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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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

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

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 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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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Differential EquationsInternational Journal of

Volume 2014

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Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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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

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Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

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