Research ArticleThe Randomized American Option as a Classical Solution tothe Penalized Problem

Guillaume Leduc

American University of Sharjah PO Box 26666 Sharjah UAE

Correspondence should be addressed to Guillaume Leduc gleducausedu

Received 13 August 2015 Accepted 7 October 2015

Academic Editor Hugo Leiva

Copyright copy 2015 Guillaume Leduc 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

We connect the exercisability randomized American option to the penalty method by showing that the randomized Americanoption value 119906 is the unique classical solution to the Cauchy problem corresponding to the canonical penalty problem for Americanoptions We also establish a uniform bound for A119906 where A is the infinitesimal generator of a geometric Brownian motion

1 Introduction

TheBlack-Scholesmodel is the lingua franca [1] the vehicularlanguage of option pricing Yet pricing American options inthe Black-Scholes remains the topic of a significant literaturebecause every known method requires significant numericalcalculations For surveys onAmerican option pricingwe referto [2ndash4]

Among all methods Howison et al [5] argue that with aquadratic convergence [6] the so-called penalty method forthe value of the American option in the Black-Scholes modelis ldquothe most efficient numerical approximation methodspresently available for American option valuationrdquo It wasused among others in [5ndash14]The penalty method transformsthe free boundary problem associated with the price of theAmerican option into a partial differential equation (PDE) ofthe form

120597

120597119905

119906 (119909 119905) = A119906 (119909 119905) minus 119903119906 (119909 119905) + 120573 (ℎ (119909) 119906 (119909 119905))

119906 (119909 0) = ℎ (119909)

(1)

where

A119891 (119909) = 1199031199091198911015840

(119909) +

1

2

1199092120590211989110158401015840

(119909) (2)

is the infinitesimal generator of a risk neutral geometricBrownian motion 120585with volatility 120590 and 119903 is the risk-free rateand where the penalty term 120573(ℎ(119909) 119906(119909 119905)) is a term which

typically is zero when ℎ(119909) le 119906(119909 119905) allowing the option tobehave like a European option and which drastically pushesthe value of the option higher when ℎ(119909) gt 119906(119909 119905) Indeedthe ldquocanonicalrdquo [5] penalty term is

120573 (ℎ (119909) 119906 (119909 119905)) = max (ℎ (119909) minus 119906 (119909 119905) 0) 119899 (3)

for some large value of 119899 gt 0In studies [5 6 9ndash12 14] where such canonical penalty

method was used for approximating the American optionvalue in the Black-Scholes model the solution to the asso-ciated PDE is seen as a viscosity solution or as a weaksolution that is the solution to a variational problem It is wellknown that in general viscosity and weak solutions do notpossess the regularity properties of classical solutions whichcan actually be differentiated in the classical sense to solvethe PDE

In this paper we connect the randomized Americanoption [15] to the penaltymethod showing that not only doesits value 119906 solve the canonical penalty problem (1) but also itis a classical solution to this Cauchy problem and for a givenmaturity A119906 is bounded

Many of the above cited papers using the canonicalpenalty method as well as other papers using penalizedproblems such as [16 17] were actually concerned not byestimating the value of American options but rather bydetermining the exact speed of convergence of option valuesunder tree schemes approximations of the Black-Scholesmodel a difficult [18] and long lasting problem still unsolved

Hindawi Publishing CorporationJournal of Function SpacesVolume 2015 Article ID 245436 5 pageshttpdxdoiorg1011552015245436

2 Journal of Function Spaces

when thematurity is not allowed to float Indeed randomizedAmerican options can be used as a tool to help determinethis exact speed of convergence It is well known that payoffsmoothness drastically affects this rate of convergence Webelieve that our resultmay contribute to solving this problemYet the submitted paper answers the very natural question ofwhether or not the canonical penalty problem has a classicalsolution

A randomly exercisable American option is an optionwhich prior to maturity can be exercised only at someexercisable times following each other independently after anexponentially distributed waiting time of average 1119899 Underthe label ldquooption with random intervention timerdquo randomlyexercisable American optionswere first introduced inDupuisand Wang [19] for American perpetuities then in Guo andLiu [20] for American lookback perpetuities and then inLeduc [15] for American options Note that the exercisabilityrandomizedAmerican option considered in this paper differsfrom Carrrsquos maturity randomized option [21] which can beexercised any time up to some random maturity In contrastthe exercisability randomized option can be exercised only atrandom times up to a fixed maturity

We denote by VR119899119905

ℎ(119909) the value of a randomly exercisableAmerican optionwithmaturity 119905 and payoff function ℎ whenthe spot price 120585

0of the underlying at time 0 is 119909 The value of

this randomized American option VR119899119905

ℎ(119909) is given by

VR119899119905

ℎ (119909)

def= sup120591isinT119899[0119905]

119864119909(119890minus119903120591

ℎ (120585120591)) (4)

whereT119899[0 119905] is the set of exercisable stopping times in [0 119905]and where 119864

119909is the expectation of 120585 given that 120585

0= 119909 As

shown in [15] VR119899119905

ℎ(119909) is the only solution to the followingevolution equation

VR119899119905

ℎ (119909) = 119890minus119899119905

E119905ℎ (119909)

+ int

119905

0

E119904(max (ℎ VR119899

119905minus119904ℎ)) (119909) 119899119890

minus119899119904119889119904

(5)

where for functions 120595 R rarr R the expression E119905120595(119909)

denotes the discounted expectation

E119905120595 (119909)

def= 119890minus119903119905

119864119909(120595 (120585119905)) (6)

It is also shown in [15] that VR119899119905

ℎ(119909) solves

VR119899119905

ℎ (119909) = U119905ℎ (119909) + int

119905

0

U119904(G119899119905minus119904ℎ) (119909) 119889119904 (7)

where

G119899119905minus119904ℎ (119910)

def= max (ℎ (119910) minus VR119899

119905minus119904ℎ (119910) 0) 119899 minus VR119899

119905minus119904ℎ (119910) 119903 (8)

and where U is the semigroup associated with 120585 that is forfunctions 120595 R rarr R

U119905120595 (119909)

def= 119864119909(120595 (120585119905)) (9)

Recall that a Lipschitz function ℎ R+

rarr R+is

absolutely continuous and almost surely differentiable In aslight abuse of notion we replace the Lipschitz constant 119862 ofℎ by ℎ1015840

infinso that for every 119909 119910 isin R

+

1003816100381610038161003816ℎ (119909) minus ℎ (119910)

1003816100381610038161003816le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 (10)

Finally we denote by 119868 the identity function 119868(119911) = 119911 forevery 119911

Theorem 1 If ℎ is a Lipschitz function and 119868ℎ1015840infin

lt infin thenVR119899119905

ℎ(119909) is the unique classical solution to the Cauchy problem

120597

120597119905

119906 (119909 119905) = 119860119906 (119909 119905) minus 119903119906 (119909 119905)

+max (ℎ (119909) minus 119906 (119909 119905) 0) 119899

119906 (119909 0) = ℎ (119909)

(11)

Furthermore

10038161003816100381610038161003816100381610038161003816

120597

120597119909

VR119899119905

ℎ (119909)

10038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

(12)

and for every 119879 there exists a constant 119876 depending only on 119903120590 119879 ℎ

infin and 119868ℎ

1015840infin

such that for every 0 lt 119905 le 119879 and0 le 119909

10038161003816100381610038161003816100381610038161003816

119909

120597

120597119909

VR119899119905

ℎ (119909)

10038161003816100381610038161003816100381610038161003816

le 119876

100381610038161003816100381610038161003816100381610038161003816

1199092 1205972

1205971199092VR119899119905

ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

119876

radic119905

(13)

The proof of our main result is divided into severalsteps In Section 2 we show that VR119899

119905ℎ(119909) is continuous In

Section 3 we prove that VR119899119905

ℎ(119909) is Lipschitzwith respect to119909In Section 4 we show that VR119899

119905ℎ(119909) is twice continuously dif-

ferentiable with respect to 119909 and the bounds for 119868(120597119889119909)VR119899119905

and 1198682(12059721198891199092)VR119899119905

are established In Section 5 we show thatVR119899119905

ℎ(119909) is a classical solution to (11)

2 Continuity

Fix some value 119879 gt 0 and for every function 119891 [0 119879] times

[0infin) rarr R set

1003817100381710038171003817119891 (119905 119909)

1003817100381710038171003817infin

def= sup119905isin[0119879]

sup119909ge0

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816 (14)

Furthermore define

VR119899119905

ℎ (119909)

def= lim(119904119906)rarr (1199051)

VR119899119904

ℎ (119906119909)

VR119899119905

ℎ (119909)

def= lim(119904119906)rarr (1199051)

VR119899119904

ℎ (119906119909)

(15)

Journal of Function Spaces 3

where (119904 119906) isin [0 119879] times [0infin) One easily gets from (5) that

VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

le int

119905

0

119890minus119899119904

119890minus119903119904

119864max (ℎ (119909120585119904) VR119899119905minus119904

ℎ (119909120585119904)) 119899 119889119904

minus int

119905

0

119890minus119899119904

119890minus119903119904

119864max (ℎ (119909120585119904) VR119899119905minus119904

ℎ (119909120585119904)) 119899 119889119904

le int

119879

0

119890minus119899119904 10038171003817

100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

119899 119889119904

(16)

yielding

10038171003817100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

le (1 minus 119890minus119899119879

)

10038171003817100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

(17)

which is possible only if VR119899119905

ℎ(119909) minus VR119899119905

ℎ(119909)infin

= 0 showingthat VR119899

119905ℎ(119909) is continuous

3 Lipschitz Property

We show here that VR119899119905

ℎ(119909) is Lipschitz with respect to 119909

Lemma 2 Under the assumptions of Theorem 1 for every 119909 ge

0 and every 120576 gt 0

10038161003816100381610038161003816VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909)

10038161003816100381610038161003816le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576 (18)

Proof Note first that because ℎ is Lipschitz for every 0 le 120572 120573

there exists a quantity 120574 with |120574| le ℎ1015840infin

such that

ℎ (120572) = ℎ (120573) + 120574 (120572 minus 120573) (19)

Let 120591119909and 120591

119909+120576be respectively the optimal (exercisable)

stopping time for the randomized American option withmaturity 119905 when 120585

0= 119909 and when 120585

0= 119909 + 120576

Step 1 Note first that when 1205850= 119909 + 120576 the stopping time 120591

119909is

suboptimal and therefore

VR119899119905

ℎ (119909 + 120576) = 119864 (119890minus119903120591119909+120576

ℎ ((119909 + 120576) 120585120591119909+120576

))

ge 119864 (119890minus119903120591119909

ℎ ((119909 + 120576) 120585120591119909

))

= 119864 (119890minus119903120591119909

ℎ (119909120585120591119909

) + 120574119890minus119903120591119909

120585120591119909

120576)

= VR119899119905

ℎ (119909) + 119864 (120574119890minus119903120591119909

120585120591119909

120576)

(20)

for some random variable 120574 satisfying |120574| le ℎ1015840infin Hence

since 120585 is risk neutral

VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909) ge minus

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

119864 (119890minus119903120591119909

120585120591119909

120576)

= minus

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576

(21)

Step 2 By definition

VR119899119905

ℎ (119909 + 120576) = 119864 (119890minus119903120591119909+120576

ℎ ((119909 + 120576) 120585120591119909+120576

))

= 119864 (119890minus119903120591119909+120576

ℎ (119909120585120591119909+120576

) + 120574119890minus119903120591119909+120576

120576120585120591119909+120576

)

(22)

for some random variable 120574 satisfying |120574| le ℎ1015840infin By

suboptimality of 120591119909+120576

when 1205850= 119909 we conclude that

VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909) le 119864 (120574119890minus119903120591119909+120576

120576120585120591119909+120576

)

le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

119864 (119890minus119903120591119909+120576

120576120585120591119909+120576

) =

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576

(23)

4 119862(2) Solution

Define

120577119904(119911)

def= exp (radic119904120590119911 + (119903 minus

1

2

1205902) 119904) (24)

and for functions 119891 R rarr R and integers ℓ ge 0 and 119904 gt 0let

E(ℓ)

119904(119891) (119909)

def= 119890minus119903119904

int

infin

minusinfin

119891 (119909120577119904(119911)) 120601

(ℓ)

(119911) 119889119911 (25)

where 120601 is the probability density function of a standardnormal random variable Note that if 119891 is bounded then

10038171003817100381710038171003817E(0)

119904(119891)

10038171003817100381710038171003817infin

10038171003817100381710038171003817E(1)

119904(119891)

10038171003817100381710038171003817infin

le10038171003817100381710038171198911003817100381710038171003817infin

(26)

For any family of functions 119891119905 R rarr R 0 le 119905 le 119879 set

1003817100381710038171003817119891sdot

1003817100381710038171003817infin

def= sup0le119905le119879

1003817100381710038171003817119891119905

1003817100381710038171003817infin

(27)

A consequence of Lemma 2 is that since ℎ is bounded andLipschitz

m119905ℎ (119909)

def= max (ℎ (119909) VR119899

119905ℎ (119909)) (28)

is also a bounded Lipschitz function It follows fromTheorem A1 in Appendix that 119864

119909(m119906(120585119906)) is infinitely differ-

entiable and that there exist constants 1205720such that

119909119890minus119903119906 120597

120597119909

119864119909(m119906(120585119906)) = 120572

0radic119906

minus1

E(1)

119906m119906ℎ (119909)

le10038161003816100381610038161205720

1003816100381610038161003816radic119906

minus1 1003817100381710038171003817m119906ℎ (119909)

1003817100381710038171003817infin

(29)

Because msdotℎinfin

le ℎinfin it follows from (5) that

119909

120597

120597119909

VR119899119905

ℎ (119909)

= 119890minus119899119905

119909

120597

120597119909

E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

(119909119890minus119903119906 120597

120597119909

119864119909max (m

119906(120585119906))) 119889119906

= 119890minus119899119905

119909

120597

120597119909

E119905ℎ (119909)

+ int

119905

0

119890minus119899119904

1205720radic119906

minus1

E(1)

119906m119905minus119906

ℎ (119909) 119899 119889119906

(30)

4 Journal of Function Spaces

Using Lemma A2 in Appendix we get10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

E119905ℎ

10038171003817100381710038171003817100381710038171003817infin

=

10038171003817100381710038171003817E119905(119868ℎ1015840) (119909)

10038171003817100381710038171003817infin

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

(31)

From this we obtain10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

VR119899sdot

ℎ

10038171003817100381710038171003817100381710038171003817infin

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

+ ℎinfinradic119879

10038161003816100381610038161205720

1003816100381610038161003816 (32)

For a fixed 119899 set

m1015840

119906ℎ (119911)

def=

120597

120597119911

max (ℎ (119911) VR119899119906

ℎ (119911)) (33)

which is Lebesgue almost everywhere sincem119906ℎ is Lipschitz

Note that

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

le (

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

VR119899sdot

ℎ

10038171003817100381710038171003817100381710038171003817infin

) lt infin (34)

Again from Theorem A1 in Appendix there exist constants1205730and 120573

1such that

1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))

= 1205730E(0)

119906(119868m1015840

119906ℎ) (119909) + 120573

1radic119906

minus1

E(1)

119906(119868m1015840

119906ℎ) (119909)

(35)

The fact that 119868m1015840sdotℎinfin

lt infin implies that100381610038161003816100381610038161003816100381610038161003816

1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))

100381610038161003816100381610038161003816100381610038161003816

le (10038161003816100381610038161205730

1003816100381610038161003816+10038161003816100381610038161205731

1003816100381610038161003816radic119906

minus1

)

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

(36)

This in turn yields

1199092 1205972

1205971199092VR119899119905

ℎ (119909)

= 119890minus119899119905

1199092 1205972

1205971199092E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

(1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))) 119889119906

= 119890minus119899119905

1199092 1205972

1205971199092E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

1

sum

ℓ=0

120573ℓradic119906

minusℓ

E(ℓ)

119906(119868m1015840

119905minus119906) (119909) 119889119906

(37)

As 119864(m119906(119909120585119906)) is infinitely differentiable with respect to 119909 gt

0 function 1199092 exp(minus119903119906)(12059721205971199092)119864

119909max(m

119906(120585119906)) is continu-

ous in 119909 gt 0 and with (36) dominated convergence givesthat 1199092(12059721205971199092)VR119899

119905ℎ(119909) is continuous FromTheorem A1 in

Appendix we obtain that for some constant119870

sup0le119905le119879

sup0lt119909

100381610038161003816100381610038161003816100381610038161003816

radic1199051199092 1205972

1205971199092E119905ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

119870 (38)

and hence

sup0le119905le119879

sup0lt119909

100381610038161003816100381610038161003816100381610038161003816

radic1199051199092 1205972

1205971199092VR119899119905

ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

119870 +10038161003816100381610038161205730

1003816100381610038161003816radic119879

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

+10038161003816100381610038161205731

1003816100381610038161003816119879

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

(39)

5 Classical Solution

Recall G119899119905ℎ from (8) Equation (7) can be rewritten as

VR119899119905

ℎ (119909) = U119905(ℎ) (119909) + int

119905

0

U119905minus119906

(G119899119906ℎ) (119909) 119889119906 (40)

from which it follows that for 0 lt 119905 119905 + 120576 le 119879

(

U120576minus 119868

120576

) VR119899119905

ℎ (119909) =

VR119899119905+120576

ℎ minus VR119899119905

ℎ (119909)

120576

minus

1

120576

int

119905+120576

119905

U119905+120576minus119906

(G119899119906ℎ) 119889119906

(41)

Letting 120576 go to zero we get

AVR119899119905

ℎ (119909) =

120597

120597119905

VR119899119905

ℎ (119909) minus G119899119905ℎ (119909) (42)

where A is the infinitesimal generator of the GBM In otherwords

120597

120597119905

VR119899119905

ℎ (119909) = AVR119899119905

ℎ (119909)

+max (ℎ (119909) minus VR119899119905

ℎ (119909) 0) 119899

minus VR119899119905

ℎ (119909) 119903

(43)

6 Conclusion

In this paper we showed that the exercisability randomizedAmerican option is a classical solution (and therefore the onlyclassical solution) to the canonical penalized problem Thisrelationship can be extended to a broader class ofmodels thanthe Black-Scholesmodel Indeed the key property is to obtainuniform bounds for 119909119896(120597119896120597119909119896)E

119904ℎ(119909) in terms of powers of

1radic119904 as inTheorem A1 in Appendix

Appendix

Theorem A1 If ℎ is a bounded Lipschitz function then forevery integer 119896 ge 0 and every 119904 gt 0 there exist real numbers1205721 120572

119896 such that

119909119896 120597119896

120597119909119896E119904ℎ (119909) =

119896

sum

ℓ=1

120572ℓradic119904

minusℓ

E(ℓ)

119904ℎ (119909) (A1)

If additionally 119896 ge 1 there exist real numbers1205730 120573

119896minus1 such

that

119909119896 120597119896

120597119909119896E119904ℎ (119909) =

119896minus1

sum

ℓ=0

120573ℓradic119904

minusℓ

E(ℓ)

119904(119868ℎ1015840) (119909) (A2)

Journal of Function Spaces 5

Proof See [22 Th 41] where the assumption that ℎ is poly-nomially bounded continuous and piecewise continuouslydifferentiable with polynomially bounded derivative can bereplaced by our assumptions on ℎ without any change inthe argument as a Lipschitz function is Lebesgue almosteverywhere differentiable and absolutely continuous allowingthe required integration by parts

LemmaA2 If ℎ is a bounded Lipschitz function then for every119904 gt 0

119909

120597

120597119909

E119904ℎ (119909) = E

119904(119868ℎ1015840) (119909) (A3)

Proof As ℎ is Lipschitz it is Lebesgue almost everywheredifferentiable and the proof of [22 Lemma 63] can befollowed without any change

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] P Carr ldquoDeriving derivatives of derivative securitiesrdquo in Pro-ceedings of the IEEEIAFEINFORMS Conference on Computa-tional Intelligence for Financial Engineering (CIFEr rsquo00) pp 101ndash128 IEEE New York NY USA March 2000

[2] M Broadie and J B Detemple ldquoANNIVERSARY ARTICLEoption pricing valuation models and applicationsrdquo Manage-ment Science vol 50 no 9 pp 1145ndash1177 2004

[3] S-R Ahn H-O Bae H-K Koo and K-J Lee ldquoA survey onamerican options old approaches and new trendsrdquo Bulletin ofthe KoreanMathematical Society vol 48 no 4 pp 791ndash812 2011

[4] S-P Zhu ldquoOn various quantitative approaches for pricingamerican optionsrdquoNewMathematics andNatural Computationvol 7 no 2 pp 313ndash332 2011

[5] S D Howison C Reisinger and J H Witte ldquoThe effect ofnonsmooth payoffs on the penalty approximation of americanoptionsrdquo SIAM Journal on Financial Mathematics vol 4 no 1pp 539ndash574 2013

[6] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[7] B F Nielsen O Skavhaug and A Tveito ldquoPenalty and front-fixing methods for the numerical solution of american optionproblemsrdquo Journal of Computational Finance vol 5 no 4 pp69ndash98 2002

[8] A Q M Khaliq D A Voss and S H K Kazmi ldquoA lin-early implicit predictor-corrector scheme for pricing Americanoptions using a penalty method approachrdquo Journal of Bankingamp Finance vol 30 no 2 pp 489ndash502 2006

[9] SWang X Q Yang and K L Teo ldquoPower penaltymethod for alinear complementarity problem arising from American optionvaluationrdquo Journal of OptimizationTheory andApplications vol129 no 2 pp 227ndash254 2006

[10] J Liang BHu L Jiang andB Bian ldquoOn the rate of convergenceof the binomial tree scheme for American optionsrdquoNumerischeMathematik vol 107 no 2 pp 333ndash352 2007

[11] B Hu J Liang and L Jiang ldquoOptimal convergence rate of theexplicit finite difference scheme forAmerican option valuationrdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 583ndash599 2009

[12] C Christara and D M Dang ldquoAdaptive and high-order meth-ods for valuing american optionsrdquo Journal of ComputationalFinance vol 14 no 4 pp 73ndash113 2011

[13] Z Cen A Le and A Xu ldquoA second-order difference schemefor the penalized blackndashscholes equation governing americanput option pricingrdquoComputational Economics vol 40 no 1 pp49ndash62 2012

[14] S Memon ldquoFinite element method for American option pric-ing a penalty approachrdquo International Journal of NumericalAnalysis and Modelling Series B Computing and Informationvol 3 no 3 pp 345ndash370 2012

[15] G Leduc ldquoExercisability randomization of the Americanoptionrdquo Stochastic Analysis and Applications vol 26 no 4 pp832ndash855 2008

[16] D Lamberton ldquoError estimates for the binomial approximationof American put optionsrdquo Annals of Applied Probability vol 8no 1 pp 206ndash233 1998

[17] D Lamberton ldquoBrownian optimal stopping and randomwalksrdquoApplied Mathematics and Optimization vol 45 no 3 pp 283ndash324 2002

[18] D Lamberton Optimal Stopping and American Options Sum-mer School on Financial Mathematics Ljubljana Slovenia2009

[19] P Dupuis and H Wang ldquoOptimal stopping with randomintervention timesrdquoAdvances in Applied Probability vol 34 no1 pp 141ndash157 2002

[20] X Guo and J Liu ldquoStopping at the maximum of geometricBrownian motion when signals are receivedrdquo Journal of AppliedProbability vol 42 no 3 pp 826ndash838 2005

[21] P Carr ldquoRandomization and the American putrdquo Review ofFinancial Studies vol 11 no 3 pp 597ndash626 1998

[22] G Leduc ldquoA European option general first-order error for-mulardquoThe ANZIAM Journal vol 54 no 4 pp 248ndash272 2013

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

2 Journal of Function Spaces

when thematurity is not allowed to float Indeed randomizedAmerican options can be used as a tool to help determinethis exact speed of convergence It is well known that payoffsmoothness drastically affects this rate of convergence Webelieve that our resultmay contribute to solving this problemYet the submitted paper answers the very natural question ofwhether or not the canonical penalty problem has a classicalsolution

A randomly exercisable American option is an optionwhich prior to maturity can be exercised only at someexercisable times following each other independently after anexponentially distributed waiting time of average 1119899 Underthe label ldquooption with random intervention timerdquo randomlyexercisable American optionswere first introduced inDupuisand Wang [19] for American perpetuities then in Guo andLiu [20] for American lookback perpetuities and then inLeduc [15] for American options Note that the exercisabilityrandomizedAmerican option considered in this paper differsfrom Carrrsquos maturity randomized option [21] which can beexercised any time up to some random maturity In contrastthe exercisability randomized option can be exercised only atrandom times up to a fixed maturity

We denote by VR119899119905

ℎ(119909) the value of a randomly exercisableAmerican optionwithmaturity 119905 and payoff function ℎ whenthe spot price 120585

0of the underlying at time 0 is 119909 The value of

this randomized American option VR119899119905

ℎ(119909) is given by

VR119899119905

ℎ (119909)

def= sup120591isinT119899[0119905]

119864119909(119890minus119903120591

ℎ (120585120591)) (4)

whereT119899[0 119905] is the set of exercisable stopping times in [0 119905]and where 119864

119909is the expectation of 120585 given that 120585

0= 119909 As

shown in [15] VR119899119905

ℎ(119909) is the only solution to the followingevolution equation

VR119899119905

ℎ (119909) = 119890minus119899119905

E119905ℎ (119909)

+ int

119905

0

E119904(max (ℎ VR119899

119905minus119904ℎ)) (119909) 119899119890

minus119899119904119889119904

(5)

where for functions 120595 R rarr R the expression E119905120595(119909)

denotes the discounted expectation

E119905120595 (119909)

def= 119890minus119903119905

119864119909(120595 (120585119905)) (6)

It is also shown in [15] that VR119899119905

ℎ(119909) solves

VR119899119905

ℎ (119909) = U119905ℎ (119909) + int

119905

0

U119904(G119899119905minus119904ℎ) (119909) 119889119904 (7)

where

G119899119905minus119904ℎ (119910)

def= max (ℎ (119910) minus VR119899

119905minus119904ℎ (119910) 0) 119899 minus VR119899

119905minus119904ℎ (119910) 119903 (8)

and where U is the semigroup associated with 120585 that is forfunctions 120595 R rarr R

U119905120595 (119909)

def= 119864119909(120595 (120585119905)) (9)

Recall that a Lipschitz function ℎ R+

rarr R+is

absolutely continuous and almost surely differentiable In aslight abuse of notion we replace the Lipschitz constant 119862 ofℎ by ℎ1015840

infinso that for every 119909 119910 isin R

+

1003816100381610038161003816ℎ (119909) minus ℎ (119910)

1003816100381610038161003816le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

1003816100381610038161003816119909 minus 119910

1003816100381610038161003816 (10)

Finally we denote by 119868 the identity function 119868(119911) = 119911 forevery 119911

Theorem 1 If ℎ is a Lipschitz function and 119868ℎ1015840infin

lt infin thenVR119899119905

ℎ(119909) is the unique classical solution to the Cauchy problem

120597

120597119905

119906 (119909 119905) = 119860119906 (119909 119905) minus 119903119906 (119909 119905)

+max (ℎ (119909) minus 119906 (119909 119905) 0) 119899

119906 (119909 0) = ℎ (119909)

(11)

Furthermore

10038161003816100381610038161003816100381610038161003816

120597

120597119909

VR119899119905

ℎ (119909)

10038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

(12)

and for every 119879 there exists a constant 119876 depending only on 119903120590 119879 ℎ

infin and 119868ℎ

1015840infin

such that for every 0 lt 119905 le 119879 and0 le 119909

10038161003816100381610038161003816100381610038161003816

119909

120597

120597119909

VR119899119905

ℎ (119909)

10038161003816100381610038161003816100381610038161003816

le 119876

100381610038161003816100381610038161003816100381610038161003816

1199092 1205972

1205971199092VR119899119905

ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

119876

radic119905

(13)

The proof of our main result is divided into severalsteps In Section 2 we show that VR119899

119905ℎ(119909) is continuous In

Section 3 we prove that VR119899119905

ℎ(119909) is Lipschitzwith respect to119909In Section 4 we show that VR119899

119905ℎ(119909) is twice continuously dif-

ferentiable with respect to 119909 and the bounds for 119868(120597119889119909)VR119899119905

and 1198682(12059721198891199092)VR119899119905

are established In Section 5 we show thatVR119899119905

ℎ(119909) is a classical solution to (11)

2 Continuity

Fix some value 119879 gt 0 and for every function 119891 [0 119879] times

[0infin) rarr R set

1003817100381710038171003817119891 (119905 119909)

1003817100381710038171003817infin

def= sup119905isin[0119879]

sup119909ge0

1003816100381610038161003816119891 (119905 119909)

1003816100381610038161003816 (14)

Furthermore define

VR119899119905

ℎ (119909)

def= lim(119904119906)rarr (1199051)

VR119899119904

ℎ (119906119909)

VR119899119905

ℎ (119909)

def= lim(119904119906)rarr (1199051)

VR119899119904

ℎ (119906119909)

(15)

Journal of Function Spaces 3

where (119904 119906) isin [0 119879] times [0infin) One easily gets from (5) that

VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

le int

119905

0

119890minus119899119904

119890minus119903119904

119864max (ℎ (119909120585119904) VR119899119905minus119904

ℎ (119909120585119904)) 119899 119889119904

minus int

119905

0

119890minus119899119904

119890minus119903119904

119864max (ℎ (119909120585119904) VR119899119905minus119904

ℎ (119909120585119904)) 119899 119889119904

le int

119879

0

119890minus119899119904 10038171003817

100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

119899 119889119904

(16)

yielding

10038171003817100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

le (1 minus 119890minus119899119879

)

10038171003817100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

(17)

which is possible only if VR119899119905

ℎ(119909) minus VR119899119905

ℎ(119909)infin

= 0 showingthat VR119899

119905ℎ(119909) is continuous

3 Lipschitz Property

We show here that VR119899119905

ℎ(119909) is Lipschitz with respect to 119909

Lemma 2 Under the assumptions of Theorem 1 for every 119909 ge

0 and every 120576 gt 0

10038161003816100381610038161003816VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909)

10038161003816100381610038161003816le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576 (18)

Proof Note first that because ℎ is Lipschitz for every 0 le 120572 120573

there exists a quantity 120574 with |120574| le ℎ1015840infin

such that

ℎ (120572) = ℎ (120573) + 120574 (120572 minus 120573) (19)

Let 120591119909and 120591

119909+120576be respectively the optimal (exercisable)

stopping time for the randomized American option withmaturity 119905 when 120585

0= 119909 and when 120585

0= 119909 + 120576

Step 1 Note first that when 1205850= 119909 + 120576 the stopping time 120591

119909is

suboptimal and therefore

VR119899119905

ℎ (119909 + 120576) = 119864 (119890minus119903120591119909+120576

ℎ ((119909 + 120576) 120585120591119909+120576

))

ge 119864 (119890minus119903120591119909

ℎ ((119909 + 120576) 120585120591119909

))

= 119864 (119890minus119903120591119909

ℎ (119909120585120591119909

) + 120574119890minus119903120591119909

120585120591119909

120576)

= VR119899119905

ℎ (119909) + 119864 (120574119890minus119903120591119909

120585120591119909

120576)

(20)

for some random variable 120574 satisfying |120574| le ℎ1015840infin Hence

since 120585 is risk neutral

VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909) ge minus

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

119864 (119890minus119903120591119909

120585120591119909

120576)

= minus

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576

(21)

Step 2 By definition

VR119899119905

ℎ (119909 + 120576) = 119864 (119890minus119903120591119909+120576

ℎ ((119909 + 120576) 120585120591119909+120576

))

= 119864 (119890minus119903120591119909+120576

ℎ (119909120585120591119909+120576

) + 120574119890minus119903120591119909+120576

120576120585120591119909+120576

)

(22)

for some random variable 120574 satisfying |120574| le ℎ1015840infin By

suboptimality of 120591119909+120576

when 1205850= 119909 we conclude that

VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909) le 119864 (120574119890minus119903120591119909+120576

120576120585120591119909+120576

)

le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

119864 (119890minus119903120591119909+120576

120576120585120591119909+120576

) =

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576

(23)

4 119862(2) Solution

Define

120577119904(119911)

def= exp (radic119904120590119911 + (119903 minus

1

2

1205902) 119904) (24)

and for functions 119891 R rarr R and integers ℓ ge 0 and 119904 gt 0let

E(ℓ)

119904(119891) (119909)

def= 119890minus119903119904

int

infin

minusinfin

119891 (119909120577119904(119911)) 120601

(ℓ)

(119911) 119889119911 (25)

where 120601 is the probability density function of a standardnormal random variable Note that if 119891 is bounded then

10038171003817100381710038171003817E(0)

119904(119891)

10038171003817100381710038171003817infin

10038171003817100381710038171003817E(1)

119904(119891)

10038171003817100381710038171003817infin

le10038171003817100381710038171198911003817100381710038171003817infin

(26)

For any family of functions 119891119905 R rarr R 0 le 119905 le 119879 set

1003817100381710038171003817119891sdot

1003817100381710038171003817infin

def= sup0le119905le119879

1003817100381710038171003817119891119905

1003817100381710038171003817infin

(27)

A consequence of Lemma 2 is that since ℎ is bounded andLipschitz

m119905ℎ (119909)

def= max (ℎ (119909) VR119899

119905ℎ (119909)) (28)

is also a bounded Lipschitz function It follows fromTheorem A1 in Appendix that 119864

119909(m119906(120585119906)) is infinitely differ-

entiable and that there exist constants 1205720such that

119909119890minus119903119906 120597

120597119909

119864119909(m119906(120585119906)) = 120572

0radic119906

minus1

E(1)

119906m119906ℎ (119909)

le10038161003816100381610038161205720

1003816100381610038161003816radic119906

minus1 1003817100381710038171003817m119906ℎ (119909)

1003817100381710038171003817infin

(29)

Because msdotℎinfin

le ℎinfin it follows from (5) that

119909

120597

120597119909

VR119899119905

ℎ (119909)

= 119890minus119899119905

119909

120597

120597119909

E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

(119909119890minus119903119906 120597

120597119909

119864119909max (m

119906(120585119906))) 119889119906

= 119890minus119899119905

119909

120597

120597119909

E119905ℎ (119909)

+ int

119905

0

119890minus119899119904

1205720radic119906

minus1

E(1)

119906m119905minus119906

ℎ (119909) 119899 119889119906

(30)

4 Journal of Function Spaces

Using Lemma A2 in Appendix we get10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

E119905ℎ

10038171003817100381710038171003817100381710038171003817infin

=

10038171003817100381710038171003817E119905(119868ℎ1015840) (119909)

10038171003817100381710038171003817infin

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

(31)

From this we obtain10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

VR119899sdot

ℎ

10038171003817100381710038171003817100381710038171003817infin

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

+ ℎinfinradic119879

10038161003816100381610038161205720

1003816100381610038161003816 (32)

For a fixed 119899 set

m1015840

119906ℎ (119911)

def=

120597

120597119911

max (ℎ (119911) VR119899119906

ℎ (119911)) (33)

which is Lebesgue almost everywhere sincem119906ℎ is Lipschitz

Note that

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

le (

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

VR119899sdot

ℎ

10038171003817100381710038171003817100381710038171003817infin

) lt infin (34)

Again from Theorem A1 in Appendix there exist constants1205730and 120573

1such that

1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))

= 1205730E(0)

119906(119868m1015840

119906ℎ) (119909) + 120573

1radic119906

minus1

E(1)

119906(119868m1015840

119906ℎ) (119909)

(35)

The fact that 119868m1015840sdotℎinfin

lt infin implies that100381610038161003816100381610038161003816100381610038161003816

1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))

100381610038161003816100381610038161003816100381610038161003816

le (10038161003816100381610038161205730

1003816100381610038161003816+10038161003816100381610038161205731

1003816100381610038161003816radic119906

minus1

)

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

(36)

This in turn yields

1199092 1205972

1205971199092VR119899119905

ℎ (119909)

= 119890minus119899119905

1199092 1205972

1205971199092E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

(1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))) 119889119906

= 119890minus119899119905

1199092 1205972

1205971199092E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

1

sum

ℓ=0

120573ℓradic119906

minusℓ

E(ℓ)

119906(119868m1015840

119905minus119906) (119909) 119889119906

(37)

As 119864(m119906(119909120585119906)) is infinitely differentiable with respect to 119909 gt

0 function 1199092 exp(minus119903119906)(12059721205971199092)119864

119909max(m

119906(120585119906)) is continu-

ous in 119909 gt 0 and with (36) dominated convergence givesthat 1199092(12059721205971199092)VR119899

119905ℎ(119909) is continuous FromTheorem A1 in

Appendix we obtain that for some constant119870

sup0le119905le119879

sup0lt119909

100381610038161003816100381610038161003816100381610038161003816

radic1199051199092 1205972

1205971199092E119905ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

119870 (38)

and hence

sup0le119905le119879

sup0lt119909

100381610038161003816100381610038161003816100381610038161003816

radic1199051199092 1205972

1205971199092VR119899119905

ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

119870 +10038161003816100381610038161205730

1003816100381610038161003816radic119879

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

+10038161003816100381610038161205731

1003816100381610038161003816119879

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

(39)

5 Classical Solution

Recall G119899119905ℎ from (8) Equation (7) can be rewritten as

VR119899119905

ℎ (119909) = U119905(ℎ) (119909) + int

119905

0

U119905minus119906

(G119899119906ℎ) (119909) 119889119906 (40)

from which it follows that for 0 lt 119905 119905 + 120576 le 119879

(

U120576minus 119868

120576

) VR119899119905

ℎ (119909) =

VR119899119905+120576

ℎ minus VR119899119905

ℎ (119909)

120576

minus

1

120576

int

119905+120576

119905

U119905+120576minus119906

(G119899119906ℎ) 119889119906

(41)

Letting 120576 go to zero we get

AVR119899119905

ℎ (119909) =

120597

120597119905

VR119899119905

ℎ (119909) minus G119899119905ℎ (119909) (42)

where A is the infinitesimal generator of the GBM In otherwords

120597

120597119905

VR119899119905

ℎ (119909) = AVR119899119905

ℎ (119909)

+max (ℎ (119909) minus VR119899119905

ℎ (119909) 0) 119899

minus VR119899119905

ℎ (119909) 119903

(43)

6 Conclusion

In this paper we showed that the exercisability randomizedAmerican option is a classical solution (and therefore the onlyclassical solution) to the canonical penalized problem Thisrelationship can be extended to a broader class ofmodels thanthe Black-Scholesmodel Indeed the key property is to obtainuniform bounds for 119909119896(120597119896120597119909119896)E

119904ℎ(119909) in terms of powers of

1radic119904 as inTheorem A1 in Appendix

Appendix

Theorem A1 If ℎ is a bounded Lipschitz function then forevery integer 119896 ge 0 and every 119904 gt 0 there exist real numbers1205721 120572

119896 such that

119909119896 120597119896

120597119909119896E119904ℎ (119909) =

119896

sum

ℓ=1

120572ℓradic119904

minusℓ

E(ℓ)

119904ℎ (119909) (A1)

If additionally 119896 ge 1 there exist real numbers1205730 120573

119896minus1 such

that

119909119896 120597119896

120597119909119896E119904ℎ (119909) =

119896minus1

sum

ℓ=0

120573ℓradic119904

minusℓ

E(ℓ)

119904(119868ℎ1015840) (119909) (A2)

Journal of Function Spaces 5

Proof See [22 Th 41] where the assumption that ℎ is poly-nomially bounded continuous and piecewise continuouslydifferentiable with polynomially bounded derivative can bereplaced by our assumptions on ℎ without any change inthe argument as a Lipschitz function is Lebesgue almosteverywhere differentiable and absolutely continuous allowingthe required integration by parts

LemmaA2 If ℎ is a bounded Lipschitz function then for every119904 gt 0

119909

120597

120597119909

E119904ℎ (119909) = E

119904(119868ℎ1015840) (119909) (A3)

Proof As ℎ is Lipschitz it is Lebesgue almost everywheredifferentiable and the proof of [22 Lemma 63] can befollowed without any change

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] P Carr ldquoDeriving derivatives of derivative securitiesrdquo in Pro-ceedings of the IEEEIAFEINFORMS Conference on Computa-tional Intelligence for Financial Engineering (CIFEr rsquo00) pp 101ndash128 IEEE New York NY USA March 2000

[2] M Broadie and J B Detemple ldquoANNIVERSARY ARTICLEoption pricing valuation models and applicationsrdquo Manage-ment Science vol 50 no 9 pp 1145ndash1177 2004

[3] S-R Ahn H-O Bae H-K Koo and K-J Lee ldquoA survey onamerican options old approaches and new trendsrdquo Bulletin ofthe KoreanMathematical Society vol 48 no 4 pp 791ndash812 2011

[4] S-P Zhu ldquoOn various quantitative approaches for pricingamerican optionsrdquoNewMathematics andNatural Computationvol 7 no 2 pp 313ndash332 2011

[5] S D Howison C Reisinger and J H Witte ldquoThe effect ofnonsmooth payoffs on the penalty approximation of americanoptionsrdquo SIAM Journal on Financial Mathematics vol 4 no 1pp 539ndash574 2013

[6] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[7] B F Nielsen O Skavhaug and A Tveito ldquoPenalty and front-fixing methods for the numerical solution of american optionproblemsrdquo Journal of Computational Finance vol 5 no 4 pp69ndash98 2002

[8] A Q M Khaliq D A Voss and S H K Kazmi ldquoA lin-early implicit predictor-corrector scheme for pricing Americanoptions using a penalty method approachrdquo Journal of Bankingamp Finance vol 30 no 2 pp 489ndash502 2006

[9] SWang X Q Yang and K L Teo ldquoPower penaltymethod for alinear complementarity problem arising from American optionvaluationrdquo Journal of OptimizationTheory andApplications vol129 no 2 pp 227ndash254 2006

[10] J Liang BHu L Jiang andB Bian ldquoOn the rate of convergenceof the binomial tree scheme for American optionsrdquoNumerischeMathematik vol 107 no 2 pp 333ndash352 2007

[11] B Hu J Liang and L Jiang ldquoOptimal convergence rate of theexplicit finite difference scheme forAmerican option valuationrdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 583ndash599 2009

[12] C Christara and D M Dang ldquoAdaptive and high-order meth-ods for valuing american optionsrdquo Journal of ComputationalFinance vol 14 no 4 pp 73ndash113 2011

[13] Z Cen A Le and A Xu ldquoA second-order difference schemefor the penalized blackndashscholes equation governing americanput option pricingrdquoComputational Economics vol 40 no 1 pp49ndash62 2012

[14] S Memon ldquoFinite element method for American option pric-ing a penalty approachrdquo International Journal of NumericalAnalysis and Modelling Series B Computing and Informationvol 3 no 3 pp 345ndash370 2012

[15] G Leduc ldquoExercisability randomization of the Americanoptionrdquo Stochastic Analysis and Applications vol 26 no 4 pp832ndash855 2008

[16] D Lamberton ldquoError estimates for the binomial approximationof American put optionsrdquo Annals of Applied Probability vol 8no 1 pp 206ndash233 1998

[17] D Lamberton ldquoBrownian optimal stopping and randomwalksrdquoApplied Mathematics and Optimization vol 45 no 3 pp 283ndash324 2002

[18] D Lamberton Optimal Stopping and American Options Sum-mer School on Financial Mathematics Ljubljana Slovenia2009

[19] P Dupuis and H Wang ldquoOptimal stopping with randomintervention timesrdquoAdvances in Applied Probability vol 34 no1 pp 141ndash157 2002

[20] X Guo and J Liu ldquoStopping at the maximum of geometricBrownian motion when signals are receivedrdquo Journal of AppliedProbability vol 42 no 3 pp 826ndash838 2005

[21] P Carr ldquoRandomization and the American putrdquo Review ofFinancial Studies vol 11 no 3 pp 597ndash626 1998

[22] G Leduc ldquoA European option general first-order error for-mulardquoThe ANZIAM Journal vol 54 no 4 pp 248ndash272 2013

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 Function Spaces 3

where (119904 119906) isin [0 119879] times [0infin) One easily gets from (5) that

VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

le int

119905

0

119890minus119899119904

119890minus119903119904

119864max (ℎ (119909120585119904) VR119899119905minus119904

ℎ (119909120585119904)) 119899 119889119904

minus int

119905

0

119890minus119899119904

119890minus119903119904

119864max (ℎ (119909120585119904) VR119899119905minus119904

ℎ (119909120585119904)) 119899 119889119904

le int

119879

0

119890minus119899119904 10038171003817

100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

119899 119889119904

(16)

yielding

10038171003817100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

le (1 minus 119890minus119899119879

)

10038171003817100381710038171003817VR119899119905

ℎ (119909) minus VR119899119905

ℎ (119909)

10038171003817100381710038171003817infin

(17)

which is possible only if VR119899119905

ℎ(119909) minus VR119899119905

ℎ(119909)infin

= 0 showingthat VR119899

119905ℎ(119909) is continuous

3 Lipschitz Property

We show here that VR119899119905

ℎ(119909) is Lipschitz with respect to 119909

Lemma 2 Under the assumptions of Theorem 1 for every 119909 ge

0 and every 120576 gt 0

10038161003816100381610038161003816VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909)

10038161003816100381610038161003816le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576 (18)

Proof Note first that because ℎ is Lipschitz for every 0 le 120572 120573

there exists a quantity 120574 with |120574| le ℎ1015840infin

such that

ℎ (120572) = ℎ (120573) + 120574 (120572 minus 120573) (19)

Let 120591119909and 120591

119909+120576be respectively the optimal (exercisable)

stopping time for the randomized American option withmaturity 119905 when 120585

0= 119909 and when 120585

0= 119909 + 120576

Step 1 Note first that when 1205850= 119909 + 120576 the stopping time 120591

119909is

suboptimal and therefore

VR119899119905

ℎ (119909 + 120576) = 119864 (119890minus119903120591119909+120576

ℎ ((119909 + 120576) 120585120591119909+120576

))

ge 119864 (119890minus119903120591119909

ℎ ((119909 + 120576) 120585120591119909

))

= 119864 (119890minus119903120591119909

ℎ (119909120585120591119909

) + 120574119890minus119903120591119909

120585120591119909

120576)

= VR119899119905

ℎ (119909) + 119864 (120574119890minus119903120591119909

120585120591119909

120576)

(20)

for some random variable 120574 satisfying |120574| le ℎ1015840infin Hence

since 120585 is risk neutral

VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909) ge minus

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

119864 (119890minus119903120591119909

120585120591119909

120576)

= minus

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576

(21)

Step 2 By definition

VR119899119905

ℎ (119909 + 120576) = 119864 (119890minus119903120591119909+120576

ℎ ((119909 + 120576) 120585120591119909+120576

))

= 119864 (119890minus119903120591119909+120576

ℎ (119909120585120591119909+120576

) + 120574119890minus119903120591119909+120576

120576120585120591119909+120576

)

(22)

for some random variable 120574 satisfying |120574| le ℎ1015840infin By

suboptimality of 120591119909+120576

when 1205850= 119909 we conclude that

VR119899119905

ℎ (119909 + 120576) minus VR119899119905

ℎ (119909) le 119864 (120574119890minus119903120591119909+120576

120576120585120591119909+120576

)

le

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

119864 (119890minus119903120591119909+120576

120576120585120591119909+120576

) =

10038171003817100381710038171003817ℎ101584010038171003817100381710038171003817infin

120576

(23)

4 119862(2) Solution

Define

120577119904(119911)

def= exp (radic119904120590119911 + (119903 minus

1

2

1205902) 119904) (24)

and for functions 119891 R rarr R and integers ℓ ge 0 and 119904 gt 0let

E(ℓ)

119904(119891) (119909)

def= 119890minus119903119904

int

infin

minusinfin

119891 (119909120577119904(119911)) 120601

(ℓ)

(119911) 119889119911 (25)

where 120601 is the probability density function of a standardnormal random variable Note that if 119891 is bounded then

10038171003817100381710038171003817E(0)

119904(119891)

10038171003817100381710038171003817infin

10038171003817100381710038171003817E(1)

119904(119891)

10038171003817100381710038171003817infin

le10038171003817100381710038171198911003817100381710038171003817infin

(26)

For any family of functions 119891119905 R rarr R 0 le 119905 le 119879 set

1003817100381710038171003817119891sdot

1003817100381710038171003817infin

def= sup0le119905le119879

1003817100381710038171003817119891119905

1003817100381710038171003817infin

(27)

A consequence of Lemma 2 is that since ℎ is bounded andLipschitz

m119905ℎ (119909)

def= max (ℎ (119909) VR119899

119905ℎ (119909)) (28)

is also a bounded Lipschitz function It follows fromTheorem A1 in Appendix that 119864

119909(m119906(120585119906)) is infinitely differ-

entiable and that there exist constants 1205720such that

119909119890minus119903119906 120597

120597119909

119864119909(m119906(120585119906)) = 120572

0radic119906

minus1

E(1)

119906m119906ℎ (119909)

le10038161003816100381610038161205720

1003816100381610038161003816radic119906

minus1 1003817100381710038171003817m119906ℎ (119909)

1003817100381710038171003817infin

(29)

Because msdotℎinfin

le ℎinfin it follows from (5) that

119909

120597

120597119909

VR119899119905

ℎ (119909)

= 119890minus119899119905

119909

120597

120597119909

E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

(119909119890minus119903119906 120597

120597119909

119864119909max (m

119906(120585119906))) 119889119906

= 119890minus119899119905

119909

120597

120597119909

E119905ℎ (119909)

+ int

119905

0

119890minus119899119904

1205720radic119906

minus1

E(1)

119906m119905minus119906

ℎ (119909) 119899 119889119906

(30)

4 Journal of Function Spaces

Using Lemma A2 in Appendix we get10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

E119905ℎ

10038171003817100381710038171003817100381710038171003817infin

=

10038171003817100381710038171003817E119905(119868ℎ1015840) (119909)

10038171003817100381710038171003817infin

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

(31)

From this we obtain10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

VR119899sdot

ℎ

10038171003817100381710038171003817100381710038171003817infin

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

+ ℎinfinradic119879

10038161003816100381610038161205720

1003816100381610038161003816 (32)

For a fixed 119899 set

m1015840

119906ℎ (119911)

def=

120597

120597119911

max (ℎ (119911) VR119899119906

ℎ (119911)) (33)

which is Lebesgue almost everywhere sincem119906ℎ is Lipschitz

Note that

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

le (

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

VR119899sdot

ℎ

10038171003817100381710038171003817100381710038171003817infin

) lt infin (34)

Again from Theorem A1 in Appendix there exist constants1205730and 120573

1such that

1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))

= 1205730E(0)

119906(119868m1015840

119906ℎ) (119909) + 120573

1radic119906

minus1

E(1)

119906(119868m1015840

119906ℎ) (119909)

(35)

The fact that 119868m1015840sdotℎinfin

lt infin implies that100381610038161003816100381610038161003816100381610038161003816

1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))

100381610038161003816100381610038161003816100381610038161003816

le (10038161003816100381610038161205730

1003816100381610038161003816+10038161003816100381610038161205731

1003816100381610038161003816radic119906

minus1

)

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

(36)

This in turn yields

1199092 1205972

1205971199092VR119899119905

ℎ (119909)

= 119890minus119899119905

1199092 1205972

1205971199092E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

(1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))) 119889119906

= 119890minus119899119905

1199092 1205972

1205971199092E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

1

sum

ℓ=0

120573ℓradic119906

minusℓ

E(ℓ)

119906(119868m1015840

119905minus119906) (119909) 119889119906

(37)

As 119864(m119906(119909120585119906)) is infinitely differentiable with respect to 119909 gt

0 function 1199092 exp(minus119903119906)(12059721205971199092)119864

119909max(m

119906(120585119906)) is continu-

ous in 119909 gt 0 and with (36) dominated convergence givesthat 1199092(12059721205971199092)VR119899

119905ℎ(119909) is continuous FromTheorem A1 in

Appendix we obtain that for some constant119870

sup0le119905le119879

sup0lt119909

100381610038161003816100381610038161003816100381610038161003816

radic1199051199092 1205972

1205971199092E119905ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

119870 (38)

and hence

sup0le119905le119879

sup0lt119909

100381610038161003816100381610038161003816100381610038161003816

radic1199051199092 1205972

1205971199092VR119899119905

ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

119870 +10038161003816100381610038161205730

1003816100381610038161003816radic119879

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

+10038161003816100381610038161205731

1003816100381610038161003816119879

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

(39)

5 Classical Solution

Recall G119899119905ℎ from (8) Equation (7) can be rewritten as

VR119899119905

ℎ (119909) = U119905(ℎ) (119909) + int

119905

0

U119905minus119906

(G119899119906ℎ) (119909) 119889119906 (40)

from which it follows that for 0 lt 119905 119905 + 120576 le 119879

(

U120576minus 119868

120576

) VR119899119905

ℎ (119909) =

VR119899119905+120576

ℎ minus VR119899119905

ℎ (119909)

120576

minus

1

120576

int

119905+120576

119905

U119905+120576minus119906

(G119899119906ℎ) 119889119906

(41)

Letting 120576 go to zero we get

AVR119899119905

ℎ (119909) =

120597

120597119905

VR119899119905

ℎ (119909) minus G119899119905ℎ (119909) (42)

where A is the infinitesimal generator of the GBM In otherwords

120597

120597119905

VR119899119905

ℎ (119909) = AVR119899119905

ℎ (119909)

+max (ℎ (119909) minus VR119899119905

ℎ (119909) 0) 119899

minus VR119899119905

ℎ (119909) 119903

(43)

6 Conclusion

In this paper we showed that the exercisability randomizedAmerican option is a classical solution (and therefore the onlyclassical solution) to the canonical penalized problem Thisrelationship can be extended to a broader class ofmodels thanthe Black-Scholesmodel Indeed the key property is to obtainuniform bounds for 119909119896(120597119896120597119909119896)E

119904ℎ(119909) in terms of powers of

1radic119904 as inTheorem A1 in Appendix

Appendix

Theorem A1 If ℎ is a bounded Lipschitz function then forevery integer 119896 ge 0 and every 119904 gt 0 there exist real numbers1205721 120572

119896 such that

119909119896 120597119896

120597119909119896E119904ℎ (119909) =

119896

sum

ℓ=1

120572ℓradic119904

minusℓ

E(ℓ)

119904ℎ (119909) (A1)

If additionally 119896 ge 1 there exist real numbers1205730 120573

119896minus1 such

that

119909119896 120597119896

120597119909119896E119904ℎ (119909) =

119896minus1

sum

ℓ=0

120573ℓradic119904

minusℓ

E(ℓ)

119904(119868ℎ1015840) (119909) (A2)

Journal of Function Spaces 5

Proof See [22 Th 41] where the assumption that ℎ is poly-nomially bounded continuous and piecewise continuouslydifferentiable with polynomially bounded derivative can bereplaced by our assumptions on ℎ without any change inthe argument as a Lipschitz function is Lebesgue almosteverywhere differentiable and absolutely continuous allowingthe required integration by parts

LemmaA2 If ℎ is a bounded Lipschitz function then for every119904 gt 0

119909

120597

120597119909

E119904ℎ (119909) = E

119904(119868ℎ1015840) (119909) (A3)

Proof As ℎ is Lipschitz it is Lebesgue almost everywheredifferentiable and the proof of [22 Lemma 63] can befollowed without any change

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] P Carr ldquoDeriving derivatives of derivative securitiesrdquo in Pro-ceedings of the IEEEIAFEINFORMS Conference on Computa-tional Intelligence for Financial Engineering (CIFEr rsquo00) pp 101ndash128 IEEE New York NY USA March 2000

[2] M Broadie and J B Detemple ldquoANNIVERSARY ARTICLEoption pricing valuation models and applicationsrdquo Manage-ment Science vol 50 no 9 pp 1145ndash1177 2004

[3] S-R Ahn H-O Bae H-K Koo and K-J Lee ldquoA survey onamerican options old approaches and new trendsrdquo Bulletin ofthe KoreanMathematical Society vol 48 no 4 pp 791ndash812 2011

[4] S-P Zhu ldquoOn various quantitative approaches for pricingamerican optionsrdquoNewMathematics andNatural Computationvol 7 no 2 pp 313ndash332 2011

[5] S D Howison C Reisinger and J H Witte ldquoThe effect ofnonsmooth payoffs on the penalty approximation of americanoptionsrdquo SIAM Journal on Financial Mathematics vol 4 no 1pp 539ndash574 2013

[6] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[7] B F Nielsen O Skavhaug and A Tveito ldquoPenalty and front-fixing methods for the numerical solution of american optionproblemsrdquo Journal of Computational Finance vol 5 no 4 pp69ndash98 2002

[8] A Q M Khaliq D A Voss and S H K Kazmi ldquoA lin-early implicit predictor-corrector scheme for pricing Americanoptions using a penalty method approachrdquo Journal of Bankingamp Finance vol 30 no 2 pp 489ndash502 2006

[9] SWang X Q Yang and K L Teo ldquoPower penaltymethod for alinear complementarity problem arising from American optionvaluationrdquo Journal of OptimizationTheory andApplications vol129 no 2 pp 227ndash254 2006

[10] J Liang BHu L Jiang andB Bian ldquoOn the rate of convergenceof the binomial tree scheme for American optionsrdquoNumerischeMathematik vol 107 no 2 pp 333ndash352 2007

[11] B Hu J Liang and L Jiang ldquoOptimal convergence rate of theexplicit finite difference scheme forAmerican option valuationrdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 583ndash599 2009

[12] C Christara and D M Dang ldquoAdaptive and high-order meth-ods for valuing american optionsrdquo Journal of ComputationalFinance vol 14 no 4 pp 73ndash113 2011

[13] Z Cen A Le and A Xu ldquoA second-order difference schemefor the penalized blackndashscholes equation governing americanput option pricingrdquoComputational Economics vol 40 no 1 pp49ndash62 2012

[14] S Memon ldquoFinite element method for American option pric-ing a penalty approachrdquo International Journal of NumericalAnalysis and Modelling Series B Computing and Informationvol 3 no 3 pp 345ndash370 2012

[15] G Leduc ldquoExercisability randomization of the Americanoptionrdquo Stochastic Analysis and Applications vol 26 no 4 pp832ndash855 2008

[16] D Lamberton ldquoError estimates for the binomial approximationof American put optionsrdquo Annals of Applied Probability vol 8no 1 pp 206ndash233 1998

[17] D Lamberton ldquoBrownian optimal stopping and randomwalksrdquoApplied Mathematics and Optimization vol 45 no 3 pp 283ndash324 2002

[18] D Lamberton Optimal Stopping and American Options Sum-mer School on Financial Mathematics Ljubljana Slovenia2009

[19] P Dupuis and H Wang ldquoOptimal stopping with randomintervention timesrdquoAdvances in Applied Probability vol 34 no1 pp 141ndash157 2002

[20] X Guo and J Liu ldquoStopping at the maximum of geometricBrownian motion when signals are receivedrdquo Journal of AppliedProbability vol 42 no 3 pp 826ndash838 2005

[21] P Carr ldquoRandomization and the American putrdquo Review ofFinancial Studies vol 11 no 3 pp 597ndash626 1998

[22] G Leduc ldquoA European option general first-order error for-mulardquoThe ANZIAM Journal vol 54 no 4 pp 248ndash272 2013

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

4 Journal of Function Spaces

Using Lemma A2 in Appendix we get10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

E119905ℎ

10038171003817100381710038171003817100381710038171003817infin

=

10038171003817100381710038171003817E119905(119868ℎ1015840) (119909)

10038171003817100381710038171003817infin

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

(31)

From this we obtain10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

VR119899sdot

ℎ

10038171003817100381710038171003817100381710038171003817infin

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

+ ℎinfinradic119879

10038161003816100381610038161205720

1003816100381610038161003816 (32)

For a fixed 119899 set

m1015840

119906ℎ (119911)

def=

120597

120597119911

max (ℎ (119911) VR119899119906

ℎ (119911)) (33)

which is Lebesgue almost everywhere sincem119906ℎ is Lipschitz

Note that

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

le (

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

10038171003817100381710038171003817100381710038171003817

119868

120597

120597119909

VR119899sdot

ℎ

10038171003817100381710038171003817100381710038171003817infin

) lt infin (34)

Again from Theorem A1 in Appendix there exist constants1205730and 120573

1such that

1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))

= 1205730E(0)

119906(119868m1015840

119906ℎ) (119909) + 120573

1radic119906

minus1

E(1)

119906(119868m1015840

119906ℎ) (119909)

(35)

The fact that 119868m1015840sdotℎinfin

lt infin implies that100381610038161003816100381610038161003816100381610038161003816

1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))

100381610038161003816100381610038161003816100381610038161003816

le (10038161003816100381610038161205730

1003816100381610038161003816+10038161003816100381610038161205731

1003816100381610038161003816radic119906

minus1

)

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

(36)

This in turn yields

1199092 1205972

1205971199092VR119899119905

ℎ (119909)

= 119890minus119899119905

1199092 1205972

1205971199092E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

(1199092119890minus119903119906 1205972

1205971199092119864119909max (m

119906(120585119906))) 119889119906

= 119890minus119899119905

1199092 1205972

1205971199092E119905ℎ (119909)

+ int

119905

0

119890minus119899119906

1

sum

ℓ=0

120573ℓradic119906

minusℓ

E(ℓ)

119906(119868m1015840

119905minus119906) (119909) 119889119906

(37)

As 119864(m119906(119909120585119906)) is infinitely differentiable with respect to 119909 gt

0 function 1199092 exp(minus119903119906)(12059721205971199092)119864

119909max(m

119906(120585119906)) is continu-

ous in 119909 gt 0 and with (36) dominated convergence givesthat 1199092(12059721205971199092)VR119899

119905ℎ(119909) is continuous FromTheorem A1 in

Appendix we obtain that for some constant119870

sup0le119905le119879

sup0lt119909

100381610038161003816100381610038161003816100381610038161003816

radic1199051199092 1205972

1205971199092E119905ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

119870 (38)

and hence

sup0le119905le119879

sup0lt119909

100381610038161003816100381610038161003816100381610038161003816

radic1199051199092 1205972

1205971199092VR119899119905

ℎ (119909)

100381610038161003816100381610038161003816100381610038161003816

le

10038171003817100381710038171003817119868ℎ101584010038171003817100381710038171003817infin

119870 +10038161003816100381610038161205730

1003816100381610038161003816radic119879

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

+10038161003816100381610038161205731

1003816100381610038161003816119879

10038171003817100381710038171003817119868m1015840

sdotℎ

10038171003817100381710038171003817infin

(39)

5 Classical Solution

Recall G119899119905ℎ from (8) Equation (7) can be rewritten as

VR119899119905

ℎ (119909) = U119905(ℎ) (119909) + int

119905

0

U119905minus119906

(G119899119906ℎ) (119909) 119889119906 (40)

from which it follows that for 0 lt 119905 119905 + 120576 le 119879

(

U120576minus 119868

120576

) VR119899119905

ℎ (119909) =

VR119899119905+120576

ℎ minus VR119899119905

ℎ (119909)

120576

minus

1

120576

int

119905+120576

119905

U119905+120576minus119906

(G119899119906ℎ) 119889119906

(41)

Letting 120576 go to zero we get

AVR119899119905

ℎ (119909) =

120597

120597119905

VR119899119905

ℎ (119909) minus G119899119905ℎ (119909) (42)

where A is the infinitesimal generator of the GBM In otherwords

120597

120597119905

VR119899119905

ℎ (119909) = AVR119899119905

ℎ (119909)

+max (ℎ (119909) minus VR119899119905

ℎ (119909) 0) 119899

minus VR119899119905

ℎ (119909) 119903

(43)

6 Conclusion

In this paper we showed that the exercisability randomizedAmerican option is a classical solution (and therefore the onlyclassical solution) to the canonical penalized problem Thisrelationship can be extended to a broader class ofmodels thanthe Black-Scholesmodel Indeed the key property is to obtainuniform bounds for 119909119896(120597119896120597119909119896)E

119904ℎ(119909) in terms of powers of

1radic119904 as inTheorem A1 in Appendix

Appendix

Theorem A1 If ℎ is a bounded Lipschitz function then forevery integer 119896 ge 0 and every 119904 gt 0 there exist real numbers1205721 120572

119896 such that

119909119896 120597119896

120597119909119896E119904ℎ (119909) =

119896

sum

ℓ=1

120572ℓradic119904

minusℓ

E(ℓ)

119904ℎ (119909) (A1)

If additionally 119896 ge 1 there exist real numbers1205730 120573

119896minus1 such

that

119909119896 120597119896

120597119909119896E119904ℎ (119909) =

119896minus1

sum

ℓ=0

120573ℓradic119904

minusℓ

E(ℓ)

119904(119868ℎ1015840) (119909) (A2)

Journal of Function Spaces 5

Proof See [22 Th 41] where the assumption that ℎ is poly-nomially bounded continuous and piecewise continuouslydifferentiable with polynomially bounded derivative can bereplaced by our assumptions on ℎ without any change inthe argument as a Lipschitz function is Lebesgue almosteverywhere differentiable and absolutely continuous allowingthe required integration by parts

LemmaA2 If ℎ is a bounded Lipschitz function then for every119904 gt 0

119909

120597

120597119909

E119904ℎ (119909) = E

119904(119868ℎ1015840) (119909) (A3)

Proof As ℎ is Lipschitz it is Lebesgue almost everywheredifferentiable and the proof of [22 Lemma 63] can befollowed without any change

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] P Carr ldquoDeriving derivatives of derivative securitiesrdquo in Pro-ceedings of the IEEEIAFEINFORMS Conference on Computa-tional Intelligence for Financial Engineering (CIFEr rsquo00) pp 101ndash128 IEEE New York NY USA March 2000

[2] M Broadie and J B Detemple ldquoANNIVERSARY ARTICLEoption pricing valuation models and applicationsrdquo Manage-ment Science vol 50 no 9 pp 1145ndash1177 2004

[3] S-R Ahn H-O Bae H-K Koo and K-J Lee ldquoA survey onamerican options old approaches and new trendsrdquo Bulletin ofthe KoreanMathematical Society vol 48 no 4 pp 791ndash812 2011

[4] S-P Zhu ldquoOn various quantitative approaches for pricingamerican optionsrdquoNewMathematics andNatural Computationvol 7 no 2 pp 313ndash332 2011

[5] S D Howison C Reisinger and J H Witte ldquoThe effect ofnonsmooth payoffs on the penalty approximation of americanoptionsrdquo SIAM Journal on Financial Mathematics vol 4 no 1pp 539ndash574 2013

[6] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[7] B F Nielsen O Skavhaug and A Tveito ldquoPenalty and front-fixing methods for the numerical solution of american optionproblemsrdquo Journal of Computational Finance vol 5 no 4 pp69ndash98 2002

[8] A Q M Khaliq D A Voss and S H K Kazmi ldquoA lin-early implicit predictor-corrector scheme for pricing Americanoptions using a penalty method approachrdquo Journal of Bankingamp Finance vol 30 no 2 pp 489ndash502 2006

[9] SWang X Q Yang and K L Teo ldquoPower penaltymethod for alinear complementarity problem arising from American optionvaluationrdquo Journal of OptimizationTheory andApplications vol129 no 2 pp 227ndash254 2006

[10] J Liang BHu L Jiang andB Bian ldquoOn the rate of convergenceof the binomial tree scheme for American optionsrdquoNumerischeMathematik vol 107 no 2 pp 333ndash352 2007

[11] B Hu J Liang and L Jiang ldquoOptimal convergence rate of theexplicit finite difference scheme forAmerican option valuationrdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 583ndash599 2009

[12] C Christara and D M Dang ldquoAdaptive and high-order meth-ods for valuing american optionsrdquo Journal of ComputationalFinance vol 14 no 4 pp 73ndash113 2011

[13] Z Cen A Le and A Xu ldquoA second-order difference schemefor the penalized blackndashscholes equation governing americanput option pricingrdquoComputational Economics vol 40 no 1 pp49ndash62 2012

[14] S Memon ldquoFinite element method for American option pric-ing a penalty approachrdquo International Journal of NumericalAnalysis and Modelling Series B Computing and Informationvol 3 no 3 pp 345ndash370 2012

[15] G Leduc ldquoExercisability randomization of the Americanoptionrdquo Stochastic Analysis and Applications vol 26 no 4 pp832ndash855 2008

[16] D Lamberton ldquoError estimates for the binomial approximationof American put optionsrdquo Annals of Applied Probability vol 8no 1 pp 206ndash233 1998

[17] D Lamberton ldquoBrownian optimal stopping and randomwalksrdquoApplied Mathematics and Optimization vol 45 no 3 pp 283ndash324 2002

[18] D Lamberton Optimal Stopping and American Options Sum-mer School on Financial Mathematics Ljubljana Slovenia2009

[19] P Dupuis and H Wang ldquoOptimal stopping with randomintervention timesrdquoAdvances in Applied Probability vol 34 no1 pp 141ndash157 2002

[20] X Guo and J Liu ldquoStopping at the maximum of geometricBrownian motion when signals are receivedrdquo Journal of AppliedProbability vol 42 no 3 pp 826ndash838 2005

[21] P Carr ldquoRandomization and the American putrdquo Review ofFinancial Studies vol 11 no 3 pp 597ndash626 1998

[22] G Leduc ldquoA European option general first-order error for-mulardquoThe ANZIAM Journal vol 54 no 4 pp 248ndash272 2013

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 Function Spaces 5

Proof See [22 Th 41] where the assumption that ℎ is poly-nomially bounded continuous and piecewise continuouslydifferentiable with polynomially bounded derivative can bereplaced by our assumptions on ℎ without any change inthe argument as a Lipschitz function is Lebesgue almosteverywhere differentiable and absolutely continuous allowingthe required integration by parts

LemmaA2 If ℎ is a bounded Lipschitz function then for every119904 gt 0

119909

120597

120597119909

E119904ℎ (119909) = E

119904(119868ℎ1015840) (119909) (A3)

Proof As ℎ is Lipschitz it is Lebesgue almost everywheredifferentiable and the proof of [22 Lemma 63] can befollowed without any change

Conflict of Interests

The author declares that there is no conflict of interestsregarding the publication of this paper

References

[1] P Carr ldquoDeriving derivatives of derivative securitiesrdquo in Pro-ceedings of the IEEEIAFEINFORMS Conference on Computa-tional Intelligence for Financial Engineering (CIFEr rsquo00) pp 101ndash128 IEEE New York NY USA March 2000

[2] M Broadie and J B Detemple ldquoANNIVERSARY ARTICLEoption pricing valuation models and applicationsrdquo Manage-ment Science vol 50 no 9 pp 1145ndash1177 2004

[3] S-R Ahn H-O Bae H-K Koo and K-J Lee ldquoA survey onamerican options old approaches and new trendsrdquo Bulletin ofthe KoreanMathematical Society vol 48 no 4 pp 791ndash812 2011

[4] S-P Zhu ldquoOn various quantitative approaches for pricingamerican optionsrdquoNewMathematics andNatural Computationvol 7 no 2 pp 313ndash332 2011

[5] S D Howison C Reisinger and J H Witte ldquoThe effect ofnonsmooth payoffs on the penalty approximation of americanoptionsrdquo SIAM Journal on Financial Mathematics vol 4 no 1pp 539ndash574 2013

[6] P A Forsyth and K R Vetzal ldquoQuadratic convergence forvaluing American options using a penalty methodrdquo SIAMJournal on Scientific Computing vol 23 no 6 pp 2095ndash21222002

[7] B F Nielsen O Skavhaug and A Tveito ldquoPenalty and front-fixing methods for the numerical solution of american optionproblemsrdquo Journal of Computational Finance vol 5 no 4 pp69ndash98 2002

[8] A Q M Khaliq D A Voss and S H K Kazmi ldquoA lin-early implicit predictor-corrector scheme for pricing Americanoptions using a penalty method approachrdquo Journal of Bankingamp Finance vol 30 no 2 pp 489ndash502 2006

[9] SWang X Q Yang and K L Teo ldquoPower penaltymethod for alinear complementarity problem arising from American optionvaluationrdquo Journal of OptimizationTheory andApplications vol129 no 2 pp 227ndash254 2006

[10] J Liang BHu L Jiang andB Bian ldquoOn the rate of convergenceof the binomial tree scheme for American optionsrdquoNumerischeMathematik vol 107 no 2 pp 333ndash352 2007

[11] B Hu J Liang and L Jiang ldquoOptimal convergence rate of theexplicit finite difference scheme forAmerican option valuationrdquoJournal of Computational andAppliedMathematics vol 230 no2 pp 583ndash599 2009

[12] C Christara and D M Dang ldquoAdaptive and high-order meth-ods for valuing american optionsrdquo Journal of ComputationalFinance vol 14 no 4 pp 73ndash113 2011

[13] Z Cen A Le and A Xu ldquoA second-order difference schemefor the penalized blackndashscholes equation governing americanput option pricingrdquoComputational Economics vol 40 no 1 pp49ndash62 2012

[14] S Memon ldquoFinite element method for American option pric-ing a penalty approachrdquo International Journal of NumericalAnalysis and Modelling Series B Computing and Informationvol 3 no 3 pp 345ndash370 2012

[15] G Leduc ldquoExercisability randomization of the Americanoptionrdquo Stochastic Analysis and Applications vol 26 no 4 pp832ndash855 2008

[16] D Lamberton ldquoError estimates for the binomial approximationof American put optionsrdquo Annals of Applied Probability vol 8no 1 pp 206ndash233 1998

[17] D Lamberton ldquoBrownian optimal stopping and randomwalksrdquoApplied Mathematics and Optimization vol 45 no 3 pp 283ndash324 2002

[18] D Lamberton Optimal Stopping and American Options Sum-mer School on Financial Mathematics Ljubljana Slovenia2009

[19] P Dupuis and H Wang ldquoOptimal stopping with randomintervention timesrdquoAdvances in Applied Probability vol 34 no1 pp 141ndash157 2002

[20] X Guo and J Liu ldquoStopping at the maximum of geometricBrownian motion when signals are receivedrdquo Journal of AppliedProbability vol 42 no 3 pp 826ndash838 2005

[21] P Carr ldquoRandomization and the American putrdquo Review ofFinancial Studies vol 11 no 3 pp 597ndash626 1998

[22] G Leduc ldquoA European option general first-order error for-mulardquoThe ANZIAM Journal vol 54 no 4 pp 248ndash272 2013

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

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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