875(119905 119869
119905 119878
119905) = minus(119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minusint119879
119905119903120579119889120579
119873 (minus119889
2)
(42)
6 Journal of Applied Mathematics
In particular if 119867 = 12 119896 = 0 119903 119902 120590 are all constant andprice formula (17) is reduced to the following formula [19]
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22minus119903)(119879minus119905)
119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
(43)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119903
lowast= (119903 minus 119902 minus
120590
2
2
)
119879 minus 119905
2119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
(44)
The following results are consequences of the above theorems
Corollary 9 If risk-free interest rate 119903 dividend yield 119902 andvolatility120590 are all constant then the price formulas of geometricaverage Asian call and put option with strike price 119870 maturity119879 and transaction fee rate 119896 under fractional Brownianmotionat time 119905 are respectively
119881
119862(119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
119881
119875(119905 119869
119905 119878
119905) = minus (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minus119903(119879minus119905)119873 (minus119889
2)
(45)
where
119903
lowast=
(119903 minus 119902)
2119879
(119879 minus 119905) minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
119871119890 (119867) 120590
2119879 minus 119905
119879
(46)
The rest of symbols are the same as Theorem 6
Noticing that if 119867 = 12 119861
119867(119905) is standard Brownian
motion119861(119905) the corresponding underlying asset price followsgeometric Brownian motion one has the following results
Corollary 10 If risk-free interest rate 119903
119905and dividend yield
119902
119905are the functions of time 119905 and volatility 120590 is constant
then the price formula 119881(119905 119869
119905 119878
119905) with respect to geometric
average Asian call option with strike price 119870 maturity 119879 andtransaction fee rate 119896 under standard Brownianmotion at time119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22)(119879minus119905)minusint
119879
119905119903120579119889120579
119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(47)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119871119890 (
1
2
) =radic
2
120587
119896
120590
(120575119905)
minus12
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879 minus 119905)
4119879
+
1
2
119871119890 (
1
2
) 120590
2119879 minus 119905
119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
radic1 minus 2119871119890 (
1
2
)
(48)
4 Numerical Example
Wediscuss the impact ofHurst index and transaction rates onthe Asian option value by numerical examples Assume thatthe parameter selection is as follows
119878
119905= 80 119905 = 0 119879 = 1
119903 = 005 119902 = 001 120590 = 04
119870 = 80 119896 = 0003 120575119905 = 002
(49)
We calculate the value of the option by using the priceformula of (45) The relationships between the value of calloption or put option and the underlying asset price withdifferentHurst index are given in Figures 1 and 2 respectivelyFrom Figures 1 and 2 the relationship between Hurst indexand Asian option value is negative Furthermore the impacton the call option value decreases with the increase of theunderlying asset price but the impact on the put option valuedecreases with the decrease of the underlying asset price
By Figures 3 and 4 we can get the change trend ofthe value of Asian call and put options with the change ofmaturity and Hurst index at the same time The option valueincreases with the maturity increases but the value of the calloption increases faster than the value of a put option increase
5 Conclusions
Asian options are popular financial derivatives that play anessential role in financial market Pricing them efficiently andaccurately is very important both in theory and practice Wehave investigated geometric average Asian option valuationproblems with transaction costs under the fractional Brow-nian motion By no-arbitrage principle and fractional Itorsquosformula this paper has deduced PDE satisfied by the optionrsquosvalue Meanwhile the pricing formula and call-put parity ofthe geometric average Asian option with transaction costsare derived by solving PDE At last the influences of Hurstexponent andmaturity on option value are discussed throughnumerical examples
Journal of Applied Mathematics 7
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
35
40
45
S
H =
H =
H =
H =
H =
50 60 70 80 90 100 110 120S
H =
H =
H =
H =
H =
VC
Figure 1 The values of the call option with different 119867
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
S
H =
H =
H =
H =
H =
VP
Figure 2 The values of the put option with different 119867
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous referees forvaluable suggestions for the improvement of this paper Allremaining errors are the responsibility of the authors Thisresearch was supported by ldquothe Fundamental Research Fundsfor the Central Universitiesrdquo (2010 LKSX03)
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity
VC
Figure 3 Relationship amongHurst indexmaturity and call optionvalue
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity0 2
0406
08
051
15
t index
Time to matu
VP
Figure 4 Relationship amongHurst indexmaturity and put optionvalue
References
[1] F Black andM S Scholes ldquoThe pricing of option and corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973
[2] G Fusai and A Meucci ldquoPricing discretely monitored Asianoptions under Levy processesrdquo Journal of Banking and Financevol 32 no 10 pp 2076ndash2088 2008
[3] J Vecer ldquoUnified pricing of Asian optionsrdquo Risk vol 6 no 15pp 113ndash116 2002
[4] J Vecer and M Xu ldquoPricing Asian options in a semimartingalemodelrdquo Quantitative Finance vol 4 no 2 pp 170ndash175 2004
[5] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 no 4 pp 422ndash437 1968
[6] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000
[7] R J Elliott and J van derHoek ldquoA general fractional white noisetheory and applications to financerdquoMathematical Finance vol13 no 2 pp 301ndash330 2003
8 Journal of Applied Mathematics
[8] H E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 pp 1283ndash1301 1985
[9] T Hoggard A E Whalley and P Wilmott ldquoHedging optionportfolios in the presence of transaction costsrdquo AdvancedFutures and Options Research vol 7 pp 21ndash35 1994
[10] P Guasoni ldquoNo arbitrage under transaction costs with frac-tional Brownian motion and beyondrdquo Mathematical Financevol 16 no 3 pp 569ndash582 2006
[11] H-K Liu and J-J Chang ldquoA closed-form approximation forthe fractional Black-Scholes model with transaction costsrdquoComputers amp Mathematics with Applications vol 65 no 11 pp1719ndash1726 2013
[12] X-T Wang ldquoScaling and long-range dependence in optionpricing I pricing European option with transaction costs underthe fractional Black-Scholes modelrdquo Physica A vol 389 no 3pp 438ndash444 2010
[13] X-T Wang E-H Zhu M-M Tang and H-G Yan ldquoScalingand long-range dependence in option pricing II pricing Euro-pean option with transaction costs under the mixed Brownian-fractional Brownian modelrdquo Physica A vol 389 no 3 pp 445ndash451 2010
[14] X-T Wang H-G Yan M-M Tang and E-H Zhu ldquoScalingand long-range dependence in option pricing III a fractionalversion of the Merton model with transaction costsrdquo Physica Avol 389 no 3 pp 452ndash458 2010
[15] X-T Wang ldquoScaling and long range dependence in optionpricing IV pricing European options with transaction costsunder the multifractional Black-Scholes modelrdquo Physica A vol389 no 4 pp 789ndash796 2010
[16] X-T Wang M Wu Z-M Zhou and W-S Jing ldquoPricingEuropean option with transaction costs under the fractionallong memory stochastic volatility modelrdquo Physica A vol 391no 4 pp 1469ndash1480 2012
[17] Y Hu and B Oslashksendal ldquoFractional white noise calculus andapplications to financerdquo Infinite Dimensional Analysis Quan-tum Probability and Related Topics vol 6 no 1 pp 1ndash32 2003
[18] C Bender ldquoAn Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameterrdquoStochastic Processes and Their Applications vol 104 no 1 pp81ndash106 2003
[19] J Li-Shang Mathematical Modeling and Methods of OptionPricing Higher Education Press Beijing China 2nd edition2008
Submit your manuscripts athttpwwwhindawicom
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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Differential EquationsInternational Journal of
Volume 2014
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OptimizationJournal of
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International 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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Algebra
Discrete Dynamics in Nature and Society
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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
2 Journal of Applied Mathematics
options To the authorsrsquo knowledge there does not existsystematic research about Asian options under time-varyingfractional Brownian motion
In this paper Asian option pricing problemswith transac-tion costs and dividends under fractional Brownian motionare studied Firstly the partial differential equation satisfiedby geometric average Asian option value is obtained on thebasis of no-arbitrage principle Then the analytic expressionsof option value and parity formula are presented by solvingthe partial differential equation At last the influences ofHurst exponent and maturity on option value are discussedby numerical examples
2 Geometric Average Asian Options PricingModel under Fractional Brownian Motion
Definition 1 (see [17]) Let (Ω 119865 119875) be a complete probabilityspace on which a standard fractional Brownian motion withHurst exponent 119867 (0 lt 119867 lt 1) is continuous centeredGaussian processes 119861
119867(119905) 119905 ge 0 with covariance functions
Cov(119861
119867(119905) 119861
119867(119904)) = (12)(|119905|
2119867+ |119904|
2119867minus |119905 minus 119904|
2119867) 119904 119905 gt 0
In 2003 Bender has obtained an Ito formula for gen-eralized functionals of a fractional Brownian motion witharbitrary Hurst parameter [18] The following lemma isobtained by using the integral Ito formula
Lemma 2 Suppose that stochastic process 119878
119905satisfied the
following equation
119889119878
119905= 120583
119905119878
119905119889119905 + 120590
119905119878
119905119889119861
119867 (119905) (1)
where 120583
119905and 120590
119905are respectively drift coefficient and diffusion
coefficient Suppose that stochastic process 119891 = 119891(119905 119869
119905 119878
119905)
then for any 119905 isin [0 119879] one has
119889119891 = (
120597119891
120597119905
+ 120583
119905119878
119905
120597119891
120597119878
119905
+
120597119891
120597119869
119905
119889119869
119905
119889119905
+ 119867(120590
119905119878
119905)
2119905
2119867minus1 1205972119891
120597119878
2
119905
) 119889119905
+ 120590
119905119878
119905
120597119891
120597119878
119905
119889119861
119867 (119905)
(2)
where 119869
119905= 119890
(1119905) int119905
0ln 119878120591119889120591 is geometric average of 119878
119905between the
time period of [0 119905]
In this paper the following basic assumptions wereneeded
(i) Underlying asset price 119878
119905 satisfied the stochastic
differential equations
119889119878
119905= (120583
119905minus 119902
119905) 119878
119905119889119905 + 120590119878
119905119889119861
119867 (119905) (3)
where 120583
119905is the expected return 119902
119905denotes dividend
yield120590 is volatility and119861
119867(119905) is a fractional Brownian
motion(ii) Risk-free interest rate 119903
119905is a certain function of time
119905
(iii) Transaction costs are proportional to the value ofthe transaction in the underlying Let 119896 denote thetransaction cost per unit dollar of transaction where119896 is a constant To buy or sell ]
119905shares of the
underlying asset need pay proportional transactioncosts (119896|]
119905|119878
119905) note that ]
119905gt 0 denotes buying the
underlying asset and ]119905
lt 0 denotes selling(iv) The expected return of the hedge portfolio equals the
risk-free rate 119903
119905
Let 119881 = 119881(119905 119869
119905 119878
119905) denote the value of the geometric
average Asian call at time 119905 where 119869
119905= 119890
(1119905) int119905
0ln 119878120591119889120591 is
geometric average of underlying asset in [0 119905] Construct aportfolioΠ long one position of the geometric average Asiancall and sell Δ shares of the underlying asset Then the valueof the portfolio at time 119905 is
Π
119905= 119881
119905minus Δ
119905119878
119905 (4)
After the time interval 120575119905 the change in the value of theportfolio Π is as follows
120575Π
119905= 120575119881
119905minus Δ
119905120575119878
119905minus Δ
119905119902
119905119878
119905120575119905 minus 119896
1003816
1003816
1003816
1003816
]119905
1003816
1003816
1003816
1003816
119878
119905+120575119905
= (
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
minus Δ
119905119902
119905119878
119905) 120575119905
+ (
120597119881
120597119878
119905
minus Δ
119905) 120575119878
119905+
120597119881
120597119869
119905
120575119869
119905minus 119896
1003816
1003816
1003816
1003816
]119905
1003816
1003816
1003816
1003816
119878
119905+120575119905
(5)
where 120575119878
119905denotes the change in the underlying asset price
and ]119905
= Δ
119905+120575119905minusΔ
119905is the change of the underlying asset share
in [119905 119905 + 120575119905] Choose Δ
119905= 120597119881120597119878
119905 then (5) becomes
120575Π
119905= (
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
minus
120597119881
120597119878
119905
119902
119905119878
119905) 120575119905
+
120597119881
120597119869
119905
120575119869
119905minus 119896
1003816
1003816
1003816
1003816
]119905
1003816
1003816
1003816
1003816
119878
119905+120575119905
(6)
where
]119905
= Δ
119905+120575119905minus Δ
119905=
120597119881
120597119878
119905+120575119905
minus
120597119881
120597119878
119905
=
120597
2119881
120597119878
2
119905
120575119878
119905+
120597
2119881
120597119878
119905120597119869
119905
120575119869
119905+ 119874 (120575119905)
=
120597
2119881
120597119878
2
119905
120590119878
119905120575119861
119867 (119905) + 119874 (120575119905)
(7)
Themathematical expectation of transaction costs is obtainedin the following form
119864 (119896
1003816
1003816
1003816
1003816
]119905
1003816
1003816
1003816
1003816
119878
119905+120575119905) = 119896
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120590119878
119905119864 (
1003816
1003816
1003816
1003816
120575119861
119867 (119905)
1003816
1003816
1003816
1003816
119878
119905+120575119889) + 119874 (120575119905)
=radic
2
120587
119896120590119878
2
119905(120575119905)
119867
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
+ 119874 (120575119905)
(8)
Journal of Applied Mathematics 3
By assumption (iv) the following relation holds
119864 (120575Π
119905) = 119903
119905Π
119905120575119905 (9)
By (6) and (8) one has
119864 (120575Π
119905) = (
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
minus
120597119881
120597119878
119905
119902
119905119878
119905) 120575119905
+
120597119881
120597119869
119905
120575119869
119905minus
radic
2
120587
119896120590119878
2
119905(120575119905)
119867
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
+ 119874 (120575119905)
(10)
Substituting (10) into (9) the following partial differentialequation is obtained
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119889119869
119905
119889119905
minusradic
2
120587
119896120590119878
2
119905(120575119905)
119867minus1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus 119903
119905119881 = 0
(11)
Substituting 119869
119905= 119890
(1119905) int119905
0ln 119878120591119889120591 and 119889119869
119905119889119905 = 119869
119905ln(119878
119905119869
119905)119905 into
(11) the following equation is obtained
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minusradic
2
120587
119896120590119878
2
119905(120575119905)
119867minus1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus 119903
119905119881 = 0
(12)
denoting
2= 2120590
2(119867119905
2119867minus1minus
radic
2
120587
119896
120590
(120575119905)
119867minus1 sign (119881
119878119878)) (13)
Substituting (13) into (12) the following result is obtained
Theorem3 Suppose that the underlying asset price 119878
119905satisfied
(3) then the value of the geometric average Asian call at time119905 (0 le 119905 le 119879) 119881(119905 119869
119905 119878
119905) satisfies the following mathematical
model
120597119881
120597119905
+
1
2
2119878
2
119905
120597
2119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minus 119903
119905119881 = 0
(14)
Remark 4 Theorem 3 is obtained for the long position of theoption If the short position of option is considered similarlywe can also get the mathematical model (14) and only thecorresponding modified volatility is given by the followingform
2= 2120590
2(119867119905
2119867minus1+
radic
2
120587
119896
120590
(120575119905)
119867minus1 sign (119881
119878119878)) (15)
Let Le(119867) =radic
2120587(119896120590)(120575119905)
119867minus1 which is called fractionalLeland number [8]
Remark 5 For the long position of a single European Asianoption its final payoff is (119869
119879minus 119870)
+ or (119870 minus 119869
119879)
+ and theyare both convex function so 119881
119869119869gt 0 and noticing 119869
119905=
119890
(1119905) int119905
0ln 119878120591119889120591 thus 119881
119878119878gt 0 However for the short position
of a single European Asian option its final payoff at maturityis minus(119869
119879minus 119870)
+ or minus(119870 minus 119869
119879)
+ and they are both concavefunction so that 119881
119869119869lt 0 119881
119878119878lt 0 So for a single European
Asian option (13) and (15) can be represented as
2= 2120590
2(119867119905
2119867minus1minus
radic
2
120587
119896
120590
(120575119905)
119867minus1) (16)
3 Option Pricing Formula
Theorem6 Suppose that the underlying asset price 119878
119905satisfied
(3) then the value 119881(119905 119869
119905 119878
119905) of the geometric average Asian
call with strike price 119870 maturity 119879 and transaction fee rate 119896
at time 119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(17)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + 119903
lowast(119879 minus 119905) + 120590
lowast2(119879
2119867minus 119905
2119867)
120590
lowastradic
119879
2119867minus 119905
2119867
119889
2= 119889
1minus 120590
lowastradic
119879
2119867minus 119905
2119867
119903
lowast=
int
119879
119905(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
Le (119867) 120590
2119879 minus 119905
119879
120590
lowast= 120590 times (1 minus
4119867 (119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879
2119867minus 119905
2119867)
+
119867 (119879
2119867+2minus 119905
2119867+2)
119879
2(119867 + 1) (119879
2119867minus 119905
2119867)
minus2 Le (119867)
(119879 minus 119905)
3
3119879
2(119879
2119867minus 119905
2119867)
)
12
119873 (119909) = int
119909
minusinfin
1
radic2120587
119890
minus11990522
119889119905
(18)
4 Journal of Applied Mathematics
Proof By Theorem 3 the value 119881(119905 119869
119905 119878
119905) of the geometric
average Asian call satisfies the following model
120597119881
120597119905
+
1
2
2119878
2
119905
120597
2119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minus 119903
119905119881 = 0
119881 (119879 119869
119879 119878
119879) = (119869
119879minus 119870)
+
(19)
Let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] 119881(119905 119869
119905 119878
119905) = 119880(119905 120585
119905)
then
120597119881
120597119905
=
ln (119869
119905119878
119905)
119879
120597119880
120597120585
119905
+
120597119880
120597119905
120597119881
120597119878
119905
=
119879 minus 119905
119879119878
119905
120597119880
120597120585
119905
120597
2119881
120597119878
2
119905
= (
119879 minus 119905
119879119878
119905
)
2120597
2119880
120597120585
2
119905
minus
119879 minus 119905
119879119878
2
119905
120597119880
120597120585
119905
120597119881
120597119869
119905
=
119905
119879119869
119905
120597119880
120597120585
119905
(20)
Combined with the boundary conditions of the call option119881(119879 119869
119879 119878
119879) = (119869
119879minus 119870)
+ the model (19) can be converted to
120597119880
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119880
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119880
120597120585
2
119905
minus 119903
119905119880 = 0
119880 (120585
119879 119879) = (119890
120585119879
minus 119870)
+
(21)
Let 120591 = 120574(119905) 120578
120591= 120585
119905+ 120572(119905) 119882(120591 120578
120591) = 119880(119905 120585
119905)119890
120573(119905) whichsatisfied the conditions
120572 (119879) = 120573 (119879) = 120574 (119879) = 0 (22)
and then we have
120597119880
120597119905
= 119890
minus120573(119905)(
120597119882
120597120591
120574
1015840(119905) minus 120573
1015840(119905) 119882 +
120597119882
120597120578
120591
120572
1015840(119905))
120597119880
120597120585
119905
= 119890
minus120573(119905) 120597119882
120597120578
120591
120597
2119880
120597120585
2
119905
= 119890
minus120573(119905) 1205972119882
120597120578
2
120591
(23)
Substituting (23) into (21) we can get
120574
1015840(119905)
120597119882
120597120591
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120578
2
120591
+ [(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
+ 120572
1015840(119905)]
120597119882
120597120578
120591
minus (119903
119905+ 120573
1015840(119905)) 119882 = 0
(24)
Set
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
+ 120572
1015840(119905) = 0 119903
119905+ 120573
1015840(119905) = 0
120574
1015840(119905) = minus
2
2
(
119879 minus 119905
119879
)
2
(25)
Combining with the terminal conditions 120572(119879) = 120573(119879) =
120574(119879) = 0 we have
120572 (119905) = int
119879
119905
(119903
120579minus 119902
120579)
119879 minus 120579
119879
119889120579 minus int
119879
119905
1
2
2(
119879 minus 120579
119879
) 119889120579
= 119903 lowast (119879 minus 119905)
120573 (119905) = int
119879
119905
119903
120579119889120579
120574 (119905) = int
119879
119905
1
2
2(
119879 minus 120579
119879
)
2
119889120579 =
120590
lowast2
2
(119879
2119867minus 119905
2119867)
(26)
where
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
Le (119867) 120590
2119879 minus 119905
119879
120590
lowast= 120590 times (1 minus
4119867 (119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879
2119867minus 119905
2119867)
+
119867 (119879
2119867+2minus 119905
2119867+2)
119879
2(119867 + 1) (119879
2119867minus 119905
2119867)
minus2Le (119867)
(119879 minus 119905)
3
3119879
2(119879
2119867minus 119905
2119867)
)
12
(27)
Thus themodel (21) is converted into the classic heat conduc-tion equation
120597119882
120597120591
=
120597
2119882
120597120578
2
120591
119882 (120578
0 0) = (119890
1205780
minus 119870)
+
(28)
Its solution is
119882 (120578
120591 120591) =
1
2radic120587120591
int
+infin
minusinfin
(119890
119910minus 119870)
+119890
minus(119910minus120578120591)24120591
119889119910
=
1
2radic120587120591
int
+infin
ln119870(119890
119910minus 119870) 119890
minus(119910minus120578120591)24120591
119889119910
= 119890
120578120591+120591
119873 (
2120591 + 120578
120591minus ln119870
radic2120591
) minus 119870119873 (
120578
120591minus ln119870
radic2120591
)
(29)
After variable reduction we have
119882 (120578
120591 120591) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119873 (119889
2)
(30)
Journal of Applied Mathematics 5
where
2120591 + 120578
120591minus ln119870
radic2120591
= (ln[
[
(119869
119905
119905119878
119879minus119905
119905)
1119879
119870
]
]
+ 119903
lowast(119879 minus 119905)
+ 120590
lowast2(119879
2119867minus 119905
2119867) )
times (120590
lowast(119879
2119867minus 119905
2119867)
12
)
minus1
≜ 119889
1
120578
120591minus ln119870
radic2120591
=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + 119903
lowast(119879 minus 119905)
120590
lowastradic
119879
2119867minus 119905
2119867
= 119889
1minus 120590
lowastradic
119879
2119867minus 119905
2119867≜ 119889
2
(31)
So the value of geometric average Asian call option at time 119905
is obtained
119881 (119905 119869
119905 119878
119905) = 119880 (120585
119905 119905) = 119882 (120578
120591 120591) 119890
minus120573(119905)
= (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(32)
Theorem 7 Suppose the underlying asset price 119878
119905satisfies
(3) then the relationship between 119881
119862(119905 119869
119905 119878
119905) the value of
geometric average Asian call option and 119881
119875(119905 119869
119905 119878
119905) the value
of put option with strike price 119870 maturity 119879 and transactionfee rate 119896 at time 119905 is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(33)
where 119903
lowast 120590
lowast are the same as above
Proof Let
119882 (119905 119869
119905 119878
119905) = 119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905) (34)
Then119882 is suitable for the following terminal question in 0 le
119878 lt infin 0 le 119869 lt infin 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119882
120597119878
119905
+
2
2
119878
2
119905
120597
2119882
120597119878
2
119905
+
120597119882
120597119869
119905
119869
119905
ln 119878
119905minus ln 119869
119905
119905
minus 119903
119905119882 = 0
119882|119905=119879= (119869
119879minus 119870)
+minus (119870 minus 119869
119879)
+= 119869
119879minus 119870
(35)
We let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] then 119882 is suitable for
the following in 120585
119905isin 119877 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119882
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120585
2
119905
minus 119903119882 = 0
119882|119905=119879= 119890
120585119879
minus 119870
(36)
Set the form solution of problem of (36) is
119882 (119905 119869
119905 119878
119905) = 119886 (119905) 119890
120585119905+ 119887 (119905)
(37)
Substituting (37) into (36) and comparing coefficients onehas
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
119886 (119905) +
2
2
(
119879 minus 119905
119879
)
2
119886 (119905)
+ 119886
1015840(119905) minus 119903
119905119886 (119905) = 0
119887
1015840(119905) minus 119903
119905119887 (119905) = 0
(38)
Taking 119886(119879) = 1 119887(119879) = minus119870 the solutions of (38) are
119886 (119905) = 119890
120572(119905)minusint119879
119905119903120591119889120591+120574(119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120591119889120591+(120590
lowast22)(1198792119867minus1199052119867)
119887 (119905) = minus 119870119890
minusint119879
119905119903120579119889120579
(39)
Thus
119882 = 119886 (119905) 119890
120585119905+ 119887 (119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879119878
(119879minus119905)119879
minus 119870119890
minusint119879
119905119903120579119889120579
(40)
The parity formula between call option and put option is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(41)
Pricing formula of the geometric average Asian putoptions can be obtained byTheorems 6 and 7
Theorem 8 Suppose 119878
119905satisfies (3) then the value 119881
119875(119905 119869
119905
119878
119905) of the geometric average Asian put option with strike price
119870 maturity 119879 and transaction fee rate 119896 at time 119905 is
119881
119875(119905 119869
119905 119878
119905) = minus(119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minusint119879
119905119903120579119889120579
119873 (minus119889
2)
(42)
6 Journal of Applied Mathematics
In particular if 119867 = 12 119896 = 0 119903 119902 120590 are all constant andprice formula (17) is reduced to the following formula [19]
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22minus119903)(119879minus119905)
119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
(43)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119903
lowast= (119903 minus 119902 minus
120590
2
2
)
119879 minus 119905
2119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
(44)
The following results are consequences of the above theorems
Corollary 9 If risk-free interest rate 119903 dividend yield 119902 andvolatility120590 are all constant then the price formulas of geometricaverage Asian call and put option with strike price 119870 maturity119879 and transaction fee rate 119896 under fractional Brownianmotionat time 119905 are respectively
119881
119862(119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
119881
119875(119905 119869
119905 119878
119905) = minus (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minus119903(119879minus119905)119873 (minus119889
2)
(45)
where
119903
lowast=
(119903 minus 119902)
2119879
(119879 minus 119905) minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
119871119890 (119867) 120590
2119879 minus 119905
119879
(46)
The rest of symbols are the same as Theorem 6
Noticing that if 119867 = 12 119861
119867(119905) is standard Brownian
motion119861(119905) the corresponding underlying asset price followsgeometric Brownian motion one has the following results
Corollary 10 If risk-free interest rate 119903
119905and dividend yield
119902
119905are the functions of time 119905 and volatility 120590 is constant
then the price formula 119881(119905 119869
119905 119878
119905) with respect to geometric
average Asian call option with strike price 119870 maturity 119879 andtransaction fee rate 119896 under standard Brownianmotion at time119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22)(119879minus119905)minusint
119879
119905119903120579119889120579
119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(47)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119871119890 (
1
2
) =radic
2
120587
119896
120590
(120575119905)
minus12
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879 minus 119905)
4119879
+
1
2
119871119890 (
1
2
) 120590
2119879 minus 119905
119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
radic1 minus 2119871119890 (
1
2
)
(48)
4 Numerical Example
Wediscuss the impact ofHurst index and transaction rates onthe Asian option value by numerical examples Assume thatthe parameter selection is as follows
119878
119905= 80 119905 = 0 119879 = 1
119903 = 005 119902 = 001 120590 = 04
119870 = 80 119896 = 0003 120575119905 = 002
(49)
We calculate the value of the option by using the priceformula of (45) The relationships between the value of calloption or put option and the underlying asset price withdifferentHurst index are given in Figures 1 and 2 respectivelyFrom Figures 1 and 2 the relationship between Hurst indexand Asian option value is negative Furthermore the impacton the call option value decreases with the increase of theunderlying asset price but the impact on the put option valuedecreases with the decrease of the underlying asset price
By Figures 3 and 4 we can get the change trend ofthe value of Asian call and put options with the change ofmaturity and Hurst index at the same time The option valueincreases with the maturity increases but the value of the calloption increases faster than the value of a put option increase
5 Conclusions
Asian options are popular financial derivatives that play anessential role in financial market Pricing them efficiently andaccurately is very important both in theory and practice Wehave investigated geometric average Asian option valuationproblems with transaction costs under the fractional Brow-nian motion By no-arbitrage principle and fractional Itorsquosformula this paper has deduced PDE satisfied by the optionrsquosvalue Meanwhile the pricing formula and call-put parity ofthe geometric average Asian option with transaction costsare derived by solving PDE At last the influences of Hurstexponent andmaturity on option value are discussed throughnumerical examples
Journal of Applied Mathematics 7
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
35
40
45
S
H =
H =
H =
H =
H =
50 60 70 80 90 100 110 120S
H =
H =
H =
H =
H =
VC
Figure 1 The values of the call option with different 119867
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
S
H =
H =
H =
H =
H =
VP
Figure 2 The values of the put option with different 119867
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous referees forvaluable suggestions for the improvement of this paper Allremaining errors are the responsibility of the authors Thisresearch was supported by ldquothe Fundamental Research Fundsfor the Central Universitiesrdquo (2010 LKSX03)
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity
VC
Figure 3 Relationship amongHurst indexmaturity and call optionvalue
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity0 2
0406
08
051
15
t index
Time to matu
VP
Figure 4 Relationship amongHurst indexmaturity and put optionvalue
References
[1] F Black andM S Scholes ldquoThe pricing of option and corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973
[2] G Fusai and A Meucci ldquoPricing discretely monitored Asianoptions under Levy processesrdquo Journal of Banking and Financevol 32 no 10 pp 2076ndash2088 2008
[3] J Vecer ldquoUnified pricing of Asian optionsrdquo Risk vol 6 no 15pp 113ndash116 2002
[4] J Vecer and M Xu ldquoPricing Asian options in a semimartingalemodelrdquo Quantitative Finance vol 4 no 2 pp 170ndash175 2004
[5] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 no 4 pp 422ndash437 1968
[6] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000
[7] R J Elliott and J van derHoek ldquoA general fractional white noisetheory and applications to financerdquoMathematical Finance vol13 no 2 pp 301ndash330 2003
8 Journal of Applied Mathematics
[8] H E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 pp 1283ndash1301 1985
[9] T Hoggard A E Whalley and P Wilmott ldquoHedging optionportfolios in the presence of transaction costsrdquo AdvancedFutures and Options Research vol 7 pp 21ndash35 1994
[10] P Guasoni ldquoNo arbitrage under transaction costs with frac-tional Brownian motion and beyondrdquo Mathematical Financevol 16 no 3 pp 569ndash582 2006
[11] H-K Liu and J-J Chang ldquoA closed-form approximation forthe fractional Black-Scholes model with transaction costsrdquoComputers amp Mathematics with Applications vol 65 no 11 pp1719ndash1726 2013
[12] X-T Wang ldquoScaling and long-range dependence in optionpricing I pricing European option with transaction costs underthe fractional Black-Scholes modelrdquo Physica A vol 389 no 3pp 438ndash444 2010
[13] X-T Wang E-H Zhu M-M Tang and H-G Yan ldquoScalingand long-range dependence in option pricing II pricing Euro-pean option with transaction costs under the mixed Brownian-fractional Brownian modelrdquo Physica A vol 389 no 3 pp 445ndash451 2010
[14] X-T Wang H-G Yan M-M Tang and E-H Zhu ldquoScalingand long-range dependence in option pricing III a fractionalversion of the Merton model with transaction costsrdquo Physica Avol 389 no 3 pp 452ndash458 2010
[15] X-T Wang ldquoScaling and long range dependence in optionpricing IV pricing European options with transaction costsunder the multifractional Black-Scholes modelrdquo Physica A vol389 no 4 pp 789ndash796 2010
[16] X-T Wang M Wu Z-M Zhou and W-S Jing ldquoPricingEuropean option with transaction costs under the fractionallong memory stochastic volatility modelrdquo Physica A vol 391no 4 pp 1469ndash1480 2012
[17] Y Hu and B Oslashksendal ldquoFractional white noise calculus andapplications to financerdquo Infinite Dimensional Analysis Quan-tum Probability and Related Topics vol 6 no 1 pp 1ndash32 2003
[18] C Bender ldquoAn Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameterrdquoStochastic Processes and Their Applications vol 104 no 1 pp81ndash106 2003
[19] J Li-Shang Mathematical Modeling and Methods of OptionPricing Higher Education Press Beijing China 2nd edition2008
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
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OptimizationJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
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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
Journal of Applied Mathematics 3
By assumption (iv) the following relation holds
119864 (120575Π
119905) = 119903
119905Π
119905120575119905 (9)
By (6) and (8) one has
119864 (120575Π
119905) = (
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
minus
120597119881
120597119878
119905
119902
119905119878
119905) 120575119905
+
120597119881
120597119869
119905
120575119869
119905minus
radic
2
120587
119896120590119878
2
119905(120575119905)
119867
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
+ 119874 (120575119905)
(10)
Substituting (10) into (9) the following partial differentialequation is obtained
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119889119869
119905
119889119905
minusradic
2
120587
119896120590119878
2
119905(120575119905)
119867minus1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus 119903
119905119881 = 0
(11)
Substituting 119869
119905= 119890
(1119905) int119905
0ln 119878120591119889120591 and 119889119869
119905119889119905 = 119869
119905ln(119878
119905119869
119905)119905 into
(11) the following equation is obtained
120597119881
120597119905
+ 119867120590
2119878
2
119905119905
2119867minus1 1205972119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minusradic
2
120587
119896120590119878
2
119905(120575119905)
119867minus1
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
120597
2119881
120597119878
2
119905
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
1003816
minus 119903
119905119881 = 0
(12)
denoting
2= 2120590
2(119867119905
2119867minus1minus
radic
2
120587
119896
120590
(120575119905)
119867minus1 sign (119881
119878119878)) (13)
Substituting (13) into (12) the following result is obtained
Theorem3 Suppose that the underlying asset price 119878
119905satisfied
(3) then the value of the geometric average Asian call at time119905 (0 le 119905 le 119879) 119881(119905 119869
119905 119878
119905) satisfies the following mathematical
model
120597119881
120597119905
+
1
2
2119878
2
119905
120597
2119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minus 119903
119905119881 = 0
(14)
Remark 4 Theorem 3 is obtained for the long position of theoption If the short position of option is considered similarlywe can also get the mathematical model (14) and only thecorresponding modified volatility is given by the followingform
2= 2120590
2(119867119905
2119867minus1+
radic
2
120587
119896
120590
(120575119905)
119867minus1 sign (119881
119878119878)) (15)
Let Le(119867) =radic
2120587(119896120590)(120575119905)
119867minus1 which is called fractionalLeland number [8]
Remark 5 For the long position of a single European Asianoption its final payoff is (119869
119879minus 119870)
+ or (119870 minus 119869
119879)
+ and theyare both convex function so 119881
119869119869gt 0 and noticing 119869
119905=
119890
(1119905) int119905
0ln 119878120591119889120591 thus 119881
119878119878gt 0 However for the short position
of a single European Asian option its final payoff at maturityis minus(119869
119879minus 119870)
+ or minus(119870 minus 119869
119879)
+ and they are both concavefunction so that 119881
119869119869lt 0 119881
119878119878lt 0 So for a single European
Asian option (13) and (15) can be represented as
2= 2120590
2(119867119905
2119867minus1minus
radic
2
120587
119896
120590
(120575119905)
119867minus1) (16)
3 Option Pricing Formula
Theorem6 Suppose that the underlying asset price 119878
119905satisfied
(3) then the value 119881(119905 119869
119905 119878
119905) of the geometric average Asian
call with strike price 119870 maturity 119879 and transaction fee rate 119896
at time 119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(17)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + 119903
lowast(119879 minus 119905) + 120590
lowast2(119879
2119867minus 119905
2119867)
120590
lowastradic
119879
2119867minus 119905
2119867
119889
2= 119889
1minus 120590
lowastradic
119879
2119867minus 119905
2119867
119903
lowast=
int
119879
119905(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
Le (119867) 120590
2119879 minus 119905
119879
120590
lowast= 120590 times (1 minus
4119867 (119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879
2119867minus 119905
2119867)
+
119867 (119879
2119867+2minus 119905
2119867+2)
119879
2(119867 + 1) (119879
2119867minus 119905
2119867)
minus2 Le (119867)
(119879 minus 119905)
3
3119879
2(119879
2119867minus 119905
2119867)
)
12
119873 (119909) = int
119909
minusinfin
1
radic2120587
119890
minus11990522
119889119905
(18)
4 Journal of Applied Mathematics
Proof By Theorem 3 the value 119881(119905 119869
119905 119878
119905) of the geometric
average Asian call satisfies the following model
120597119881
120597119905
+
1
2
2119878
2
119905
120597
2119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minus 119903
119905119881 = 0
119881 (119879 119869
119879 119878
119879) = (119869
119879minus 119870)
+
(19)
Let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] 119881(119905 119869
119905 119878
119905) = 119880(119905 120585
119905)
then
120597119881
120597119905
=
ln (119869
119905119878
119905)
119879
120597119880
120597120585
119905
+
120597119880
120597119905
120597119881
120597119878
119905
=
119879 minus 119905
119879119878
119905
120597119880
120597120585
119905
120597
2119881
120597119878
2
119905
= (
119879 minus 119905
119879119878
119905
)
2120597
2119880
120597120585
2
119905
minus
119879 minus 119905
119879119878
2
119905
120597119880
120597120585
119905
120597119881
120597119869
119905
=
119905
119879119869
119905
120597119880
120597120585
119905
(20)
Combined with the boundary conditions of the call option119881(119879 119869
119879 119878
119879) = (119869
119879minus 119870)
+ the model (19) can be converted to
120597119880
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119880
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119880
120597120585
2
119905
minus 119903
119905119880 = 0
119880 (120585
119879 119879) = (119890
120585119879
minus 119870)
+
(21)
Let 120591 = 120574(119905) 120578
120591= 120585
119905+ 120572(119905) 119882(120591 120578
120591) = 119880(119905 120585
119905)119890
120573(119905) whichsatisfied the conditions
120572 (119879) = 120573 (119879) = 120574 (119879) = 0 (22)
and then we have
120597119880
120597119905
= 119890
minus120573(119905)(
120597119882
120597120591
120574
1015840(119905) minus 120573
1015840(119905) 119882 +
120597119882
120597120578
120591
120572
1015840(119905))
120597119880
120597120585
119905
= 119890
minus120573(119905) 120597119882
120597120578
120591
120597
2119880
120597120585
2
119905
= 119890
minus120573(119905) 1205972119882
120597120578
2
120591
(23)
Substituting (23) into (21) we can get
120574
1015840(119905)
120597119882
120597120591
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120578
2
120591
+ [(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
+ 120572
1015840(119905)]
120597119882
120597120578
120591
minus (119903
119905+ 120573
1015840(119905)) 119882 = 0
(24)
Set
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
+ 120572
1015840(119905) = 0 119903
119905+ 120573
1015840(119905) = 0
120574
1015840(119905) = minus
2
2
(
119879 minus 119905
119879
)
2
(25)
Combining with the terminal conditions 120572(119879) = 120573(119879) =
120574(119879) = 0 we have
120572 (119905) = int
119879
119905
(119903
120579minus 119902
120579)
119879 minus 120579
119879
119889120579 minus int
119879
119905
1
2
2(
119879 minus 120579
119879
) 119889120579
= 119903 lowast (119879 minus 119905)
120573 (119905) = int
119879
119905
119903
120579119889120579
120574 (119905) = int
119879
119905
1
2
2(
119879 minus 120579
119879
)
2
119889120579 =
120590
lowast2
2
(119879
2119867minus 119905
2119867)
(26)
where
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
Le (119867) 120590
2119879 minus 119905
119879
120590
lowast= 120590 times (1 minus
4119867 (119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879
2119867minus 119905
2119867)
+
119867 (119879
2119867+2minus 119905
2119867+2)
119879
2(119867 + 1) (119879
2119867minus 119905
2119867)
minus2Le (119867)
(119879 minus 119905)
3
3119879
2(119879
2119867minus 119905
2119867)
)
12
(27)
Thus themodel (21) is converted into the classic heat conduc-tion equation
120597119882
120597120591
=
120597
2119882
120597120578
2
120591
119882 (120578
0 0) = (119890
1205780
minus 119870)
+
(28)
Its solution is
119882 (120578
120591 120591) =
1
2radic120587120591
int
+infin
minusinfin
(119890
119910minus 119870)
+119890
minus(119910minus120578120591)24120591
119889119910
=
1
2radic120587120591
int
+infin
ln119870(119890
119910minus 119870) 119890
minus(119910minus120578120591)24120591
119889119910
= 119890
120578120591+120591
119873 (
2120591 + 120578
120591minus ln119870
radic2120591
) minus 119870119873 (
120578
120591minus ln119870
radic2120591
)
(29)
After variable reduction we have
119882 (120578
120591 120591) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119873 (119889
2)
(30)
Journal of Applied Mathematics 5
where
2120591 + 120578
120591minus ln119870
radic2120591
= (ln[
[
(119869
119905
119905119878
119879minus119905
119905)
1119879
119870
]
]
+ 119903
lowast(119879 minus 119905)
+ 120590
lowast2(119879
2119867minus 119905
2119867) )
times (120590
lowast(119879
2119867minus 119905
2119867)
12
)
minus1
≜ 119889
1
120578
120591minus ln119870
radic2120591
=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + 119903
lowast(119879 minus 119905)
120590
lowastradic
119879
2119867minus 119905
2119867
= 119889
1minus 120590
lowastradic
119879
2119867minus 119905
2119867≜ 119889
2
(31)
So the value of geometric average Asian call option at time 119905
is obtained
119881 (119905 119869
119905 119878
119905) = 119880 (120585
119905 119905) = 119882 (120578
120591 120591) 119890
minus120573(119905)
= (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(32)
Theorem 7 Suppose the underlying asset price 119878
119905satisfies
(3) then the relationship between 119881
119862(119905 119869
119905 119878
119905) the value of
geometric average Asian call option and 119881
119875(119905 119869
119905 119878
119905) the value
of put option with strike price 119870 maturity 119879 and transactionfee rate 119896 at time 119905 is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(33)
where 119903
lowast 120590
lowast are the same as above
Proof Let
119882 (119905 119869
119905 119878
119905) = 119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905) (34)
Then119882 is suitable for the following terminal question in 0 le
119878 lt infin 0 le 119869 lt infin 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119882
120597119878
119905
+
2
2
119878
2
119905
120597
2119882
120597119878
2
119905
+
120597119882
120597119869
119905
119869
119905
ln 119878
119905minus ln 119869
119905
119905
minus 119903
119905119882 = 0
119882|119905=119879= (119869
119879minus 119870)
+minus (119870 minus 119869
119879)
+= 119869
119879minus 119870
(35)
We let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] then 119882 is suitable for
the following in 120585
119905isin 119877 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119882
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120585
2
119905
minus 119903119882 = 0
119882|119905=119879= 119890
120585119879
minus 119870
(36)
Set the form solution of problem of (36) is
119882 (119905 119869
119905 119878
119905) = 119886 (119905) 119890
120585119905+ 119887 (119905)
(37)
Substituting (37) into (36) and comparing coefficients onehas
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
119886 (119905) +
2
2
(
119879 minus 119905
119879
)
2
119886 (119905)
+ 119886
1015840(119905) minus 119903
119905119886 (119905) = 0
119887
1015840(119905) minus 119903
119905119887 (119905) = 0
(38)
Taking 119886(119879) = 1 119887(119879) = minus119870 the solutions of (38) are
119886 (119905) = 119890
120572(119905)minusint119879
119905119903120591119889120591+120574(119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120591119889120591+(120590
lowast22)(1198792119867minus1199052119867)
119887 (119905) = minus 119870119890
minusint119879
119905119903120579119889120579
(39)
Thus
119882 = 119886 (119905) 119890
120585119905+ 119887 (119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879119878
(119879minus119905)119879
minus 119870119890
minusint119879
119905119903120579119889120579
(40)
The parity formula between call option and put option is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(41)
Pricing formula of the geometric average Asian putoptions can be obtained byTheorems 6 and 7
Theorem 8 Suppose 119878
119905satisfies (3) then the value 119881
119875(119905 119869
119905
119878
119905) of the geometric average Asian put option with strike price
119870 maturity 119879 and transaction fee rate 119896 at time 119905 is
119881
119875(119905 119869
119905 119878
119905) = minus(119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minusint119879
119905119903120579119889120579
119873 (minus119889
2)
(42)
6 Journal of Applied Mathematics
In particular if 119867 = 12 119896 = 0 119903 119902 120590 are all constant andprice formula (17) is reduced to the following formula [19]
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22minus119903)(119879minus119905)
119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
(43)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119903
lowast= (119903 minus 119902 minus
120590
2
2
)
119879 minus 119905
2119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
(44)
The following results are consequences of the above theorems
Corollary 9 If risk-free interest rate 119903 dividend yield 119902 andvolatility120590 are all constant then the price formulas of geometricaverage Asian call and put option with strike price 119870 maturity119879 and transaction fee rate 119896 under fractional Brownianmotionat time 119905 are respectively
119881
119862(119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
119881
119875(119905 119869
119905 119878
119905) = minus (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minus119903(119879minus119905)119873 (minus119889
2)
(45)
where
119903
lowast=
(119903 minus 119902)
2119879
(119879 minus 119905) minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
119871119890 (119867) 120590
2119879 minus 119905
119879
(46)
The rest of symbols are the same as Theorem 6
Noticing that if 119867 = 12 119861
119867(119905) is standard Brownian
motion119861(119905) the corresponding underlying asset price followsgeometric Brownian motion one has the following results
Corollary 10 If risk-free interest rate 119903
119905and dividend yield
119902
119905are the functions of time 119905 and volatility 120590 is constant
then the price formula 119881(119905 119869
119905 119878
119905) with respect to geometric
average Asian call option with strike price 119870 maturity 119879 andtransaction fee rate 119896 under standard Brownianmotion at time119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22)(119879minus119905)minusint
119879
119905119903120579119889120579
119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(47)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119871119890 (
1
2
) =radic
2
120587
119896
120590
(120575119905)
minus12
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879 minus 119905)
4119879
+
1
2
119871119890 (
1
2
) 120590
2119879 minus 119905
119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
radic1 minus 2119871119890 (
1
2
)
(48)
4 Numerical Example
Wediscuss the impact ofHurst index and transaction rates onthe Asian option value by numerical examples Assume thatthe parameter selection is as follows
119878
119905= 80 119905 = 0 119879 = 1
119903 = 005 119902 = 001 120590 = 04
119870 = 80 119896 = 0003 120575119905 = 002
(49)
We calculate the value of the option by using the priceformula of (45) The relationships between the value of calloption or put option and the underlying asset price withdifferentHurst index are given in Figures 1 and 2 respectivelyFrom Figures 1 and 2 the relationship between Hurst indexand Asian option value is negative Furthermore the impacton the call option value decreases with the increase of theunderlying asset price but the impact on the put option valuedecreases with the decrease of the underlying asset price
By Figures 3 and 4 we can get the change trend ofthe value of Asian call and put options with the change ofmaturity and Hurst index at the same time The option valueincreases with the maturity increases but the value of the calloption increases faster than the value of a put option increase
5 Conclusions
Asian options are popular financial derivatives that play anessential role in financial market Pricing them efficiently andaccurately is very important both in theory and practice Wehave investigated geometric average Asian option valuationproblems with transaction costs under the fractional Brow-nian motion By no-arbitrage principle and fractional Itorsquosformula this paper has deduced PDE satisfied by the optionrsquosvalue Meanwhile the pricing formula and call-put parity ofthe geometric average Asian option with transaction costsare derived by solving PDE At last the influences of Hurstexponent andmaturity on option value are discussed throughnumerical examples
Journal of Applied Mathematics 7
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
35
40
45
S
H =
H =
H =
H =
H =
50 60 70 80 90 100 110 120S
H =
H =
H =
H =
H =
VC
Figure 1 The values of the call option with different 119867
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
S
H =
H =
H =
H =
H =
VP
Figure 2 The values of the put option with different 119867
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous referees forvaluable suggestions for the improvement of this paper Allremaining errors are the responsibility of the authors Thisresearch was supported by ldquothe Fundamental Research Fundsfor the Central Universitiesrdquo (2010 LKSX03)
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity
VC
Figure 3 Relationship amongHurst indexmaturity and call optionvalue
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity0 2
0406
08
051
15
t index
Time to matu
VP
Figure 4 Relationship amongHurst indexmaturity and put optionvalue
References
[1] F Black andM S Scholes ldquoThe pricing of option and corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973
[2] G Fusai and A Meucci ldquoPricing discretely monitored Asianoptions under Levy processesrdquo Journal of Banking and Financevol 32 no 10 pp 2076ndash2088 2008
[3] J Vecer ldquoUnified pricing of Asian optionsrdquo Risk vol 6 no 15pp 113ndash116 2002
[4] J Vecer and M Xu ldquoPricing Asian options in a semimartingalemodelrdquo Quantitative Finance vol 4 no 2 pp 170ndash175 2004
[5] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 no 4 pp 422ndash437 1968
[6] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000
[7] R J Elliott and J van derHoek ldquoA general fractional white noisetheory and applications to financerdquoMathematical Finance vol13 no 2 pp 301ndash330 2003
8 Journal of Applied Mathematics
[8] H E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 pp 1283ndash1301 1985
[9] T Hoggard A E Whalley and P Wilmott ldquoHedging optionportfolios in the presence of transaction costsrdquo AdvancedFutures and Options Research vol 7 pp 21ndash35 1994
[10] P Guasoni ldquoNo arbitrage under transaction costs with frac-tional Brownian motion and beyondrdquo Mathematical Financevol 16 no 3 pp 569ndash582 2006
[11] H-K Liu and J-J Chang ldquoA closed-form approximation forthe fractional Black-Scholes model with transaction costsrdquoComputers amp Mathematics with Applications vol 65 no 11 pp1719ndash1726 2013
[12] X-T Wang ldquoScaling and long-range dependence in optionpricing I pricing European option with transaction costs underthe fractional Black-Scholes modelrdquo Physica A vol 389 no 3pp 438ndash444 2010
[13] X-T Wang E-H Zhu M-M Tang and H-G Yan ldquoScalingand long-range dependence in option pricing II pricing Euro-pean option with transaction costs under the mixed Brownian-fractional Brownian modelrdquo Physica A vol 389 no 3 pp 445ndash451 2010
[14] X-T Wang H-G Yan M-M Tang and E-H Zhu ldquoScalingand long-range dependence in option pricing III a fractionalversion of the Merton model with transaction costsrdquo Physica Avol 389 no 3 pp 452ndash458 2010
[15] X-T Wang ldquoScaling and long range dependence in optionpricing IV pricing European options with transaction costsunder the multifractional Black-Scholes modelrdquo Physica A vol389 no 4 pp 789ndash796 2010
[16] X-T Wang M Wu Z-M Zhou and W-S Jing ldquoPricingEuropean option with transaction costs under the fractionallong memory stochastic volatility modelrdquo Physica A vol 391no 4 pp 1469ndash1480 2012
[17] Y Hu and B Oslashksendal ldquoFractional white noise calculus andapplications to financerdquo Infinite Dimensional Analysis Quan-tum Probability and Related Topics vol 6 no 1 pp 1ndash32 2003
[18] C Bender ldquoAn Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameterrdquoStochastic Processes and Their Applications vol 104 no 1 pp81ndash106 2003
[19] J Li-Shang Mathematical Modeling and Methods of OptionPricing Higher Education Press Beijing China 2nd edition2008
Submit your manuscripts athttpwwwhindawicom
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Function Spaces
Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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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
4 Journal of Applied Mathematics
Proof By Theorem 3 the value 119881(119905 119869
119905 119878
119905) of the geometric
average Asian call satisfies the following model
120597119881
120597119905
+
1
2
2119878
2
119905
120597
2119881
120597119878
2
119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119881
120597119878
119905
+
120597119881
120597119869
119905
119869
119905ln (119878
119905119869
119905)
119905
minus 119903
119905119881 = 0
119881 (119879 119869
119879 119878
119879) = (119869
119879minus 119870)
+
(19)
Let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] 119881(119905 119869
119905 119878
119905) = 119880(119905 120585
119905)
then
120597119881
120597119905
=
ln (119869
119905119878
119905)
119879
120597119880
120597120585
119905
+
120597119880
120597119905
120597119881
120597119878
119905
=
119879 minus 119905
119879119878
119905
120597119880
120597120585
119905
120597
2119881
120597119878
2
119905
= (
119879 minus 119905
119879119878
119905
)
2120597
2119880
120597120585
2
119905
minus
119879 minus 119905
119879119878
2
119905
120597119880
120597120585
119905
120597119881
120597119869
119905
=
119905
119879119869
119905
120597119880
120597120585
119905
(20)
Combined with the boundary conditions of the call option119881(119879 119869
119879 119878
119879) = (119869
119879minus 119870)
+ the model (19) can be converted to
120597119880
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119880
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119880
120597120585
2
119905
minus 119903
119905119880 = 0
119880 (120585
119879 119879) = (119890
120585119879
minus 119870)
+
(21)
Let 120591 = 120574(119905) 120578
120591= 120585
119905+ 120572(119905) 119882(120591 120578
120591) = 119880(119905 120585
119905)119890
120573(119905) whichsatisfied the conditions
120572 (119879) = 120573 (119879) = 120574 (119879) = 0 (22)
and then we have
120597119880
120597119905
= 119890
minus120573(119905)(
120597119882
120597120591
120574
1015840(119905) minus 120573
1015840(119905) 119882 +
120597119882
120597120578
120591
120572
1015840(119905))
120597119880
120597120585
119905
= 119890
minus120573(119905) 120597119882
120597120578
120591
120597
2119880
120597120585
2
119905
= 119890
minus120573(119905) 1205972119882
120597120578
2
120591
(23)
Substituting (23) into (21) we can get
120574
1015840(119905)
120597119882
120597120591
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120578
2
120591
+ [(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
+ 120572
1015840(119905)]
120597119882
120597120578
120591
minus (119903
119905+ 120573
1015840(119905)) 119882 = 0
(24)
Set
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
+ 120572
1015840(119905) = 0 119903
119905+ 120573
1015840(119905) = 0
120574
1015840(119905) = minus
2
2
(
119879 minus 119905
119879
)
2
(25)
Combining with the terminal conditions 120572(119879) = 120573(119879) =
120574(119879) = 0 we have
120572 (119905) = int
119879
119905
(119903
120579minus 119902
120579)
119879 minus 120579
119879
119889120579 minus int
119879
119905
1
2
2(
119879 minus 120579
119879
) 119889120579
= 119903 lowast (119879 minus 119905)
120573 (119905) = int
119879
119905
119903
120579119889120579
120574 (119905) = int
119879
119905
1
2
2(
119879 minus 120579
119879
)
2
119889120579 =
120590
lowast2
2
(119879
2119867minus 119905
2119867)
(26)
where
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
Le (119867) 120590
2119879 minus 119905
119879
120590
lowast= 120590 times (1 minus
4119867 (119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879
2119867minus 119905
2119867)
+
119867 (119879
2119867+2minus 119905
2119867+2)
119879
2(119867 + 1) (119879
2119867minus 119905
2119867)
minus2Le (119867)
(119879 minus 119905)
3
3119879
2(119879
2119867minus 119905
2119867)
)
12
(27)
Thus themodel (21) is converted into the classic heat conduc-tion equation
120597119882
120597120591
=
120597
2119882
120597120578
2
120591
119882 (120578
0 0) = (119890
1205780
minus 119870)
+
(28)
Its solution is
119882 (120578
120591 120591) =
1
2radic120587120591
int
+infin
minusinfin
(119890
119910minus 119870)
+119890
minus(119910minus120578120591)24120591
119889119910
=
1
2radic120587120591
int
+infin
ln119870(119890
119910minus 119870) 119890
minus(119910minus120578120591)24120591
119889119910
= 119890
120578120591+120591
119873 (
2120591 + 120578
120591minus ln119870
radic2120591
) minus 119870119873 (
120578
120591minus ln119870
radic2120591
)
(29)
After variable reduction we have
119882 (120578
120591 120591) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119873 (119889
2)
(30)
Journal of Applied Mathematics 5
where
2120591 + 120578
120591minus ln119870
radic2120591
= (ln[
[
(119869
119905
119905119878
119879minus119905
119905)
1119879
119870
]
]
+ 119903
lowast(119879 minus 119905)
+ 120590
lowast2(119879
2119867minus 119905
2119867) )
times (120590
lowast(119879
2119867minus 119905
2119867)
12
)
minus1
≜ 119889
1
120578
120591minus ln119870
radic2120591
=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + 119903
lowast(119879 minus 119905)
120590
lowastradic
119879
2119867minus 119905
2119867
= 119889
1minus 120590
lowastradic
119879
2119867minus 119905
2119867≜ 119889
2
(31)
So the value of geometric average Asian call option at time 119905
is obtained
119881 (119905 119869
119905 119878
119905) = 119880 (120585
119905 119905) = 119882 (120578
120591 120591) 119890
minus120573(119905)
= (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(32)
Theorem 7 Suppose the underlying asset price 119878
119905satisfies
(3) then the relationship between 119881
119862(119905 119869
119905 119878
119905) the value of
geometric average Asian call option and 119881
119875(119905 119869
119905 119878
119905) the value
of put option with strike price 119870 maturity 119879 and transactionfee rate 119896 at time 119905 is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(33)
where 119903
lowast 120590
lowast are the same as above
Proof Let
119882 (119905 119869
119905 119878
119905) = 119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905) (34)
Then119882 is suitable for the following terminal question in 0 le
119878 lt infin 0 le 119869 lt infin 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119882
120597119878
119905
+
2
2
119878
2
119905
120597
2119882
120597119878
2
119905
+
120597119882
120597119869
119905
119869
119905
ln 119878
119905minus ln 119869
119905
119905
minus 119903
119905119882 = 0
119882|119905=119879= (119869
119879minus 119870)
+minus (119870 minus 119869
119879)
+= 119869
119879minus 119870
(35)
We let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] then 119882 is suitable for
the following in 120585
119905isin 119877 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119882
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120585
2
119905
minus 119903119882 = 0
119882|119905=119879= 119890
120585119879
minus 119870
(36)
Set the form solution of problem of (36) is
119882 (119905 119869
119905 119878
119905) = 119886 (119905) 119890
120585119905+ 119887 (119905)
(37)
Substituting (37) into (36) and comparing coefficients onehas
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
119886 (119905) +
2
2
(
119879 minus 119905
119879
)
2
119886 (119905)
+ 119886
1015840(119905) minus 119903
119905119886 (119905) = 0
119887
1015840(119905) minus 119903
119905119887 (119905) = 0
(38)
Taking 119886(119879) = 1 119887(119879) = minus119870 the solutions of (38) are
119886 (119905) = 119890
120572(119905)minusint119879
119905119903120591119889120591+120574(119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120591119889120591+(120590
lowast22)(1198792119867minus1199052119867)
119887 (119905) = minus 119870119890
minusint119879
119905119903120579119889120579
(39)
Thus
119882 = 119886 (119905) 119890
120585119905+ 119887 (119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879119878
(119879minus119905)119879
minus 119870119890
minusint119879
119905119903120579119889120579
(40)
The parity formula between call option and put option is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(41)
Pricing formula of the geometric average Asian putoptions can be obtained byTheorems 6 and 7
Theorem 8 Suppose 119878
119905satisfies (3) then the value 119881
119875(119905 119869
119905
119878
119905) of the geometric average Asian put option with strike price
119870 maturity 119879 and transaction fee rate 119896 at time 119905 is
119881
119875(119905 119869
119905 119878
119905) = minus(119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minusint119879
119905119903120579119889120579
119873 (minus119889
2)
(42)
6 Journal of Applied Mathematics
In particular if 119867 = 12 119896 = 0 119903 119902 120590 are all constant andprice formula (17) is reduced to the following formula [19]
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22minus119903)(119879minus119905)
119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
(43)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119903
lowast= (119903 minus 119902 minus
120590
2
2
)
119879 minus 119905
2119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
(44)
The following results are consequences of the above theorems
Corollary 9 If risk-free interest rate 119903 dividend yield 119902 andvolatility120590 are all constant then the price formulas of geometricaverage Asian call and put option with strike price 119870 maturity119879 and transaction fee rate 119896 under fractional Brownianmotionat time 119905 are respectively
119881
119862(119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
119881
119875(119905 119869
119905 119878
119905) = minus (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minus119903(119879minus119905)119873 (minus119889
2)
(45)
where
119903
lowast=
(119903 minus 119902)
2119879
(119879 minus 119905) minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
119871119890 (119867) 120590
2119879 minus 119905
119879
(46)
The rest of symbols are the same as Theorem 6
Noticing that if 119867 = 12 119861
119867(119905) is standard Brownian
motion119861(119905) the corresponding underlying asset price followsgeometric Brownian motion one has the following results
Corollary 10 If risk-free interest rate 119903
119905and dividend yield
119902
119905are the functions of time 119905 and volatility 120590 is constant
then the price formula 119881(119905 119869
119905 119878
119905) with respect to geometric
average Asian call option with strike price 119870 maturity 119879 andtransaction fee rate 119896 under standard Brownianmotion at time119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22)(119879minus119905)minusint
119879
119905119903120579119889120579
119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(47)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119871119890 (
1
2
) =radic
2
120587
119896
120590
(120575119905)
minus12
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879 minus 119905)
4119879
+
1
2
119871119890 (
1
2
) 120590
2119879 minus 119905
119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
radic1 minus 2119871119890 (
1
2
)
(48)
4 Numerical Example
Wediscuss the impact ofHurst index and transaction rates onthe Asian option value by numerical examples Assume thatthe parameter selection is as follows
119878
119905= 80 119905 = 0 119879 = 1
119903 = 005 119902 = 001 120590 = 04
119870 = 80 119896 = 0003 120575119905 = 002
(49)
We calculate the value of the option by using the priceformula of (45) The relationships between the value of calloption or put option and the underlying asset price withdifferentHurst index are given in Figures 1 and 2 respectivelyFrom Figures 1 and 2 the relationship between Hurst indexand Asian option value is negative Furthermore the impacton the call option value decreases with the increase of theunderlying asset price but the impact on the put option valuedecreases with the decrease of the underlying asset price
By Figures 3 and 4 we can get the change trend ofthe value of Asian call and put options with the change ofmaturity and Hurst index at the same time The option valueincreases with the maturity increases but the value of the calloption increases faster than the value of a put option increase
5 Conclusions
Asian options are popular financial derivatives that play anessential role in financial market Pricing them efficiently andaccurately is very important both in theory and practice Wehave investigated geometric average Asian option valuationproblems with transaction costs under the fractional Brow-nian motion By no-arbitrage principle and fractional Itorsquosformula this paper has deduced PDE satisfied by the optionrsquosvalue Meanwhile the pricing formula and call-put parity ofthe geometric average Asian option with transaction costsare derived by solving PDE At last the influences of Hurstexponent andmaturity on option value are discussed throughnumerical examples
Journal of Applied Mathematics 7
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
35
40
45
S
H =
H =
H =
H =
H =
50 60 70 80 90 100 110 120S
H =
H =
H =
H =
H =
VC
Figure 1 The values of the call option with different 119867
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
S
H =
H =
H =
H =
H =
VP
Figure 2 The values of the put option with different 119867
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous referees forvaluable suggestions for the improvement of this paper Allremaining errors are the responsibility of the authors Thisresearch was supported by ldquothe Fundamental Research Fundsfor the Central Universitiesrdquo (2010 LKSX03)
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity
VC
Figure 3 Relationship amongHurst indexmaturity and call optionvalue
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity0 2
0406
08
051
15
t index
Time to matu
VP
Figure 4 Relationship amongHurst indexmaturity and put optionvalue
References
[1] F Black andM S Scholes ldquoThe pricing of option and corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973
[2] G Fusai and A Meucci ldquoPricing discretely monitored Asianoptions under Levy processesrdquo Journal of Banking and Financevol 32 no 10 pp 2076ndash2088 2008
[3] J Vecer ldquoUnified pricing of Asian optionsrdquo Risk vol 6 no 15pp 113ndash116 2002
[4] J Vecer and M Xu ldquoPricing Asian options in a semimartingalemodelrdquo Quantitative Finance vol 4 no 2 pp 170ndash175 2004
[5] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 no 4 pp 422ndash437 1968
[6] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000
[7] R J Elliott and J van derHoek ldquoA general fractional white noisetheory and applications to financerdquoMathematical Finance vol13 no 2 pp 301ndash330 2003
8 Journal of Applied Mathematics
[8] H E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 pp 1283ndash1301 1985
[9] T Hoggard A E Whalley and P Wilmott ldquoHedging optionportfolios in the presence of transaction costsrdquo AdvancedFutures and Options Research vol 7 pp 21ndash35 1994
[10] P Guasoni ldquoNo arbitrage under transaction costs with frac-tional Brownian motion and beyondrdquo Mathematical Financevol 16 no 3 pp 569ndash582 2006
[11] H-K Liu and J-J Chang ldquoA closed-form approximation forthe fractional Black-Scholes model with transaction costsrdquoComputers amp Mathematics with Applications vol 65 no 11 pp1719ndash1726 2013
[12] X-T Wang ldquoScaling and long-range dependence in optionpricing I pricing European option with transaction costs underthe fractional Black-Scholes modelrdquo Physica A vol 389 no 3pp 438ndash444 2010
[13] X-T Wang E-H Zhu M-M Tang and H-G Yan ldquoScalingand long-range dependence in option pricing II pricing Euro-pean option with transaction costs under the mixed Brownian-fractional Brownian modelrdquo Physica A vol 389 no 3 pp 445ndash451 2010
[14] X-T Wang H-G Yan M-M Tang and E-H Zhu ldquoScalingand long-range dependence in option pricing III a fractionalversion of the Merton model with transaction costsrdquo Physica Avol 389 no 3 pp 452ndash458 2010
[15] X-T Wang ldquoScaling and long range dependence in optionpricing IV pricing European options with transaction costsunder the multifractional Black-Scholes modelrdquo Physica A vol389 no 4 pp 789ndash796 2010
[16] X-T Wang M Wu Z-M Zhou and W-S Jing ldquoPricingEuropean option with transaction costs under the fractionallong memory stochastic volatility modelrdquo Physica A vol 391no 4 pp 1469ndash1480 2012
[17] Y Hu and B Oslashksendal ldquoFractional white noise calculus andapplications to financerdquo Infinite Dimensional Analysis Quan-tum Probability and Related Topics vol 6 no 1 pp 1ndash32 2003
[18] C Bender ldquoAn Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameterrdquoStochastic Processes and Their Applications vol 104 no 1 pp81ndash106 2003
[19] J Li-Shang Mathematical Modeling and Methods of OptionPricing Higher Education Press Beijing China 2nd edition2008
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
Journal of Applied Mathematics 5
where
2120591 + 120578
120591minus ln119870
radic2120591
= (ln[
[
(119869
119905
119905119878
119879minus119905
119905)
1119879
119870
]
]
+ 119903
lowast(119879 minus 119905)
+ 120590
lowast2(119879
2119867minus 119905
2119867) )
times (120590
lowast(119879
2119867minus 119905
2119867)
12
)
minus1
≜ 119889
1
120578
120591minus ln119870
radic2120591
=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + 119903
lowast(119879 minus 119905)
120590
lowastradic
119879
2119867minus 119905
2119867
= 119889
1minus 120590
lowastradic
119879
2119867minus 119905
2119867≜ 119889
2
(31)
So the value of geometric average Asian call option at time 119905
is obtained
119881 (119905 119869
119905 119878
119905) = 119880 (120585
119905 119905) = 119882 (120578
120591 120591) 119890
minus120573(119905)
= (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(32)
Theorem 7 Suppose the underlying asset price 119878
119905satisfies
(3) then the relationship between 119881
119862(119905 119869
119905 119878
119905) the value of
geometric average Asian call option and 119881
119875(119905 119869
119905 119878
119905) the value
of put option with strike price 119870 maturity 119879 and transactionfee rate 119896 at time 119905 is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(33)
where 119903
lowast 120590
lowast are the same as above
Proof Let
119882 (119905 119869
119905 119878
119905) = 119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905) (34)
Then119882 is suitable for the following terminal question in 0 le
119878 lt infin 0 le 119869 lt infin 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905) 119878
119905
120597119882
120597119878
119905
+
2
2
119878
2
119905
120597
2119882
120597119878
2
119905
+
120597119882
120597119869
119905
119869
119905
ln 119878
119905minus ln 119869
119905
119905
minus 119903
119905119882 = 0
119882|119905=119879= (119869
119879minus 119870)
+minus (119870 minus 119869
119879)
+= 119869
119879minus 119870
(35)
We let 120585
119905= (1119879)[119905 ln 119869
119905+ (119879 minus 119905) ln 119878
119905] then 119882 is suitable for
the following in 120585
119905isin 119877 0 le 119905 le 119879
120597119882
120597119905
+ (119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
120597119882
120597120585
119905
+
2
2
(
119879 minus 119905
119879
)
2120597
2119882
120597120585
2
119905
minus 119903119882 = 0
119882|119905=119879= 119890
120585119879
minus 119870
(36)
Set the form solution of problem of (36) is
119882 (119905 119869
119905 119878
119905) = 119886 (119905) 119890
120585119905+ 119887 (119905)
(37)
Substituting (37) into (36) and comparing coefficients onehas
(119903
119905minus 119902
119905minus
2
2
)
119879 minus 119905
119879
119886 (119905) +
2
2
(
119879 minus 119905
119879
)
2
119886 (119905)
+ 119886
1015840(119905) minus 119903
119905119886 (119905) = 0
119887
1015840(119905) minus 119903
119905119887 (119905) = 0
(38)
Taking 119886(119879) = 1 119887(119879) = minus119870 the solutions of (38) are
119886 (119905) = 119890
120572(119905)minusint119879
119905119903120591119889120591+120574(119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120591119889120591+(120590
lowast22)(1198792119867minus1199052119867)
119887 (119905) = minus 119870119890
minusint119879
119905119903120579119889120579
(39)
Thus
119882 = 119886 (119905) 119890
120585119905+ 119887 (119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879119878
(119879minus119905)119879
minus 119870119890
minusint119879
119905119903120579119889120579
(40)
The parity formula between call option and put option is
119881
119862(119905 119869
119905 119878
119905) minus 119881
119875(119905 119869
119905 119878
119905)
= 119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)119869
119905119879
119905119878
(119879minus119905)119879
119905
minus 119870119890
minusint119879
119905119903120579119889120579
(41)
Pricing formula of the geometric average Asian putoptions can be obtained byTheorems 6 and 7
Theorem 8 Suppose 119878
119905satisfies (3) then the value 119881
119875(119905 119869
119905
119878
119905) of the geometric average Asian put option with strike price
119870 maturity 119879 and transaction fee rate 119896 at time 119905 is
119881
119875(119905 119869
119905 119878
119905) = minus(119869
119905
119905119878
119879minus119905
119905)
1119879
119890
119903lowast(119879minus119905)minusint
119879
119905119903120579119889120579+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minusint119879
119905119903120579119889120579
119873 (minus119889
2)
(42)
6 Journal of Applied Mathematics
In particular if 119867 = 12 119896 = 0 119903 119902 120590 are all constant andprice formula (17) is reduced to the following formula [19]
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22minus119903)(119879minus119905)
119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
(43)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119903
lowast= (119903 minus 119902 minus
120590
2
2
)
119879 minus 119905
2119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
(44)
The following results are consequences of the above theorems
Corollary 9 If risk-free interest rate 119903 dividend yield 119902 andvolatility120590 are all constant then the price formulas of geometricaverage Asian call and put option with strike price 119870 maturity119879 and transaction fee rate 119896 under fractional Brownianmotionat time 119905 are respectively
119881
119862(119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
119881
119875(119905 119869
119905 119878
119905) = minus (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minus119903(119879minus119905)119873 (minus119889
2)
(45)
where
119903
lowast=
(119903 minus 119902)
2119879
(119879 minus 119905) minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
119871119890 (119867) 120590
2119879 minus 119905
119879
(46)
The rest of symbols are the same as Theorem 6
Noticing that if 119867 = 12 119861
119867(119905) is standard Brownian
motion119861(119905) the corresponding underlying asset price followsgeometric Brownian motion one has the following results
Corollary 10 If risk-free interest rate 119903
119905and dividend yield
119902
119905are the functions of time 119905 and volatility 120590 is constant
then the price formula 119881(119905 119869
119905 119878
119905) with respect to geometric
average Asian call option with strike price 119870 maturity 119879 andtransaction fee rate 119896 under standard Brownianmotion at time119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22)(119879minus119905)minusint
119879
119905119903120579119889120579
119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(47)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119871119890 (
1
2
) =radic
2
120587
119896
120590
(120575119905)
minus12
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879 minus 119905)
4119879
+
1
2
119871119890 (
1
2
) 120590
2119879 minus 119905
119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
radic1 minus 2119871119890 (
1
2
)
(48)
4 Numerical Example
Wediscuss the impact ofHurst index and transaction rates onthe Asian option value by numerical examples Assume thatthe parameter selection is as follows
119878
119905= 80 119905 = 0 119879 = 1
119903 = 005 119902 = 001 120590 = 04
119870 = 80 119896 = 0003 120575119905 = 002
(49)
We calculate the value of the option by using the priceformula of (45) The relationships between the value of calloption or put option and the underlying asset price withdifferentHurst index are given in Figures 1 and 2 respectivelyFrom Figures 1 and 2 the relationship between Hurst indexand Asian option value is negative Furthermore the impacton the call option value decreases with the increase of theunderlying asset price but the impact on the put option valuedecreases with the decrease of the underlying asset price
By Figures 3 and 4 we can get the change trend ofthe value of Asian call and put options with the change ofmaturity and Hurst index at the same time The option valueincreases with the maturity increases but the value of the calloption increases faster than the value of a put option increase
5 Conclusions
Asian options are popular financial derivatives that play anessential role in financial market Pricing them efficiently andaccurately is very important both in theory and practice Wehave investigated geometric average Asian option valuationproblems with transaction costs under the fractional Brow-nian motion By no-arbitrage principle and fractional Itorsquosformula this paper has deduced PDE satisfied by the optionrsquosvalue Meanwhile the pricing formula and call-put parity ofthe geometric average Asian option with transaction costsare derived by solving PDE At last the influences of Hurstexponent andmaturity on option value are discussed throughnumerical examples
Journal of Applied Mathematics 7
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
35
40
45
S
H =
H =
H =
H =
H =
50 60 70 80 90 100 110 120S
H =
H =
H =
H =
H =
VC
Figure 1 The values of the call option with different 119867
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
S
H =
H =
H =
H =
H =
VP
Figure 2 The values of the put option with different 119867
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous referees forvaluable suggestions for the improvement of this paper Allremaining errors are the responsibility of the authors Thisresearch was supported by ldquothe Fundamental Research Fundsfor the Central Universitiesrdquo (2010 LKSX03)
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity
VC
Figure 3 Relationship amongHurst indexmaturity and call optionvalue
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity0 2
0406
08
051
15
t index
Time to matu
VP
Figure 4 Relationship amongHurst indexmaturity and put optionvalue
References
[1] F Black andM S Scholes ldquoThe pricing of option and corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973
[2] G Fusai and A Meucci ldquoPricing discretely monitored Asianoptions under Levy processesrdquo Journal of Banking and Financevol 32 no 10 pp 2076ndash2088 2008
[3] J Vecer ldquoUnified pricing of Asian optionsrdquo Risk vol 6 no 15pp 113ndash116 2002
[4] J Vecer and M Xu ldquoPricing Asian options in a semimartingalemodelrdquo Quantitative Finance vol 4 no 2 pp 170ndash175 2004
[5] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 no 4 pp 422ndash437 1968
[6] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000
[7] R J Elliott and J van derHoek ldquoA general fractional white noisetheory and applications to financerdquoMathematical Finance vol13 no 2 pp 301ndash330 2003
8 Journal of Applied Mathematics
[8] H E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 pp 1283ndash1301 1985
[9] T Hoggard A E Whalley and P Wilmott ldquoHedging optionportfolios in the presence of transaction costsrdquo AdvancedFutures and Options Research vol 7 pp 21ndash35 1994
[10] P Guasoni ldquoNo arbitrage under transaction costs with frac-tional Brownian motion and beyondrdquo Mathematical Financevol 16 no 3 pp 569ndash582 2006
[11] H-K Liu and J-J Chang ldquoA closed-form approximation forthe fractional Black-Scholes model with transaction costsrdquoComputers amp Mathematics with Applications vol 65 no 11 pp1719ndash1726 2013
[12] X-T Wang ldquoScaling and long-range dependence in optionpricing I pricing European option with transaction costs underthe fractional Black-Scholes modelrdquo Physica A vol 389 no 3pp 438ndash444 2010
[13] X-T Wang E-H Zhu M-M Tang and H-G Yan ldquoScalingand long-range dependence in option pricing II pricing Euro-pean option with transaction costs under the mixed Brownian-fractional Brownian modelrdquo Physica A vol 389 no 3 pp 445ndash451 2010
[14] X-T Wang H-G Yan M-M Tang and E-H Zhu ldquoScalingand long-range dependence in option pricing III a fractionalversion of the Merton model with transaction costsrdquo Physica Avol 389 no 3 pp 452ndash458 2010
[15] X-T Wang ldquoScaling and long range dependence in optionpricing IV pricing European options with transaction costsunder the multifractional Black-Scholes modelrdquo Physica A vol389 no 4 pp 789ndash796 2010
[16] X-T Wang M Wu Z-M Zhou and W-S Jing ldquoPricingEuropean option with transaction costs under the fractionallong memory stochastic volatility modelrdquo Physica A vol 391no 4 pp 1469ndash1480 2012
[17] Y Hu and B Oslashksendal ldquoFractional white noise calculus andapplications to financerdquo Infinite Dimensional Analysis Quan-tum Probability and Related Topics vol 6 no 1 pp 1ndash32 2003
[18] C Bender ldquoAn Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameterrdquoStochastic Processes and Their Applications vol 104 no 1 pp81ndash106 2003
[19] J Li-Shang Mathematical Modeling and Methods of OptionPricing Higher Education Press Beijing China 2nd edition2008
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
6 Journal of Applied Mathematics
In particular if 119867 = 12 119896 = 0 119903 119902 120590 are all constant andprice formula (17) is reduced to the following formula [19]
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22minus119903)(119879minus119905)
119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
(43)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119903
lowast= (119903 minus 119902 minus
120590
2
2
)
119879 minus 119905
2119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
(44)
The following results are consequences of the above theorems
Corollary 9 If risk-free interest rate 119903 dividend yield 119902 andvolatility120590 are all constant then the price formulas of geometricaverage Asian call and put option with strike price 119870 maturity119879 and transaction fee rate 119896 under fractional Brownianmotionat time 119905 are respectively
119881
119862(119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)119873 (119889
1)
minus 119870119890
minus119903(119879minus119905)119873 (119889
2)
119881
119875(119905 119869
119905 119878
119905) = minus (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowastminus119903)(119879minus119905)+(120590
lowast22)(1198792119867minus1199052119867)
times 119873 (minus119889
1) + 119870119890
minus119903(119879minus119905)119873 (minus119889
2)
(45)
where
119903
lowast=
(119903 minus 119902)
2119879
(119879 minus 119905) minus
120590
2(119879
2119867minus 119905
2119867)
2 (119879 minus 119905)
+
119867120590
2(119879
2119867+1minus 119905
2119867+1)
119879 (2119867 + 1) (119879 minus 119905)
+
1
2
119871119890 (119867) 120590
2119879 minus 119905
119879
(46)
The rest of symbols are the same as Theorem 6
Noticing that if 119867 = 12 119861
119867(119905) is standard Brownian
motion119861(119905) the corresponding underlying asset price followsgeometric Brownian motion one has the following results
Corollary 10 If risk-free interest rate 119903
119905and dividend yield
119902
119905are the functions of time 119905 and volatility 120590 is constant
then the price formula 119881(119905 119869
119905 119878
119905) with respect to geometric
average Asian call option with strike price 119870 maturity 119879 andtransaction fee rate 119896 under standard Brownianmotion at time119905 is
119881 (119905 119869
119905 119878
119905) = (119869
119905
119905119878
119879minus119905
119905)
1119879
119890
(119903lowast+120590lowast22)(119879minus119905)minusint
119879
119905119903120579119889120579
119873 (119889
1)
minus 119870119890
minusint119879
119905119903120579119889120579
119873 (119889
2)
(47)
where
119889
1=
ln [(119869
119905
119905119878
119879minus119905
119905)
1119879
119870] + (119903
lowast+ 120590
lowast2) (119879 minus 119905)
120590
lowastradic
119879 minus 119905
119889
2= 119889
1minus 120590
lowastradic
119879 minus 119905
119871119890 (
1
2
) =radic
2
120587
119896
120590
(120575119905)
minus12
119903
lowast=
int
119879
119905
(119903
120579minus 119902
120579) ((119879 minus 120579) 119879) 119889120579
119879 minus 119905
minus
120590
2(119879 minus 119905)
4119879
+
1
2
119871119890 (
1
2
) 120590
2119879 minus 119905
119879
120590
lowast= 120590
(119879 minus 119905)
radic3119879
radic1 minus 2119871119890 (
1
2
)
(48)
4 Numerical Example
Wediscuss the impact ofHurst index and transaction rates onthe Asian option value by numerical examples Assume thatthe parameter selection is as follows
119878
119905= 80 119905 = 0 119879 = 1
119903 = 005 119902 = 001 120590 = 04
119870 = 80 119896 = 0003 120575119905 = 002
(49)
We calculate the value of the option by using the priceformula of (45) The relationships between the value of calloption or put option and the underlying asset price withdifferentHurst index are given in Figures 1 and 2 respectivelyFrom Figures 1 and 2 the relationship between Hurst indexand Asian option value is negative Furthermore the impacton the call option value decreases with the increase of theunderlying asset price but the impact on the put option valuedecreases with the decrease of the underlying asset price
By Figures 3 and 4 we can get the change trend ofthe value of Asian call and put options with the change ofmaturity and Hurst index at the same time The option valueincreases with the maturity increases but the value of the calloption increases faster than the value of a put option increase
5 Conclusions
Asian options are popular financial derivatives that play anessential role in financial market Pricing them efficiently andaccurately is very important both in theory and practice Wehave investigated geometric average Asian option valuationproblems with transaction costs under the fractional Brow-nian motion By no-arbitrage principle and fractional Itorsquosformula this paper has deduced PDE satisfied by the optionrsquosvalue Meanwhile the pricing formula and call-put parity ofthe geometric average Asian option with transaction costsare derived by solving PDE At last the influences of Hurstexponent andmaturity on option value are discussed throughnumerical examples
Journal of Applied Mathematics 7
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
35
40
45
S
H =
H =
H =
H =
H =
50 60 70 80 90 100 110 120S
H =
H =
H =
H =
H =
VC
Figure 1 The values of the call option with different 119867
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
S
H =
H =
H =
H =
H =
VP
Figure 2 The values of the put option with different 119867
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous referees forvaluable suggestions for the improvement of this paper Allremaining errors are the responsibility of the authors Thisresearch was supported by ldquothe Fundamental Research Fundsfor the Central Universitiesrdquo (2010 LKSX03)
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity
VC
Figure 3 Relationship amongHurst indexmaturity and call optionvalue
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity0 2
0406
08
051
15
t index
Time to matu
VP
Figure 4 Relationship amongHurst indexmaturity and put optionvalue
References
[1] F Black andM S Scholes ldquoThe pricing of option and corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973
[2] G Fusai and A Meucci ldquoPricing discretely monitored Asianoptions under Levy processesrdquo Journal of Banking and Financevol 32 no 10 pp 2076ndash2088 2008
[3] J Vecer ldquoUnified pricing of Asian optionsrdquo Risk vol 6 no 15pp 113ndash116 2002
[4] J Vecer and M Xu ldquoPricing Asian options in a semimartingalemodelrdquo Quantitative Finance vol 4 no 2 pp 170ndash175 2004
[5] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 no 4 pp 422ndash437 1968
[6] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000
[7] R J Elliott and J van derHoek ldquoA general fractional white noisetheory and applications to financerdquoMathematical Finance vol13 no 2 pp 301ndash330 2003
8 Journal of Applied Mathematics
[8] H E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 pp 1283ndash1301 1985
[9] T Hoggard A E Whalley and P Wilmott ldquoHedging optionportfolios in the presence of transaction costsrdquo AdvancedFutures and Options Research vol 7 pp 21ndash35 1994
[10] P Guasoni ldquoNo arbitrage under transaction costs with frac-tional Brownian motion and beyondrdquo Mathematical Financevol 16 no 3 pp 569ndash582 2006
[11] H-K Liu and J-J Chang ldquoA closed-form approximation forthe fractional Black-Scholes model with transaction costsrdquoComputers amp Mathematics with Applications vol 65 no 11 pp1719ndash1726 2013
[12] X-T Wang ldquoScaling and long-range dependence in optionpricing I pricing European option with transaction costs underthe fractional Black-Scholes modelrdquo Physica A vol 389 no 3pp 438ndash444 2010
[13] X-T Wang E-H Zhu M-M Tang and H-G Yan ldquoScalingand long-range dependence in option pricing II pricing Euro-pean option with transaction costs under the mixed Brownian-fractional Brownian modelrdquo Physica A vol 389 no 3 pp 445ndash451 2010
[14] X-T Wang H-G Yan M-M Tang and E-H Zhu ldquoScalingand long-range dependence in option pricing III a fractionalversion of the Merton model with transaction costsrdquo Physica Avol 389 no 3 pp 452ndash458 2010
[15] X-T Wang ldquoScaling and long range dependence in optionpricing IV pricing European options with transaction costsunder the multifractional Black-Scholes modelrdquo Physica A vol389 no 4 pp 789ndash796 2010
[16] X-T Wang M Wu Z-M Zhou and W-S Jing ldquoPricingEuropean option with transaction costs under the fractionallong memory stochastic volatility modelrdquo Physica A vol 391no 4 pp 1469ndash1480 2012
[17] Y Hu and B Oslashksendal ldquoFractional white noise calculus andapplications to financerdquo Infinite Dimensional Analysis Quan-tum Probability and Related Topics vol 6 no 1 pp 1ndash32 2003
[18] C Bender ldquoAn Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameterrdquoStochastic Processes and Their Applications vol 104 no 1 pp81ndash106 2003
[19] J Li-Shang Mathematical Modeling and Methods of OptionPricing Higher Education Press Beijing China 2nd edition2008
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
Journal of Applied Mathematics 7
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
35
40
45
S
H =
H =
H =
H =
H =
50 60 70 80 90 100 110 120S
H =
H =
H =
H =
H =
VC
Figure 1 The values of the call option with different 119867
50 60 70 80 90 100 110 1200
5
10
15
20
25
30
S
H =
H =
H =
H =
H =
VP
Figure 2 The values of the put option with different 119867
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to thank the anonymous referees forvaluable suggestions for the improvement of this paper Allremaining errors are the responsibility of the authors Thisresearch was supported by ldquothe Fundamental Research Fundsfor the Central Universitiesrdquo (2010 LKSX03)
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity
VC
Figure 3 Relationship amongHurst indexmaturity and call optionvalue
002
0406
081
005
115
20
2
4
6
8
10
12
Hurst index
Time to maturity0 2
0406
08
051
15
t index
Time to matu
VP
Figure 4 Relationship amongHurst indexmaturity and put optionvalue
References
[1] F Black andM S Scholes ldquoThe pricing of option and corporateliabilitiesrdquo Journal of Political Economy vol 81 pp 637ndash6591973
[2] G Fusai and A Meucci ldquoPricing discretely monitored Asianoptions under Levy processesrdquo Journal of Banking and Financevol 32 no 10 pp 2076ndash2088 2008
[3] J Vecer ldquoUnified pricing of Asian optionsrdquo Risk vol 6 no 15pp 113ndash116 2002
[4] J Vecer and M Xu ldquoPricing Asian options in a semimartingalemodelrdquo Quantitative Finance vol 4 no 2 pp 170ndash175 2004
[5] B B Mandelbrot and J W Van Ness ldquoFractional Brownianmotions fractional noises and applicationsrdquo SIAM Review vol10 no 4 pp 422ndash437 1968
[6] T E Duncan Y Hu and B Pasik-Duncan ldquoStochastic calculusfor fractional Brownian motion I Theoryrdquo SIAM Journal onControl and Optimization vol 38 no 2 pp 582ndash612 2000
[7] R J Elliott and J van derHoek ldquoA general fractional white noisetheory and applications to financerdquoMathematical Finance vol13 no 2 pp 301ndash330 2003
8 Journal of Applied Mathematics
[8] H E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 pp 1283ndash1301 1985
[9] T Hoggard A E Whalley and P Wilmott ldquoHedging optionportfolios in the presence of transaction costsrdquo AdvancedFutures and Options Research vol 7 pp 21ndash35 1994
[10] P Guasoni ldquoNo arbitrage under transaction costs with frac-tional Brownian motion and beyondrdquo Mathematical Financevol 16 no 3 pp 569ndash582 2006
[11] H-K Liu and J-J Chang ldquoA closed-form approximation forthe fractional Black-Scholes model with transaction costsrdquoComputers amp Mathematics with Applications vol 65 no 11 pp1719ndash1726 2013
[12] X-T Wang ldquoScaling and long-range dependence in optionpricing I pricing European option with transaction costs underthe fractional Black-Scholes modelrdquo Physica A vol 389 no 3pp 438ndash444 2010
[13] X-T Wang E-H Zhu M-M Tang and H-G Yan ldquoScalingand long-range dependence in option pricing II pricing Euro-pean option with transaction costs under the mixed Brownian-fractional Brownian modelrdquo Physica A vol 389 no 3 pp 445ndash451 2010
[14] X-T Wang H-G Yan M-M Tang and E-H Zhu ldquoScalingand long-range dependence in option pricing III a fractionalversion of the Merton model with transaction costsrdquo Physica Avol 389 no 3 pp 452ndash458 2010
[15] X-T Wang ldquoScaling and long range dependence in optionpricing IV pricing European options with transaction costsunder the multifractional Black-Scholes modelrdquo Physica A vol389 no 4 pp 789ndash796 2010
[16] X-T Wang M Wu Z-M Zhou and W-S Jing ldquoPricingEuropean option with transaction costs under the fractionallong memory stochastic volatility modelrdquo Physica A vol 391no 4 pp 1469ndash1480 2012
[17] Y Hu and B Oslashksendal ldquoFractional white noise calculus andapplications to financerdquo Infinite Dimensional Analysis Quan-tum Probability and Related Topics vol 6 no 1 pp 1ndash32 2003
[18] C Bender ldquoAn Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameterrdquoStochastic Processes and Their Applications vol 104 no 1 pp81ndash106 2003
[19] J Li-Shang Mathematical Modeling and Methods of OptionPricing Higher Education Press Beijing China 2nd edition2008
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
8 Journal of Applied Mathematics
[8] H E Leland ldquoOption pricing and replication with transactionscostsrdquoThe Journal of Finance vol 40 pp 1283ndash1301 1985
[9] T Hoggard A E Whalley and P Wilmott ldquoHedging optionportfolios in the presence of transaction costsrdquo AdvancedFutures and Options Research vol 7 pp 21ndash35 1994
[10] P Guasoni ldquoNo arbitrage under transaction costs with frac-tional Brownian motion and beyondrdquo Mathematical Financevol 16 no 3 pp 569ndash582 2006
[11] H-K Liu and J-J Chang ldquoA closed-form approximation forthe fractional Black-Scholes model with transaction costsrdquoComputers amp Mathematics with Applications vol 65 no 11 pp1719ndash1726 2013
[12] X-T Wang ldquoScaling and long-range dependence in optionpricing I pricing European option with transaction costs underthe fractional Black-Scholes modelrdquo Physica A vol 389 no 3pp 438ndash444 2010
[13] X-T Wang E-H Zhu M-M Tang and H-G Yan ldquoScalingand long-range dependence in option pricing II pricing Euro-pean option with transaction costs under the mixed Brownian-fractional Brownian modelrdquo Physica A vol 389 no 3 pp 445ndash451 2010
[14] X-T Wang H-G Yan M-M Tang and E-H Zhu ldquoScalingand long-range dependence in option pricing III a fractionalversion of the Merton model with transaction costsrdquo Physica Avol 389 no 3 pp 452ndash458 2010
[15] X-T Wang ldquoScaling and long range dependence in optionpricing IV pricing European options with transaction costsunder the multifractional Black-Scholes modelrdquo Physica A vol389 no 4 pp 789ndash796 2010
[16] X-T Wang M Wu Z-M Zhou and W-S Jing ldquoPricingEuropean option with transaction costs under the fractionallong memory stochastic volatility modelrdquo Physica A vol 391no 4 pp 1469ndash1480 2012
[17] Y Hu and B Oslashksendal ldquoFractional white noise calculus andapplications to financerdquo Infinite Dimensional Analysis Quan-tum Probability and Related Topics vol 6 no 1 pp 1ndash32 2003
[18] C Bender ldquoAn Ito formula for generalized functionals of afractional Brownian motion with arbitrary Hurst parameterrdquoStochastic Processes and Their Applications vol 104 no 1 pp81ndash106 2003
[19] J Li-Shang Mathematical Modeling and Methods of OptionPricing Higher Education Press Beijing China 2nd edition2008
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
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