Research ArticleNumerical Methods for Pricing American Options withTime-Fractional PDE Models

Zhiqiang Zhou1 and Xuemei Gao2

1School of Economic Mathematics Southwestern University of Finance and Economics Chengdu Wenjiang 611130 China2School of Economic Mathematics and School of Finance Southwestern University of Finance and Economics ChengduWenjiang 611130 China

Correspondence should be addressed to Zhiqiang Zhou zqzhou2014swufeeducn

Received 26 September 2015 Accepted 22 December 2015

Academic Editor George Tsiatas

Copyright copy 2016 Z Zhou and X GaoThis 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

In this paper we develop a Laplace transform method and a finite difference method for solving American option pricing problemwhen the change of the option price with time is considered as a fractal transmission system In this scenario the option price isgoverned by a time-fractional partial differential equation (PDE) with free boundary The Laplace transform method is applied tothe time-fractional PDE It then leads to a nonlinear equation for the free boundary (ie optimal early exercise boundary) functionin Laplace space After numerically finding the solution of the nonlinear equation the Laplace inversion is used to transformthe approximate early exercise boundary into the time space Finally the approximate price of the American option is obtainedA boundary-searching finite difference method is also proposed to solve the free-boundary time-fractional PDEs for pricing theAmerican options Numerical examples are carried out to compare the Laplace approach with the finite difference method and itis confirmed that the former approach is much faster than the latter one

1 Introduction

Fractional differential equations have wide applications inthe fields of physics modelling (see eg a book [1] and thereferences therein) Based on the fact that the fractional-orderderivatives are characterized by the ldquoglobalnessrdquo and canprovide a powerful instrument for the description ofmemoryand hereditary properties of different substances recently thefractional differential equations are applied to the area ofmathematical finance by generalizing the Black-Scholes (B-S) equations to the fractional order In this subject there aremainly two types of fractional derivatives involved a space-fractional derivative and a time-fractional derivative Regard-ing a space-fractional derivative Carr andWu [2] introduceda finite moment log stable (FMLS) model and showed thatthe model outperforms other widely used financial modelsCartea and del-Castillo-Negrete [3] successfully connectedthe FMLS process with the space-fractional derivatives Lateron Chen et al [4] derived the explicit closed-form formulafor European vanilla options under the FMLS model As fortime-fractional derivative Wyss [5] derived a closed-form

solution for European vanilla options under a time-fractionalB-S equation Kumar et al [6] further explored the Europeanoption pricing under time-fractional B-S equation using theLaplace transform Later on using tick-to-tick date Cartea[7] found that the value of European-style options satisfies afractional partial differential equation with the Caputo time-fractional derivative Jumarie [8] derived a time-and-space-fractional B-S equation Liang et al [9] introduced bifrac-tional B-S models in which they assume that the underlyingasset follows a fractional Ito process and the change of optionprice with time is a fractal transmission system Chen et al[10] derived an analytic formula for pricing double barrieroptions based on a time-fractional B-S equation

To the best of our knowledge we are only aware of onepaper Chen et al [11] studying the American option pricingunder fractional derivatives In [11] they studied predictor-corrector finite difference methods for pricing Americanoptions under the FMLS model which is a kind of space-fractional derivativemodel In this paper we study the pricingof American options with time-fractional model which hasessential difference to the space-fractional model Following

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 5614950 8 pageshttpdxdoiorg10115520165614950

2 Mathematical Problems in Engineering

the model in [10] we assume that the underlying assetprice still follows the classical Brownian motion but thechange in the option price is considered as a fractal trans-mission systemThe price of such American option follows atime-fractional partial differential equation (PDE) with freeboundary The solution of the time-fractional PDE with freeboundary is much more challenging than solving the frac-tional PDEswith fixed boundary for European option pricingin [10] In this paper we develop two numerical methodsLaplace transform method and finite difference method tosolve the free-boundary problem of time-fractional PDETheLaplace transform for partial differential equation can resultin an ordinary differential equation that is it can convert par-tial differential equations into ordinary differential equationsHowever the solution of ordinary differential equations ismuch simpler than that for partial differential equationsThisis the intuition of using Laplace transformmethod for solvingthe PDEs with free boundaryThe Laplace transformmethodhas been successfully applied to the pricingAmerican optionswith classical (integer) B-S equations (see [12]) In the currentwork by applying the Laplace transform to time-fractionalPDE with respect to time constructing the general solutionsto the resulted ordinary differential equation (ODE) and thenusing the (free) boundary conditions we derive a nonlinearequation for the free boundary (or optimal early exerciseboundary) function in Laplace spaceThis nonlinear equationis solved by the secant method and the approximate freeboundary in the Laplace space is obtained Then the Laplaceinversion is applied to get the optimal early exercise boundaryin the time space Finally the approximate value of theAmerican option is obtained Also motivated by [13 14] forsolving integer-order PDEs with free boundary we developa boundary-searching finite difference method to solve thetime-fractional PDEs free boundary The comparison of theLaplace method with the finite difference method is madeby several examples The numerical results show that theLaplace transform method is much more efficient than thefinite difference method

The rest of paper is arranged as follows In Section 2we describe the American option pricing models with time-fractional derivatives in Section 3 we introduce the Laplacetransform methods for the problem in Section 4 we givethe boundary-searching finite difference methods for theproblem in Section 5 we provide numerical examples tocompare the performance of the Laplace transform methodwith FDM the conclusions are made in the final section

2 American Option Pricing Models withTime-Fractional Derivatives

Assume that the underlying asset price is governed by theconstant elasticity of variance (CEV) model (see eg Cox[15])

119889119878119905= (119903 minus 119902) 119878

119905119889119905 + 120575119878

120573+1

119905119889119882119905 (1)

where 119903 is the risk-free interest rate 119902 is the dividend yieldand 119882

119905is standard Brownian motion 120590(119878) = 120575119878

120573 representsthe local volatility function and 120573 can be interpreted as the

elasticity of 120590(119878) If 120573 = 0 then SDE (1) becomes the log-normal diffusion model

Denote by 119891(119878 119905) the price of American option with 119878

being underlying and 119905 being the current time From SDE (1)119891(119878 119905) should satisfy the modified B-S equation

120597119891

120597119905+

1

212057521198782120573+2 1205972119891

1205971198782+ (119903 minus 119902) 119878

120597119891

120597119878minus 119903119891 = 0 (2)

However it is argued that the time derivative 120597119891120597119905 shouldbe replaced by the fractional derivative 120597

120572119891120597119905120572(0 lt 120572 lt

1) under the assumption that the change in the option pricefollows a fractal transmission system (see eg [9 10]) thatis

120597120572119891

120597119905120572+

1

212057521198782120573+2 1205972119891

1205971198782+ (119903 minus 119902) 119878

120597119891

120597119878minus 119903119891 = 0 (3)

The meaning of the assumption is that the diffusion of theoption price depends on the history of the time to maturityMoreover it is suggested by [9 10] that the following right-hand side modified Riemann-Liouville derivative can beused

120597120572119891

120597119905120572=

1

Γ (1 minus 120572)

120597

120597119905int

119879

119905

119891 (119878 119906) minus 119891 (119878 119879)

(119906 minus 119905)120572

119889119906

0 lt 120572 lt 1

(4)

For convenience use the coordinate transform 120591 = 119879 minus 119905 anddenote

119881 (119878 120591) equiv 119891 (119878 119879 minus 120591) = 119891 (119878 119905) (5)

Then we calculate

120597120572119891

120597119905120572=

1

Γ (1 minus 120572)

120597

120597119905int

119879

119905

119891 (119878 119906) minus 119891 (119878 119879)

(119906 minus 119905)120572

119889119906

=1

Γ (1 minus 120572)

120597

120597120591int

0

120591

119891 (119878 119879 minus 120579) minus 119891 (119878 119879)

(120591 minus 120579)120572

119889120579

(using 119906 = 119879 minus 120579)

=minus1

Γ (1 minus 120572)

120597

120597120591int

120591

0

119881 (119878 120579) minus 119881 (119878 0)

(120591 minus 120579)120572

119889120579

= minus120597

120597120591int

120591

0

[119881 (119878 120579) minus 119881 (119878 0)] 119889(120591 minus 120579)

1minus120572

1 minus 120572

= minus120597

120597120591int

120591

0

(120591 minus 120579)1minus120572

1 minus 120572

120597119881 (119878 120579)

120597120579119889120579

=minus1

Γ (1 minus 120572)int

120591

0

1

(120591 minus 120579)120572

120597119881 (119878 120579)

120597120579119889120579

(6)

Recall from [16] that the left-hand side Caputo fractionalderivative is defined as

119862

0119863120572

120591119881 (119878 120591) equiv

1

Γ (1 minus 120572)int

120591

0

1

(120591 minus 120579)120572

120597119881 (119878 120579)

120597120579119889120579 (7)

Mathematical Problems in Engineering 3

Then (6) gives that

120597120572119891

120597119905120572= minus119862

0119863120572

120591119881 (119878 120591) (8)

According to (3) (5) (8) and the facts

120597119891

120597119878=

120597119881

120597119878

1205972119891

1205971198782=

1205972119881

1205971198782

(9)

the valuation of American put option can be formulated as atime-fractional free-boundary problem

119862

0119863120572

120591119881 (119878 120591) =

1

212057521198782120573+2 1205972119881

1205971198782+ (119903 minus 119902) 119878

120597119881

120597119878minus 119903119881 (10)

119881 (119878 0) = max (119870 minus 119878 0) (11)

119881(119878119891(120591) 120591) = 119870 minus 119878

119891(120591) (12)

120597119881 (119878119891(120591) 120591)

120597119878= minus1 (13)

lim119878rarrinfin

119881 (119878 120591) = 0 (14)

The main purpose of this paper is to solve problem (10)ndash(14)

3 Laplace Transform Methods

For 120582 gt 0 define the Laplace-Carson transform (LCT) as

(119878 120582) = int

+infin

0

119881 (119878 120591) 120582119890minus120582120591

119889120591 fl LC [119881 (119878 120591)] (120582) (15)

The LCT is essentially the same as the Laplace transform (LT)and the relationship between LCT and LT is

LC [119881 (119878 120591)] (120582) = 120582L [119881 (119878 120591)] (120582) (16)

The reason of using the LCT is to simplify the notations inthe later analysis Using the Laplace transform formula for theCaputo fractional derivative (see (2253) in [16])

L [119862

0119863120572

120591119881 (119878 120591)] (120582) = 120582

120572L [119881 (119878 120591)] (120582)

minus 120582120572minus1

119881 (119878 0)

(17)

and relationship (16) the LCT for 1198620119863120572

120591119881(119878 120591) is found as

LC [119862

0119863120572

120591119881 (119878 120591)] (120582)

= 120582120572LC [119881 (119878 120591)] (120582) minus 120582

120572119881 (119878 0)

(18)

Taking LCT to (10)ndash(14) we have

1

212057521198782120573+2 1205972

1205971198782+ (119903 minus 119902) 119878

120597

120597119878minus (120582120572+ 119903)

+ 120582120572max (119870 minus 119878 0) = 0

(19)

(119878119891(120582) 120582) = 119870 minus 119878

119891(120582) (20)

(119878119891(120582) 120582)

120597119878= minus1 (21)

lim119878rarrinfin

(119878 120582) = 0 (22)

The solution of governing equation (19) is given by

(119878 120582)

=

11986211120601120582(119878) + 119862

12120595120582(119878) when 119878 isin [119870infin)

11986221120601120582(119878) + 119862

22120595120582(119878) + 119906

120582(119878) when 119878 isin (119878

119891 119870)

(23)

For the case 120573 = 0 the basis functions 120601120582(119878) and 120595

120582(119878) have

the following forms (the derivation is analogous to that in[17])

120601120582(119878) = 119878

120574minus

120595120582(119878) = 119878

120574+

120574plusmn= minus120574 plusmn radic1205742 +

2 (120582120572+ 119903)

1205752 120574 =

119903 minus 119902

1205752minus

1

2

(24)

By simple calculation we have

Λ (119878) equiv 120585 (119878) 120596120582

120585 (119878) = 119878minus2120574minus1

120596120582= 2radic1205742 +

2 (120582120572+ 119903)

1205752

(25)

For the case 120573 = 0 the functions 120601120582(119878) and120595

120582(119878) are given by

(the derivation is similar to that in [18])

120601120582(119878)

=

119878120573+12

119890120598119909(119878)2

119882119896(120582)119898

(119909 (119878)) 120573 lt 0 120583 = 0

119878120573+12

119890120598119909(119878)2

119872119896(120582)119898

(119909 (119878)) 120573 gt 0 120583 = 0

11987812

119870] (radic2 (120582120572 + 119903)119911 (119878)) 120573 lt 0 120583 = 0

11987812

119868] (radic2 (120582120572 + 119903)119911 (119878)) 120573 gt 0 120583 = 0

120595120582(119878)

=

119878120573+12

119890120598119909(119878)2

119872119896(120582)119898

(119909 (119878)) 120573 lt 0 120583 = 0

119878120573+12

119890120598119909(119878)2

119882119896(120582)119898

(119909 (119878)) 120573 gt 0 120583 = 0

11987812

119868] (radic2 (120582120572 + 119903)119911 (119878)) 120573 lt 0 120583 = 0

11987812

119870] (radic2 (120582120572 + 119903)119911 (119878)) 120573 gt 0 120583 = 0

(26)

4 Mathematical Problems in Engineering

where 119872119896119898

(119909) and 119882119896119898

(119909) are the Whittaker functions119868](119909) and 119870](119909) are the modified Bessel functions (see eg[18]) and

120583 = 119903 minus 119902

119909 (119878) =

10038161003816100381610038161205831003816100381610038161003816

12057521003816100381610038161003816120573

1003816100381610038161003816

119878minus2120573

119911 (119878) =1

1205751003816100381610038161003816120573

1003816100381610038161003816

119878minus120573

120598 = sign (120583120573)

119898 =1

41003816100381610038161003816120573

1003816100381610038161003816

119896 (120582) = 120598 (1

2+

1

4120573) minus

120582120572+ 119903

21003816100381610038161003816120583120573

1003816100381610038161003816

] =1

21003816100381610038161003816120573

1003816100381610038161003816

(27)

Moreover it can be calculated that

Λ (119878) = 120585 (119878) 120596120582

120585 (119878) = exp(120583

1205752120573119878minus2120573

)

120596120582=

21003816100381610038161003816119903 minus 119902

1003816100381610038161003816 Γ (2119898 + 1)

1205752Γ (119898 minus 119896 + 12) 119903 minus 119902 = 0

10038161003816100381610038161205731003816100381610038161003816 119903 minus 119902 = 0

(28)

Next we give a particular solution 119906120582(119878) and derive the

nonlinear equation for 119878119891(120582) By matching conditions (20)ndash

(22) we derive that

119906120582(119878) = minus

120582120572119878

120582120572 + 119902+

120582120572119870

120582120572 + 119903

11986212

= 0

11986211

=1

1198865Λ (119870)

[(11988641198865minus 11988631198866) 1198871

+ (11988611198866minus 11988621198865minus Λ (119870)) 119887

2minus Λ (119870)]

11986221

=1

1198865Λ (119870)

[minus119886311988661198871+ (11988611198866minus Λ (119870)) 119887

2

minus Λ (119870)]

11986222

=1

Λ (119870)[11988631198871minus 11988611198872]

(29)

where1198861= 120601120582(119870)

1198862= 120595120582(119870)

1198863=

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

1198864=

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

1198865=

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

1198866=

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

1198871= 119906120582(119870)

1198872=

minus120582120572

120582120572 + 119902

(30)

Furthermore letting 119878 = 119878119891(120582) in formula (23) and using

conditions (20) and (21) we derive the nonlinear equation for119878119891(120582)

1198601Λ(119878119891(120582)) = 119860

2120601120582(119878119891(120582))

+ 1198603119878119891(120582)

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

+ 1198604

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

(31)

where

1198601= (

120582120572119870

120582120572 + 119903minus

120582120572119870

120582120572 + 119902)

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

+120582120572120601120582(119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198602= minus

119902Λ (119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198603=

119902Λ (119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198604= minus

119903119870Λ (119878)

120582120572 + 119903

10038161003816100381610038161003816100381610038161003816119878=119870

(32)

Using the secant method to solve (31) we obtain 119878119891(120582) for

different values of 120582 Finally the optimal exercise boundaryand put option price can be expressed in terms of the Laplaceinversion

119878119891(120591) = L

minus1[119878119891(120582)

120582]

119881 (119878 120591) = Lminus1

[ (119878 120582)

120582]

(33)

Mathematical Problems in Engineering 5

4 Finite Difference Methods

Suppose 119878max is a large enough positive number such that119881(119878max 120591) asymp 0 for all 120591 isin [0 119879] Define uniform time andspace mesh

Δ120591 =119879

119872

120591119896= 119896Δ120591 119896 = 0 1 119872

Δ119878 =119878max119873

119878119895= 119895Δ119878 119895 = 0 1 119873

(34)

The time-fractional derivative 1198620119863120572

120591119881(119878 120591) at 119878 = 119878

119895 120591 = 120591

119899

can be formulated as (see [19])

119862

0119863120572

120591119881 (119878 120591)

10038161003816100381610038161003816119878=119878119895

120591=120591119899

=

119899

sum

119896=1

119887(119899)

119896

119881(119878119895 120591119896) minus 119881 (119878

119895 120591119896minus1

)

Δ120591+ 119874 ((Δ120591)

2minus120572)

(35)

with

119887(119899)

119896=

1

Γ (1 minus 120572)int

120591119896

120591119896minus1

119889120596

(120591119899minus 120596)120572

=1

Γ (2 minus 120572)[(120591119899minus 120591119896minus1

)1minus120572

minus (120591119899minus 120591119896)1minus120572

]

119896 = 1 2 119899 minus 1

119887(119899)

119899=

(Δ120591)1minus120572

Γ (2 minus 120572)

(36)

We approximate the space derivatives by central difference

1205972119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895minus1

120591119899) minus 2119881 (119878

119895 120591119899) + 119881 (119878

119895+1 120591119899)

(Δ119878)2

+ 119874 ((Δ119878)2)

120597119881

120597119878

10038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895+1

120591119899) minus 119881 (119878

119895minus1 120591119899)

2Δ119878+ 119874 ((Δ119878)

2)

(37)

Inserting (35) and (37) into PDE (10) and omitting the higherterm 119874((Δ120591)

2minus120572) and 119874((Δ119878)

2) we obtain the FDM scheme

(with notation 119881119899

119895asymp 119881(119878

119895 120591119899)) as follows

119888119895minus1

119881119899

119895minus1+ 119888119895119881119899

119895+ 119888119895+1

119881119899

119895+1= 119889119899

119895 (38)

where

119888119895minus1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895+

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119888119895= 119887(119899)

119899+

Δ120591

(Δ119878)212057521198782120573+2

119895

119888119895+1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895minus

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119889119899

119895= 119887(119899)

119899119881119899minus1

119895minus

119899minus1

sum

119896=1

119887(119899)

119896(119881119896

119895minus 119881119896minus1

119895)

(39)

The initial condition is1198810119895= max(119870minus119878

119895 0) and the left-hand

side boundary and the boundary conditions are specified bythe following algorithm

Differently from fixed boundary problems how to deter-mine the moving boundaries 119878

119891(120591) is the key for pricing

American option We design a simple algorithm namelyboundary-searching method for solving time-fractionalfree-boundary problem (10)ndash(14)

Boundary-Searching Algorithm

Step 1 Let 119878119891(0) = 119870 Since 119881

0

119895= max(119870 minus 119878

119895 0) we

determine 119869(1205910) by

119869 (1205910) = min119895

119895 | 1198810

119895= 0 (40)

So we know 119878119869(1205910

)minus1le 119878119891(0) le 119878

119869(1205910

)

Step 2 For 119899 = 1 119872

Step 21 Search backward for 119869(120591119899)

For 119894 = 119869(120591119899minus1

) minus 1 119869(120591119899minus1

) minus 2 1 0

Solve (38) with boundary conditions119881119899119894

= 119870minus119878119894

and 119881119899

119873= 0

If the solutions 119881119899

119894+1and 119881

119899

119894satisfy (119881

119899

119894+1minus

119881119899

119894)Δ119878 le minus1 then119869(120591119899) = 119894 + 1 Break

End If

End For

Then we have that 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899

)] More

accurately searching of 119878119891(120591119899) can be carried out in the next

substep

Step 22 Let 119878120579119891(120591119899) equiv 120579119878

119869(120591119899

)+ (1 minus 120579)119878

119869(120591119899

)minus1 0 le 120579 le 1 Then

redefine the boundary conditions for (38) as

119871minus1

(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1minus

1198710(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1= 119870 minus 119878

120579

119891(120591119899)

(41)

6 Mathematical Problems in Engineering

with Lagrange basis functions

119871ℓ(119878) = prod

119895=minus101

119895 =ℓ

[119878 minus (119869 (120591119899) + 119895) Δ119878] ℓ = minus1 0 1

(42)

Then using the solutions of (38) with the new boundarycondition (41) we calculate the approximation

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp

1198711015840

minus1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1

minus1198711015840

0(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711015840

1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1

(43)

We find the appropriate value of 0 le 120579 le 1 such that

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp minus1 (44)

Step 3 Output early exercise boundaries 119878120579

119891(120591119899) and Ameri-

can put option prices119881119899119895(for 119895 = 0 1 119869(120591

119899)minus1119881119899

119895= 119870minus119878

119895

and for 119895 = 119869(120591119899) 119873119881119899

119895are obtained fromFDM(38)with

the new boundary condition (41))

Remark 1 (a) The boundary-searching method is based onthe fact that the early exercise boundary 119878

119891(120591) is monotoni-

cally decreasing in 120591 and the derivative 120597119881120597119878 is monotoni-cally increasing with respect to 119878 (b) Note that the left-handside of (41) is just the quadratic Lagrange approximation of119881(119878120579

119891(120591119899) 120591119899) The role of Step 22 is to numerically search a

value 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899)] such that

120597

120597119878119881 (119878119891(120591119899) 120591119899) = minus1

119881 (119878119891(120591119899) 120591119899) = 119870 minus 119878

119891(120591119899)

(45)

In fact the boundary-searching method still works withoutStep 22 however Step 22 can lead to more accurate results

5 Numerical Examples

In this section we implement and compare the Laplace trans-form method and the boundary-searching finite differencemethod (FDM)The Laplace inversion is calculated byGaver-Wynn-Rho (GWR) algorithm in [20] The mesh parametersare taken as 119872 = 200 119873 = 800 and 119878max = 200 for FDMThe volatility at time 119905 = 0 is defined by 120590

0equiv 120575119878120573

0

Table 1 lists the computational values of American putoptions Columns entitled ldquoLTMrdquo and ldquoFDMrdquo representthe Laplace transform method and the boundary-searchingFDM respectively For 120572 = 1 the time-fractional derivativereduces to the first-order derivative in time and the problembecomes the classical American option pricing problem

Table 1 Prices of American put option at 120591 = 3

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = 0

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 12189 12362 34116 3479209 11771 11912 32651 3315707 10959 11028 29817 3007104 09778 09793 25770 2582902 09005 09002 23184 23191CPU time (s) 165 23468 173 24134

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = minus1

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 11877 12020 33325 3383409 11485 11604 31918 3229707 10722 10802 29208 2940004 09609 09657 25347 2539702 08877 08922 22876 22898CPU time (s) 153 23921 198 24722

From Table 1 we can see that when 120572 = 1 the results areconsistent with the prices listed in [12] At time to maturity120591 = 3 one could observe that the option prices are decreasingas 120572 is decreasing Moreover the LTM takes much less CPUtime than the FDM as observed from Table 1

In Figure 1 we plot the early exercise boundaries com-puted by LTM and FDM for the cases of 120572 = 07 and 120572 = 04The numbers of mesh nodes are taken as 119872 = 50 119873 = 200

for FDM It can be seen that all the boundaries at differentobtained by LTM are very close time 120591 obtained by LTM arevery close to those by FDM Figure 2 illustrates the effectof different time-fractional derivative order 120572 on the earlyexercise boundaries and option prices It can be seen fromFigure 2(a) that when the value of120572 is decreasing the exerciseregion will be shrunk in the region of time to maturity 120591

close to 0 and will be enlarged in the region of 120591 closeto 119879 From Figure 2(b) we observe that the time-fractionalderivatives have little effect on the option price for the casesof deep-in-the-money (119878

0≪ 119870) and deep-out-the-money

(1198780

≫ 119870) and have significant effect near to on-the-money(1198780asymp 119870)

6 Conclusions

There have been increasing applications of fractional PDEsin the field of option pricing In the history most focuseshave been on the valuation of options without early exercisefeatures In current work we study the American optionpricing with time-fractional PDEs The option pricing prob-lem is formulated into a time-fractional PDE with freeboundary Two methods namely Laplace transform method(LTM) and finite difference method (FDM) are proposed tosolve the time-fractional free-boundary problem Numerical

Mathematical Problems in Engineering 7

28

30

32

34

36

38

40

FDMLTM

0 05 1 15 2 25 330

31

32

33

34

35

36

37

38

39

40

FDMLTM

0 05 1 15 2 25 3

120572 = 07 1205900 = 02 120572 = 04 1205900 = 02

Time to maturity (120591) Time to maturity (120591)

Sf

(120591)

Sf

(120591)

Figure 1 Early exercise boundaries computed by LTM and FDM

0 05 1 15 2 25 310

15

20

25

30

35

40

120572 = 09

120572 = 07

120572 = 05

120572 = 03

Time to maturity (120591)

Sf

(120591)

(a)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

Underlying

Opt

ion

pric

e

50 60 70 800

2

4

6

120572 = 09

120572 = 07

120572 = 05

120572 = 03

(b)

Figure 2 Early exercise boundaries (a) and option prices at 120591 = 119879 = 3 (b) with different 120572 and fixed 1205900= 04 120573 = minus1 119903 = 005 119902 = 0 and

119870 = 40

examples show that the LTM is more efficient than theFDM The methods in this paper are quite different to thepredictor-corrector approach for pricing American optionsunder space-fractional derivatives in [11]The extension of the

methods in this paper to the American option pricing underspace-fractional models will be left for the future work Inaddition it will be also interesting to extend the methods tothe newly established models [21 22]

8 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant nos 15CX141110 andJBK1307012) and National Natural Science Foundation ofChina (Grant no 11471137)

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods World ScientificPublishers London UK 2012

[2] P Carr and L Wu ldquoThe finite moment log stable process andoption pricingrdquo The Journal of Finance vol 58 no 2 pp 597ndash626 2003

[3] A Cartea and D del-Castillo-Negrete ldquoFractional diffusionmodels of option prices in markets with jumpsrdquo Physica AStatistical Mechanics and Its Applications vol 374 no 2 pp749ndash763 2007

[4] W Chen X Xu and S-P Zhu ldquoAnalytically pricing European-style options under the modified Black-Scholes equation with aspatial-fractional derivativerdquoQuarterly of AppliedMathematicsvol 72 no 3 pp 597ndash611 2014

[5] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus amp Applied Analysis vol 3 no 1 pp 51ndash61 2000

[6] S Kumar A Yildirim Y Khan H Jafari K Sayevand and LWei ldquoAnalytical solution of fractional Black-Scholes Europeanoption pricing equation by using Laplace transformrdquo FractionalCalculus and Applied Analysis vol 2 pp 1ndash9 2012

[7] A Cartea ldquoDerivatives pricing with marked point processesusing tick-by-tick datardquo Quantitative Finance vol 13 no 1 pp111ndash123 2013

[8] G Jumarie ldquoStock exchange fractional dynamics defined asfractional exponential growth driven by (usual) Gaussian whitenoise Application to fractional Black-Scholes equationsrdquo Insur-ance Mathematics and Economics vol 42 no 1 pp 271ndash2872008

[9] J-R Liang JWangW-J ZhangW-Y Qiu and F-Y Ren ldquoThesolution to a bifractional Black-Scholes-Merton differentialequationrdquo International Journal of Pure and Applied Mathemat-ics vol 58 no 1 pp 99ndash112 2010

[10] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[11] W Chen X Xu and S-P Zhu ldquoA predictor-corrector approachfor pricing American options under the finite moment log-stable modelrdquo Applied Numerical Mathematics vol 97 pp 15ndash29 2015

[12] H Y Wong and J Zhao ldquoValuing American options underthe CEV model by Laplace-Carson transformsrdquo OperationsResearch Letters vol 38 no 5 pp 474ndash481 2010

[13] K Muthuraman ldquoA moving boundary approach to Americanoption pricingrdquo Journal of Economic Dynamics amp Control vol32 no 11 pp 3520ndash3537 2008

[14] A Chockalingam and K Muthuraman ldquoAn approximate mov-ing boundary method for American option pricingrdquo EuropeanJournal of Operational Research vol 240 no 2 pp 431ndash4382015

[15] J Cox ldquoNotes on option pricing I constant elasticity of variancediffusionsrdquoThe Journal of PortfolioManagement vol 22 pp 15ndash17 1996

[16] I Podlubny Fractional Differential Equations Academic Press1999

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

[18] D Davydov and V Linetsky ldquoPricing and hedging path-dependent options under the CEV processrdquo Management Sci-ence vol 47 no 7 pp 949ndash965 2001

[19] Y Jiang and J Ma ldquoMoving finite element methods for timefractional partial differential equationsrdquo Science China Mathe-matics vol 56 no 6 pp 1287ndash1300 2013

[20] P P Valko and S Vajda ldquoInversion of noise-free Laplacetransforms towards a standardized set of test problemsrdquo InverseProblems in Engineering vol 10 no 5 pp 467ndash483 2002

[21] C Huang X Ma and Q Lan ldquoAn empirical study on listedcompanyrsquos value of cash holdings an information asymmetryperspectiverdquoDiscrete Dynamics in Nature and Society vol 2014Article ID 897278 12 pages 2014

[22] C Huang X Yang X G Yang and H Sheng ldquoAn empiricalstudy of the effect of investor sentiment on returns of differentindustriesrdquo Mathematical Problems in Engineering vol 2014Article ID 545723 11 pages 2014

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Mathematical Problems in Engineering

the model in [10] we assume that the underlying assetprice still follows the classical Brownian motion but thechange in the option price is considered as a fractal trans-mission systemThe price of such American option follows atime-fractional partial differential equation (PDE) with freeboundary The solution of the time-fractional PDE with freeboundary is much more challenging than solving the frac-tional PDEswith fixed boundary for European option pricingin [10] In this paper we develop two numerical methodsLaplace transform method and finite difference method tosolve the free-boundary problem of time-fractional PDETheLaplace transform for partial differential equation can resultin an ordinary differential equation that is it can convert par-tial differential equations into ordinary differential equationsHowever the solution of ordinary differential equations ismuch simpler than that for partial differential equationsThisis the intuition of using Laplace transformmethod for solvingthe PDEs with free boundaryThe Laplace transformmethodhas been successfully applied to the pricingAmerican optionswith classical (integer) B-S equations (see [12]) In the currentwork by applying the Laplace transform to time-fractionalPDE with respect to time constructing the general solutionsto the resulted ordinary differential equation (ODE) and thenusing the (free) boundary conditions we derive a nonlinearequation for the free boundary (or optimal early exerciseboundary) function in Laplace spaceThis nonlinear equationis solved by the secant method and the approximate freeboundary in the Laplace space is obtained Then the Laplaceinversion is applied to get the optimal early exercise boundaryin the time space Finally the approximate value of theAmerican option is obtained Also motivated by [13 14] forsolving integer-order PDEs with free boundary we developa boundary-searching finite difference method to solve thetime-fractional PDEs free boundary The comparison of theLaplace method with the finite difference method is madeby several examples The numerical results show that theLaplace transform method is much more efficient than thefinite difference method

The rest of paper is arranged as follows In Section 2we describe the American option pricing models with time-fractional derivatives in Section 3 we introduce the Laplacetransform methods for the problem in Section 4 we givethe boundary-searching finite difference methods for theproblem in Section 5 we provide numerical examples tocompare the performance of the Laplace transform methodwith FDM the conclusions are made in the final section

2 American Option Pricing Models withTime-Fractional Derivatives

Assume that the underlying asset price is governed by theconstant elasticity of variance (CEV) model (see eg Cox[15])

119889119878119905= (119903 minus 119902) 119878

119905119889119905 + 120575119878

120573+1

119905119889119882119905 (1)

where 119903 is the risk-free interest rate 119902 is the dividend yieldand 119882

119905is standard Brownian motion 120590(119878) = 120575119878

120573 representsthe local volatility function and 120573 can be interpreted as the

elasticity of 120590(119878) If 120573 = 0 then SDE (1) becomes the log-normal diffusion model

Denote by 119891(119878 119905) the price of American option with 119878

being underlying and 119905 being the current time From SDE (1)119891(119878 119905) should satisfy the modified B-S equation

120597119891

120597119905+

1

212057521198782120573+2 1205972119891

1205971198782+ (119903 minus 119902) 119878

120597119891

120597119878minus 119903119891 = 0 (2)

However it is argued that the time derivative 120597119891120597119905 shouldbe replaced by the fractional derivative 120597

120572119891120597119905120572(0 lt 120572 lt

1) under the assumption that the change in the option pricefollows a fractal transmission system (see eg [9 10]) thatis

120597120572119891

120597119905120572+

1

212057521198782120573+2 1205972119891

1205971198782+ (119903 minus 119902) 119878

120597119891

120597119878minus 119903119891 = 0 (3)

The meaning of the assumption is that the diffusion of theoption price depends on the history of the time to maturityMoreover it is suggested by [9 10] that the following right-hand side modified Riemann-Liouville derivative can beused

120597120572119891

120597119905120572=

1

Γ (1 minus 120572)

120597

120597119905int

119879

119905

119891 (119878 119906) minus 119891 (119878 119879)

(119906 minus 119905)120572

119889119906

0 lt 120572 lt 1

(4)

For convenience use the coordinate transform 120591 = 119879 minus 119905 anddenote

119881 (119878 120591) equiv 119891 (119878 119879 minus 120591) = 119891 (119878 119905) (5)

Then we calculate

120597120572119891

120597119905120572=

1

Γ (1 minus 120572)

120597

120597119905int

119879

119905

119891 (119878 119906) minus 119891 (119878 119879)

(119906 minus 119905)120572

119889119906

=1

Γ (1 minus 120572)

120597

120597120591int

0

120591

119891 (119878 119879 minus 120579) minus 119891 (119878 119879)

(120591 minus 120579)120572

119889120579

(using 119906 = 119879 minus 120579)

=minus1

Γ (1 minus 120572)

120597

120597120591int

120591

0

119881 (119878 120579) minus 119881 (119878 0)

(120591 minus 120579)120572

119889120579

= minus120597

120597120591int

120591

0

[119881 (119878 120579) minus 119881 (119878 0)] 119889(120591 minus 120579)

1minus120572

1 minus 120572

= minus120597

120597120591int

120591

0

(120591 minus 120579)1minus120572

1 minus 120572

120597119881 (119878 120579)

120597120579119889120579

=minus1

Γ (1 minus 120572)int

120591

0

1

(120591 minus 120579)120572

120597119881 (119878 120579)

120597120579119889120579

(6)

Recall from [16] that the left-hand side Caputo fractionalderivative is defined as

119862

0119863120572

120591119881 (119878 120591) equiv

1

Γ (1 minus 120572)int

120591

0

1

(120591 minus 120579)120572

120597119881 (119878 120579)

120597120579119889120579 (7)

Mathematical Problems in Engineering 3

Then (6) gives that

120597120572119891

120597119905120572= minus119862

0119863120572

120591119881 (119878 120591) (8)

According to (3) (5) (8) and the facts

120597119891

120597119878=

120597119881

120597119878

1205972119891

1205971198782=

1205972119881

1205971198782

(9)

the valuation of American put option can be formulated as atime-fractional free-boundary problem

119862

0119863120572

120591119881 (119878 120591) =

1

212057521198782120573+2 1205972119881

1205971198782+ (119903 minus 119902) 119878

120597119881

120597119878minus 119903119881 (10)

119881 (119878 0) = max (119870 minus 119878 0) (11)

119881(119878119891(120591) 120591) = 119870 minus 119878

119891(120591) (12)

120597119881 (119878119891(120591) 120591)

120597119878= minus1 (13)

lim119878rarrinfin

119881 (119878 120591) = 0 (14)

The main purpose of this paper is to solve problem (10)ndash(14)

3 Laplace Transform Methods

For 120582 gt 0 define the Laplace-Carson transform (LCT) as

(119878 120582) = int

+infin

0

119881 (119878 120591) 120582119890minus120582120591

119889120591 fl LC [119881 (119878 120591)] (120582) (15)

The LCT is essentially the same as the Laplace transform (LT)and the relationship between LCT and LT is

LC [119881 (119878 120591)] (120582) = 120582L [119881 (119878 120591)] (120582) (16)

The reason of using the LCT is to simplify the notations inthe later analysis Using the Laplace transform formula for theCaputo fractional derivative (see (2253) in [16])

L [119862

0119863120572

120591119881 (119878 120591)] (120582) = 120582

120572L [119881 (119878 120591)] (120582)

minus 120582120572minus1

119881 (119878 0)

(17)

and relationship (16) the LCT for 1198620119863120572

120591119881(119878 120591) is found as

LC [119862

0119863120572

120591119881 (119878 120591)] (120582)

= 120582120572LC [119881 (119878 120591)] (120582) minus 120582

120572119881 (119878 0)

(18)

Taking LCT to (10)ndash(14) we have

1

212057521198782120573+2 1205972

1205971198782+ (119903 minus 119902) 119878

120597

120597119878minus (120582120572+ 119903)

+ 120582120572max (119870 minus 119878 0) = 0

(19)

(119878119891(120582) 120582) = 119870 minus 119878

119891(120582) (20)

(119878119891(120582) 120582)

120597119878= minus1 (21)

lim119878rarrinfin

(119878 120582) = 0 (22)

The solution of governing equation (19) is given by

(119878 120582)

=

11986211120601120582(119878) + 119862

12120595120582(119878) when 119878 isin [119870infin)

11986221120601120582(119878) + 119862

22120595120582(119878) + 119906

120582(119878) when 119878 isin (119878

119891 119870)

(23)

For the case 120573 = 0 the basis functions 120601120582(119878) and 120595

120582(119878) have

the following forms (the derivation is analogous to that in[17])

120601120582(119878) = 119878

120574minus

120595120582(119878) = 119878

120574+

120574plusmn= minus120574 plusmn radic1205742 +

2 (120582120572+ 119903)

1205752 120574 =

119903 minus 119902

1205752minus

1

2

(24)

By simple calculation we have

Λ (119878) equiv 120585 (119878) 120596120582

120585 (119878) = 119878minus2120574minus1

120596120582= 2radic1205742 +

2 (120582120572+ 119903)

1205752

(25)

For the case 120573 = 0 the functions 120601120582(119878) and120595

120582(119878) are given by

(the derivation is similar to that in [18])

120601120582(119878)

=

119878120573+12

119890120598119909(119878)2

119882119896(120582)119898

(119909 (119878)) 120573 lt 0 120583 = 0

119878120573+12

119890120598119909(119878)2

119872119896(120582)119898

(119909 (119878)) 120573 gt 0 120583 = 0

11987812

119870] (radic2 (120582120572 + 119903)119911 (119878)) 120573 lt 0 120583 = 0

11987812

119868] (radic2 (120582120572 + 119903)119911 (119878)) 120573 gt 0 120583 = 0

120595120582(119878)

=

119878120573+12

119890120598119909(119878)2

119872119896(120582)119898

(119909 (119878)) 120573 lt 0 120583 = 0

119878120573+12

119890120598119909(119878)2

119882119896(120582)119898

(119909 (119878)) 120573 gt 0 120583 = 0

11987812

119868] (radic2 (120582120572 + 119903)119911 (119878)) 120573 lt 0 120583 = 0

11987812

119870] (radic2 (120582120572 + 119903)119911 (119878)) 120573 gt 0 120583 = 0

(26)

4 Mathematical Problems in Engineering

where 119872119896119898

(119909) and 119882119896119898

(119909) are the Whittaker functions119868](119909) and 119870](119909) are the modified Bessel functions (see eg[18]) and

120583 = 119903 minus 119902

119909 (119878) =

10038161003816100381610038161205831003816100381610038161003816

12057521003816100381610038161003816120573

1003816100381610038161003816

119878minus2120573

119911 (119878) =1

1205751003816100381610038161003816120573

1003816100381610038161003816

119878minus120573

120598 = sign (120583120573)

119898 =1

41003816100381610038161003816120573

1003816100381610038161003816

119896 (120582) = 120598 (1

2+

1

4120573) minus

120582120572+ 119903

21003816100381610038161003816120583120573

1003816100381610038161003816

] =1

21003816100381610038161003816120573

1003816100381610038161003816

(27)

Moreover it can be calculated that

Λ (119878) = 120585 (119878) 120596120582

120585 (119878) = exp(120583

1205752120573119878minus2120573

)

120596120582=

21003816100381610038161003816119903 minus 119902

1003816100381610038161003816 Γ (2119898 + 1)

1205752Γ (119898 minus 119896 + 12) 119903 minus 119902 = 0

10038161003816100381610038161205731003816100381610038161003816 119903 minus 119902 = 0

(28)

Next we give a particular solution 119906120582(119878) and derive the

nonlinear equation for 119878119891(120582) By matching conditions (20)ndash

(22) we derive that

119906120582(119878) = minus

120582120572119878

120582120572 + 119902+

120582120572119870

120582120572 + 119903

11986212

= 0

11986211

=1

1198865Λ (119870)

[(11988641198865minus 11988631198866) 1198871

+ (11988611198866minus 11988621198865minus Λ (119870)) 119887

2minus Λ (119870)]

11986221

=1

1198865Λ (119870)

[minus119886311988661198871+ (11988611198866minus Λ (119870)) 119887

2

minus Λ (119870)]

11986222

=1

Λ (119870)[11988631198871minus 11988611198872]

(29)

where1198861= 120601120582(119870)

1198862= 120595120582(119870)

1198863=

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

1198864=

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

1198865=

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

1198866=

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

1198871= 119906120582(119870)

1198872=

minus120582120572

120582120572 + 119902

(30)

Furthermore letting 119878 = 119878119891(120582) in formula (23) and using

conditions (20) and (21) we derive the nonlinear equation for119878119891(120582)

1198601Λ(119878119891(120582)) = 119860

2120601120582(119878119891(120582))

+ 1198603119878119891(120582)

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

+ 1198604

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

(31)

where

1198601= (

120582120572119870

120582120572 + 119903minus

120582120572119870

120582120572 + 119902)

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

+120582120572120601120582(119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198602= minus

119902Λ (119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198603=

119902Λ (119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198604= minus

119903119870Λ (119878)

120582120572 + 119903

10038161003816100381610038161003816100381610038161003816119878=119870

(32)

Using the secant method to solve (31) we obtain 119878119891(120582) for

different values of 120582 Finally the optimal exercise boundaryand put option price can be expressed in terms of the Laplaceinversion

119878119891(120591) = L

minus1[119878119891(120582)

120582]

119881 (119878 120591) = Lminus1

[ (119878 120582)

120582]

(33)

Mathematical Problems in Engineering 5

4 Finite Difference Methods

Suppose 119878max is a large enough positive number such that119881(119878max 120591) asymp 0 for all 120591 isin [0 119879] Define uniform time andspace mesh

Δ120591 =119879

119872

120591119896= 119896Δ120591 119896 = 0 1 119872

Δ119878 =119878max119873

119878119895= 119895Δ119878 119895 = 0 1 119873

(34)

The time-fractional derivative 1198620119863120572

120591119881(119878 120591) at 119878 = 119878

119895 120591 = 120591

119899

can be formulated as (see [19])

119862

0119863120572

120591119881 (119878 120591)

10038161003816100381610038161003816119878=119878119895

120591=120591119899

=

119899

sum

119896=1

119887(119899)

119896

119881(119878119895 120591119896) minus 119881 (119878

119895 120591119896minus1

)

Δ120591+ 119874 ((Δ120591)

2minus120572)

(35)

with

119887(119899)

119896=

1

Γ (1 minus 120572)int

120591119896

120591119896minus1

119889120596

(120591119899minus 120596)120572

=1

Γ (2 minus 120572)[(120591119899minus 120591119896minus1

)1minus120572

minus (120591119899minus 120591119896)1minus120572

]

119896 = 1 2 119899 minus 1

119887(119899)

119899=

(Δ120591)1minus120572

Γ (2 minus 120572)

(36)

We approximate the space derivatives by central difference

1205972119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895minus1

120591119899) minus 2119881 (119878

119895 120591119899) + 119881 (119878

119895+1 120591119899)

(Δ119878)2

+ 119874 ((Δ119878)2)

120597119881

120597119878

10038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895+1

120591119899) minus 119881 (119878

119895minus1 120591119899)

2Δ119878+ 119874 ((Δ119878)

2)

(37)

Inserting (35) and (37) into PDE (10) and omitting the higherterm 119874((Δ120591)

2minus120572) and 119874((Δ119878)

2) we obtain the FDM scheme

(with notation 119881119899

119895asymp 119881(119878

119895 120591119899)) as follows

119888119895minus1

119881119899

119895minus1+ 119888119895119881119899

119895+ 119888119895+1

119881119899

119895+1= 119889119899

119895 (38)

where

119888119895minus1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895+

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119888119895= 119887(119899)

119899+

Δ120591

(Δ119878)212057521198782120573+2

119895

119888119895+1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895minus

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119889119899

119895= 119887(119899)

119899119881119899minus1

119895minus

119899minus1

sum

119896=1

119887(119899)

119896(119881119896

119895minus 119881119896minus1

119895)

(39)

The initial condition is1198810119895= max(119870minus119878

119895 0) and the left-hand

side boundary and the boundary conditions are specified bythe following algorithm

Differently from fixed boundary problems how to deter-mine the moving boundaries 119878

119891(120591) is the key for pricing

American option We design a simple algorithm namelyboundary-searching method for solving time-fractionalfree-boundary problem (10)ndash(14)

Boundary-Searching Algorithm

Step 1 Let 119878119891(0) = 119870 Since 119881

0

119895= max(119870 minus 119878

119895 0) we

determine 119869(1205910) by

119869 (1205910) = min119895

119895 | 1198810

119895= 0 (40)

So we know 119878119869(1205910

)minus1le 119878119891(0) le 119878

119869(1205910

)

Step 2 For 119899 = 1 119872

Step 21 Search backward for 119869(120591119899)

For 119894 = 119869(120591119899minus1

) minus 1 119869(120591119899minus1

) minus 2 1 0

Solve (38) with boundary conditions119881119899119894

= 119870minus119878119894

and 119881119899

119873= 0

If the solutions 119881119899

119894+1and 119881

119899

119894satisfy (119881

119899

119894+1minus

119881119899

119894)Δ119878 le minus1 then119869(120591119899) = 119894 + 1 Break

End If

End For

Then we have that 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899

)] More

accurately searching of 119878119891(120591119899) can be carried out in the next

substep

Step 22 Let 119878120579119891(120591119899) equiv 120579119878

119869(120591119899

)+ (1 minus 120579)119878

119869(120591119899

)minus1 0 le 120579 le 1 Then

redefine the boundary conditions for (38) as

119871minus1

(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1minus

1198710(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1= 119870 minus 119878

120579

119891(120591119899)

(41)

6 Mathematical Problems in Engineering

with Lagrange basis functions

119871ℓ(119878) = prod

119895=minus101

119895 =ℓ

[119878 minus (119869 (120591119899) + 119895) Δ119878] ℓ = minus1 0 1

(42)

Then using the solutions of (38) with the new boundarycondition (41) we calculate the approximation

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp

1198711015840

minus1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1

minus1198711015840

0(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711015840

1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1

(43)

We find the appropriate value of 0 le 120579 le 1 such that

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp minus1 (44)

Step 3 Output early exercise boundaries 119878120579

119891(120591119899) and Ameri-

can put option prices119881119899119895(for 119895 = 0 1 119869(120591

119899)minus1119881119899

119895= 119870minus119878

119895

and for 119895 = 119869(120591119899) 119873119881119899

119895are obtained fromFDM(38)with

the new boundary condition (41))

Remark 1 (a) The boundary-searching method is based onthe fact that the early exercise boundary 119878

119891(120591) is monotoni-

cally decreasing in 120591 and the derivative 120597119881120597119878 is monotoni-cally increasing with respect to 119878 (b) Note that the left-handside of (41) is just the quadratic Lagrange approximation of119881(119878120579

119891(120591119899) 120591119899) The role of Step 22 is to numerically search a

value 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899)] such that

120597

120597119878119881 (119878119891(120591119899) 120591119899) = minus1

119881 (119878119891(120591119899) 120591119899) = 119870 minus 119878

119891(120591119899)

(45)

In fact the boundary-searching method still works withoutStep 22 however Step 22 can lead to more accurate results

5 Numerical Examples

In this section we implement and compare the Laplace trans-form method and the boundary-searching finite differencemethod (FDM)The Laplace inversion is calculated byGaver-Wynn-Rho (GWR) algorithm in [20] The mesh parametersare taken as 119872 = 200 119873 = 800 and 119878max = 200 for FDMThe volatility at time 119905 = 0 is defined by 120590

0equiv 120575119878120573

0

Table 1 lists the computational values of American putoptions Columns entitled ldquoLTMrdquo and ldquoFDMrdquo representthe Laplace transform method and the boundary-searchingFDM respectively For 120572 = 1 the time-fractional derivativereduces to the first-order derivative in time and the problembecomes the classical American option pricing problem

Table 1 Prices of American put option at 120591 = 3

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = 0

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 12189 12362 34116 3479209 11771 11912 32651 3315707 10959 11028 29817 3007104 09778 09793 25770 2582902 09005 09002 23184 23191CPU time (s) 165 23468 173 24134

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = minus1

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 11877 12020 33325 3383409 11485 11604 31918 3229707 10722 10802 29208 2940004 09609 09657 25347 2539702 08877 08922 22876 22898CPU time (s) 153 23921 198 24722

From Table 1 we can see that when 120572 = 1 the results areconsistent with the prices listed in [12] At time to maturity120591 = 3 one could observe that the option prices are decreasingas 120572 is decreasing Moreover the LTM takes much less CPUtime than the FDM as observed from Table 1

In Figure 1 we plot the early exercise boundaries com-puted by LTM and FDM for the cases of 120572 = 07 and 120572 = 04The numbers of mesh nodes are taken as 119872 = 50 119873 = 200

for FDM It can be seen that all the boundaries at differentobtained by LTM are very close time 120591 obtained by LTM arevery close to those by FDM Figure 2 illustrates the effectof different time-fractional derivative order 120572 on the earlyexercise boundaries and option prices It can be seen fromFigure 2(a) that when the value of120572 is decreasing the exerciseregion will be shrunk in the region of time to maturity 120591

close to 0 and will be enlarged in the region of 120591 closeto 119879 From Figure 2(b) we observe that the time-fractionalderivatives have little effect on the option price for the casesof deep-in-the-money (119878

0≪ 119870) and deep-out-the-money

(1198780

≫ 119870) and have significant effect near to on-the-money(1198780asymp 119870)

6 Conclusions

There have been increasing applications of fractional PDEsin the field of option pricing In the history most focuseshave been on the valuation of options without early exercisefeatures In current work we study the American optionpricing with time-fractional PDEs The option pricing prob-lem is formulated into a time-fractional PDE with freeboundary Two methods namely Laplace transform method(LTM) and finite difference method (FDM) are proposed tosolve the time-fractional free-boundary problem Numerical

Mathematical Problems in Engineering 7

28

30

32

34

36

38

40

FDMLTM

0 05 1 15 2 25 330

31

32

33

34

35

36

37

38

39

40

FDMLTM

0 05 1 15 2 25 3

120572 = 07 1205900 = 02 120572 = 04 1205900 = 02

Time to maturity (120591) Time to maturity (120591)

Sf

(120591)

Sf

(120591)

Figure 1 Early exercise boundaries computed by LTM and FDM

0 05 1 15 2 25 310

15

20

25

30

35

40

120572 = 09

120572 = 07

120572 = 05

120572 = 03

Time to maturity (120591)

Sf

(120591)

(a)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

Underlying

Opt

ion

pric

e

50 60 70 800

2

4

6

120572 = 09

120572 = 07

120572 = 05

120572 = 03

(b)

Figure 2 Early exercise boundaries (a) and option prices at 120591 = 119879 = 3 (b) with different 120572 and fixed 1205900= 04 120573 = minus1 119903 = 005 119902 = 0 and

119870 = 40

examples show that the LTM is more efficient than theFDM The methods in this paper are quite different to thepredictor-corrector approach for pricing American optionsunder space-fractional derivatives in [11]The extension of the

methods in this paper to the American option pricing underspace-fractional models will be left for the future work Inaddition it will be also interesting to extend the methods tothe newly established models [21 22]

8 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant nos 15CX141110 andJBK1307012) and National Natural Science Foundation ofChina (Grant no 11471137)

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods World ScientificPublishers London UK 2012

[2] P Carr and L Wu ldquoThe finite moment log stable process andoption pricingrdquo The Journal of Finance vol 58 no 2 pp 597ndash626 2003

[3] A Cartea and D del-Castillo-Negrete ldquoFractional diffusionmodels of option prices in markets with jumpsrdquo Physica AStatistical Mechanics and Its Applications vol 374 no 2 pp749ndash763 2007

[4] W Chen X Xu and S-P Zhu ldquoAnalytically pricing European-style options under the modified Black-Scholes equation with aspatial-fractional derivativerdquoQuarterly of AppliedMathematicsvol 72 no 3 pp 597ndash611 2014

[5] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus amp Applied Analysis vol 3 no 1 pp 51ndash61 2000

[6] S Kumar A Yildirim Y Khan H Jafari K Sayevand and LWei ldquoAnalytical solution of fractional Black-Scholes Europeanoption pricing equation by using Laplace transformrdquo FractionalCalculus and Applied Analysis vol 2 pp 1ndash9 2012

[7] A Cartea ldquoDerivatives pricing with marked point processesusing tick-by-tick datardquo Quantitative Finance vol 13 no 1 pp111ndash123 2013

[8] G Jumarie ldquoStock exchange fractional dynamics defined asfractional exponential growth driven by (usual) Gaussian whitenoise Application to fractional Black-Scholes equationsrdquo Insur-ance Mathematics and Economics vol 42 no 1 pp 271ndash2872008

[9] J-R Liang JWangW-J ZhangW-Y Qiu and F-Y Ren ldquoThesolution to a bifractional Black-Scholes-Merton differentialequationrdquo International Journal of Pure and Applied Mathemat-ics vol 58 no 1 pp 99ndash112 2010

[10] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[11] W Chen X Xu and S-P Zhu ldquoA predictor-corrector approachfor pricing American options under the finite moment log-stable modelrdquo Applied Numerical Mathematics vol 97 pp 15ndash29 2015

[12] H Y Wong and J Zhao ldquoValuing American options underthe CEV model by Laplace-Carson transformsrdquo OperationsResearch Letters vol 38 no 5 pp 474ndash481 2010

[13] K Muthuraman ldquoA moving boundary approach to Americanoption pricingrdquo Journal of Economic Dynamics amp Control vol32 no 11 pp 3520ndash3537 2008

[14] A Chockalingam and K Muthuraman ldquoAn approximate mov-ing boundary method for American option pricingrdquo EuropeanJournal of Operational Research vol 240 no 2 pp 431ndash4382015

[15] J Cox ldquoNotes on option pricing I constant elasticity of variancediffusionsrdquoThe Journal of PortfolioManagement vol 22 pp 15ndash17 1996

[16] I Podlubny Fractional Differential Equations Academic Press1999

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

[18] D Davydov and V Linetsky ldquoPricing and hedging path-dependent options under the CEV processrdquo Management Sci-ence vol 47 no 7 pp 949ndash965 2001

[19] Y Jiang and J Ma ldquoMoving finite element methods for timefractional partial differential equationsrdquo Science China Mathe-matics vol 56 no 6 pp 1287ndash1300 2013

[20] P P Valko and S Vajda ldquoInversion of noise-free Laplacetransforms towards a standardized set of test problemsrdquo InverseProblems in Engineering vol 10 no 5 pp 467ndash483 2002

[21] C Huang X Ma and Q Lan ldquoAn empirical study on listedcompanyrsquos value of cash holdings an information asymmetryperspectiverdquoDiscrete Dynamics in Nature and Society vol 2014Article ID 897278 12 pages 2014

[22] C Huang X Yang X G Yang and H Sheng ldquoAn empiricalstudy of the effect of investor sentiment on returns of differentindustriesrdquo Mathematical Problems in Engineering vol 2014Article ID 545723 11 pages 2014

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

Mathematical Problems in Engineering 3

Then (6) gives that

120597120572119891

120597119905120572= minus119862

0119863120572

120591119881 (119878 120591) (8)

According to (3) (5) (8) and the facts

120597119891

120597119878=

120597119881

120597119878

1205972119891

1205971198782=

1205972119881

1205971198782

(9)

the valuation of American put option can be formulated as atime-fractional free-boundary problem

119862

0119863120572

120591119881 (119878 120591) =

1

212057521198782120573+2 1205972119881

1205971198782+ (119903 minus 119902) 119878

120597119881

120597119878minus 119903119881 (10)

119881 (119878 0) = max (119870 minus 119878 0) (11)

119881(119878119891(120591) 120591) = 119870 minus 119878

119891(120591) (12)

120597119881 (119878119891(120591) 120591)

120597119878= minus1 (13)

lim119878rarrinfin

119881 (119878 120591) = 0 (14)

The main purpose of this paper is to solve problem (10)ndash(14)

3 Laplace Transform Methods

For 120582 gt 0 define the Laplace-Carson transform (LCT) as

(119878 120582) = int

+infin

0

119881 (119878 120591) 120582119890minus120582120591

119889120591 fl LC [119881 (119878 120591)] (120582) (15)

The LCT is essentially the same as the Laplace transform (LT)and the relationship between LCT and LT is

LC [119881 (119878 120591)] (120582) = 120582L [119881 (119878 120591)] (120582) (16)

The reason of using the LCT is to simplify the notations inthe later analysis Using the Laplace transform formula for theCaputo fractional derivative (see (2253) in [16])

L [119862

0119863120572

120591119881 (119878 120591)] (120582) = 120582

120572L [119881 (119878 120591)] (120582)

minus 120582120572minus1

119881 (119878 0)

(17)

and relationship (16) the LCT for 1198620119863120572

120591119881(119878 120591) is found as

LC [119862

0119863120572

120591119881 (119878 120591)] (120582)

= 120582120572LC [119881 (119878 120591)] (120582) minus 120582

120572119881 (119878 0)

(18)

Taking LCT to (10)ndash(14) we have

1

212057521198782120573+2 1205972

1205971198782+ (119903 minus 119902) 119878

120597

120597119878minus (120582120572+ 119903)

+ 120582120572max (119870 minus 119878 0) = 0

(19)

(119878119891(120582) 120582) = 119870 minus 119878

119891(120582) (20)

(119878119891(120582) 120582)

120597119878= minus1 (21)

lim119878rarrinfin

(119878 120582) = 0 (22)

The solution of governing equation (19) is given by

(119878 120582)

=

11986211120601120582(119878) + 119862

12120595120582(119878) when 119878 isin [119870infin)

11986221120601120582(119878) + 119862

22120595120582(119878) + 119906

120582(119878) when 119878 isin (119878

119891 119870)

(23)

For the case 120573 = 0 the basis functions 120601120582(119878) and 120595

120582(119878) have

the following forms (the derivation is analogous to that in[17])

120601120582(119878) = 119878

120574minus

120595120582(119878) = 119878

120574+

120574plusmn= minus120574 plusmn radic1205742 +

2 (120582120572+ 119903)

1205752 120574 =

119903 minus 119902

1205752minus

1

2

(24)

By simple calculation we have

Λ (119878) equiv 120585 (119878) 120596120582

120585 (119878) = 119878minus2120574minus1

120596120582= 2radic1205742 +

2 (120582120572+ 119903)

1205752

(25)

For the case 120573 = 0 the functions 120601120582(119878) and120595

120582(119878) are given by

(the derivation is similar to that in [18])

120601120582(119878)

=

119878120573+12

119890120598119909(119878)2

119882119896(120582)119898

(119909 (119878)) 120573 lt 0 120583 = 0

119878120573+12

119890120598119909(119878)2

119872119896(120582)119898

(119909 (119878)) 120573 gt 0 120583 = 0

11987812

119870] (radic2 (120582120572 + 119903)119911 (119878)) 120573 lt 0 120583 = 0

11987812

119868] (radic2 (120582120572 + 119903)119911 (119878)) 120573 gt 0 120583 = 0

120595120582(119878)

=

119878120573+12

119890120598119909(119878)2

119872119896(120582)119898

(119909 (119878)) 120573 lt 0 120583 = 0

119878120573+12

119890120598119909(119878)2

119882119896(120582)119898

(119909 (119878)) 120573 gt 0 120583 = 0

11987812

119868] (radic2 (120582120572 + 119903)119911 (119878)) 120573 lt 0 120583 = 0

11987812

119870] (radic2 (120582120572 + 119903)119911 (119878)) 120573 gt 0 120583 = 0

(26)

4 Mathematical Problems in Engineering

where 119872119896119898

(119909) and 119882119896119898

(119909) are the Whittaker functions119868](119909) and 119870](119909) are the modified Bessel functions (see eg[18]) and

120583 = 119903 minus 119902

119909 (119878) =

10038161003816100381610038161205831003816100381610038161003816

12057521003816100381610038161003816120573

1003816100381610038161003816

119878minus2120573

119911 (119878) =1

1205751003816100381610038161003816120573

1003816100381610038161003816

119878minus120573

120598 = sign (120583120573)

119898 =1

41003816100381610038161003816120573

1003816100381610038161003816

119896 (120582) = 120598 (1

2+

1

4120573) minus

120582120572+ 119903

21003816100381610038161003816120583120573

1003816100381610038161003816

] =1

21003816100381610038161003816120573

1003816100381610038161003816

(27)

Moreover it can be calculated that

Λ (119878) = 120585 (119878) 120596120582

120585 (119878) = exp(120583

1205752120573119878minus2120573

)

120596120582=

21003816100381610038161003816119903 minus 119902

1003816100381610038161003816 Γ (2119898 + 1)

1205752Γ (119898 minus 119896 + 12) 119903 minus 119902 = 0

10038161003816100381610038161205731003816100381610038161003816 119903 minus 119902 = 0

(28)

Next we give a particular solution 119906120582(119878) and derive the

nonlinear equation for 119878119891(120582) By matching conditions (20)ndash

(22) we derive that

119906120582(119878) = minus

120582120572119878

120582120572 + 119902+

120582120572119870

120582120572 + 119903

11986212

= 0

11986211

=1

1198865Λ (119870)

[(11988641198865minus 11988631198866) 1198871

+ (11988611198866minus 11988621198865minus Λ (119870)) 119887

2minus Λ (119870)]

11986221

=1

1198865Λ (119870)

[minus119886311988661198871+ (11988611198866minus Λ (119870)) 119887

2

minus Λ (119870)]

11986222

=1

Λ (119870)[11988631198871minus 11988611198872]

(29)

where1198861= 120601120582(119870)

1198862= 120595120582(119870)

1198863=

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

1198864=

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

1198865=

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

1198866=

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

1198871= 119906120582(119870)

1198872=

minus120582120572

120582120572 + 119902

(30)

Furthermore letting 119878 = 119878119891(120582) in formula (23) and using

conditions (20) and (21) we derive the nonlinear equation for119878119891(120582)

1198601Λ(119878119891(120582)) = 119860

2120601120582(119878119891(120582))

+ 1198603119878119891(120582)

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

+ 1198604

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

(31)

where

1198601= (

120582120572119870

120582120572 + 119903minus

120582120572119870

120582120572 + 119902)

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

+120582120572120601120582(119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198602= minus

119902Λ (119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198603=

119902Λ (119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198604= minus

119903119870Λ (119878)

120582120572 + 119903

10038161003816100381610038161003816100381610038161003816119878=119870

(32)

Using the secant method to solve (31) we obtain 119878119891(120582) for

different values of 120582 Finally the optimal exercise boundaryand put option price can be expressed in terms of the Laplaceinversion

119878119891(120591) = L

minus1[119878119891(120582)

120582]

119881 (119878 120591) = Lminus1

[ (119878 120582)

120582]

(33)

Mathematical Problems in Engineering 5

4 Finite Difference Methods

Suppose 119878max is a large enough positive number such that119881(119878max 120591) asymp 0 for all 120591 isin [0 119879] Define uniform time andspace mesh

Δ120591 =119879

119872

120591119896= 119896Δ120591 119896 = 0 1 119872

Δ119878 =119878max119873

119878119895= 119895Δ119878 119895 = 0 1 119873

(34)

The time-fractional derivative 1198620119863120572

120591119881(119878 120591) at 119878 = 119878

119895 120591 = 120591

119899

can be formulated as (see [19])

119862

0119863120572

120591119881 (119878 120591)

10038161003816100381610038161003816119878=119878119895

120591=120591119899

=

119899

sum

119896=1

119887(119899)

119896

119881(119878119895 120591119896) minus 119881 (119878

119895 120591119896minus1

)

Δ120591+ 119874 ((Δ120591)

2minus120572)

(35)

with

119887(119899)

119896=

1

Γ (1 minus 120572)int

120591119896

120591119896minus1

119889120596

(120591119899minus 120596)120572

=1

Γ (2 minus 120572)[(120591119899minus 120591119896minus1

)1minus120572

minus (120591119899minus 120591119896)1minus120572

]

119896 = 1 2 119899 minus 1

119887(119899)

119899=

(Δ120591)1minus120572

Γ (2 minus 120572)

(36)

We approximate the space derivatives by central difference

1205972119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895minus1

120591119899) minus 2119881 (119878

119895 120591119899) + 119881 (119878

119895+1 120591119899)

(Δ119878)2

+ 119874 ((Δ119878)2)

120597119881

120597119878

10038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895+1

120591119899) minus 119881 (119878

119895minus1 120591119899)

2Δ119878+ 119874 ((Δ119878)

2)

(37)

Inserting (35) and (37) into PDE (10) and omitting the higherterm 119874((Δ120591)

2minus120572) and 119874((Δ119878)

2) we obtain the FDM scheme

(with notation 119881119899

119895asymp 119881(119878

119895 120591119899)) as follows

119888119895minus1

119881119899

119895minus1+ 119888119895119881119899

119895+ 119888119895+1

119881119899

119895+1= 119889119899

119895 (38)

where

119888119895minus1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895+

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119888119895= 119887(119899)

119899+

Δ120591

(Δ119878)212057521198782120573+2

119895

119888119895+1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895minus

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119889119899

119895= 119887(119899)

119899119881119899minus1

119895minus

119899minus1

sum

119896=1

119887(119899)

119896(119881119896

119895minus 119881119896minus1

119895)

(39)

The initial condition is1198810119895= max(119870minus119878

119895 0) and the left-hand

side boundary and the boundary conditions are specified bythe following algorithm

Differently from fixed boundary problems how to deter-mine the moving boundaries 119878

119891(120591) is the key for pricing

American option We design a simple algorithm namelyboundary-searching method for solving time-fractionalfree-boundary problem (10)ndash(14)

Boundary-Searching Algorithm

Step 1 Let 119878119891(0) = 119870 Since 119881

0

119895= max(119870 minus 119878

119895 0) we

determine 119869(1205910) by

119869 (1205910) = min119895

119895 | 1198810

119895= 0 (40)

So we know 119878119869(1205910

)minus1le 119878119891(0) le 119878

119869(1205910

)

Step 2 For 119899 = 1 119872

Step 21 Search backward for 119869(120591119899)

For 119894 = 119869(120591119899minus1

) minus 1 119869(120591119899minus1

) minus 2 1 0

Solve (38) with boundary conditions119881119899119894

= 119870minus119878119894

and 119881119899

119873= 0

If the solutions 119881119899

119894+1and 119881

119899

119894satisfy (119881

119899

119894+1minus

119881119899

119894)Δ119878 le minus1 then119869(120591119899) = 119894 + 1 Break

End If

End For

Then we have that 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899

)] More

accurately searching of 119878119891(120591119899) can be carried out in the next

substep

Step 22 Let 119878120579119891(120591119899) equiv 120579119878

119869(120591119899

)+ (1 minus 120579)119878

119869(120591119899

)minus1 0 le 120579 le 1 Then

redefine the boundary conditions for (38) as

119871minus1

(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1minus

1198710(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1= 119870 minus 119878

120579

119891(120591119899)

(41)

6 Mathematical Problems in Engineering

with Lagrange basis functions

119871ℓ(119878) = prod

119895=minus101

119895 =ℓ

[119878 minus (119869 (120591119899) + 119895) Δ119878] ℓ = minus1 0 1

(42)

Then using the solutions of (38) with the new boundarycondition (41) we calculate the approximation

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp

1198711015840

minus1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1

minus1198711015840

0(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711015840

1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1

(43)

We find the appropriate value of 0 le 120579 le 1 such that

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp minus1 (44)

Step 3 Output early exercise boundaries 119878120579

119891(120591119899) and Ameri-

can put option prices119881119899119895(for 119895 = 0 1 119869(120591

119899)minus1119881119899

119895= 119870minus119878

119895

and for 119895 = 119869(120591119899) 119873119881119899

119895are obtained fromFDM(38)with

the new boundary condition (41))

Remark 1 (a) The boundary-searching method is based onthe fact that the early exercise boundary 119878

119891(120591) is monotoni-

cally decreasing in 120591 and the derivative 120597119881120597119878 is monotoni-cally increasing with respect to 119878 (b) Note that the left-handside of (41) is just the quadratic Lagrange approximation of119881(119878120579

119891(120591119899) 120591119899) The role of Step 22 is to numerically search a

value 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899)] such that

120597

120597119878119881 (119878119891(120591119899) 120591119899) = minus1

119881 (119878119891(120591119899) 120591119899) = 119870 minus 119878

119891(120591119899)

(45)

In fact the boundary-searching method still works withoutStep 22 however Step 22 can lead to more accurate results

5 Numerical Examples

In this section we implement and compare the Laplace trans-form method and the boundary-searching finite differencemethod (FDM)The Laplace inversion is calculated byGaver-Wynn-Rho (GWR) algorithm in [20] The mesh parametersare taken as 119872 = 200 119873 = 800 and 119878max = 200 for FDMThe volatility at time 119905 = 0 is defined by 120590

0equiv 120575119878120573

0

Table 1 lists the computational values of American putoptions Columns entitled ldquoLTMrdquo and ldquoFDMrdquo representthe Laplace transform method and the boundary-searchingFDM respectively For 120572 = 1 the time-fractional derivativereduces to the first-order derivative in time and the problembecomes the classical American option pricing problem

Table 1 Prices of American put option at 120591 = 3

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = 0

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 12189 12362 34116 3479209 11771 11912 32651 3315707 10959 11028 29817 3007104 09778 09793 25770 2582902 09005 09002 23184 23191CPU time (s) 165 23468 173 24134

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = minus1

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 11877 12020 33325 3383409 11485 11604 31918 3229707 10722 10802 29208 2940004 09609 09657 25347 2539702 08877 08922 22876 22898CPU time (s) 153 23921 198 24722

From Table 1 we can see that when 120572 = 1 the results areconsistent with the prices listed in [12] At time to maturity120591 = 3 one could observe that the option prices are decreasingas 120572 is decreasing Moreover the LTM takes much less CPUtime than the FDM as observed from Table 1

In Figure 1 we plot the early exercise boundaries com-puted by LTM and FDM for the cases of 120572 = 07 and 120572 = 04The numbers of mesh nodes are taken as 119872 = 50 119873 = 200

for FDM It can be seen that all the boundaries at differentobtained by LTM are very close time 120591 obtained by LTM arevery close to those by FDM Figure 2 illustrates the effectof different time-fractional derivative order 120572 on the earlyexercise boundaries and option prices It can be seen fromFigure 2(a) that when the value of120572 is decreasing the exerciseregion will be shrunk in the region of time to maturity 120591

close to 0 and will be enlarged in the region of 120591 closeto 119879 From Figure 2(b) we observe that the time-fractionalderivatives have little effect on the option price for the casesof deep-in-the-money (119878

0≪ 119870) and deep-out-the-money

(1198780

≫ 119870) and have significant effect near to on-the-money(1198780asymp 119870)

6 Conclusions

There have been increasing applications of fractional PDEsin the field of option pricing In the history most focuseshave been on the valuation of options without early exercisefeatures In current work we study the American optionpricing with time-fractional PDEs The option pricing prob-lem is formulated into a time-fractional PDE with freeboundary Two methods namely Laplace transform method(LTM) and finite difference method (FDM) are proposed tosolve the time-fractional free-boundary problem Numerical

Mathematical Problems in Engineering 7

28

30

32

34

36

38

40

FDMLTM

0 05 1 15 2 25 330

31

32

33

34

35

36

37

38

39

40

FDMLTM

0 05 1 15 2 25 3

120572 = 07 1205900 = 02 120572 = 04 1205900 = 02

Time to maturity (120591) Time to maturity (120591)

Sf

(120591)

Sf

(120591)

Figure 1 Early exercise boundaries computed by LTM and FDM

0 05 1 15 2 25 310

15

20

25

30

35

40

120572 = 09

120572 = 07

120572 = 05

120572 = 03

Time to maturity (120591)

Sf

(120591)

(a)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

Underlying

Opt

ion

pric

e

50 60 70 800

2

4

6

120572 = 09

120572 = 07

120572 = 05

120572 = 03

(b)

Figure 2 Early exercise boundaries (a) and option prices at 120591 = 119879 = 3 (b) with different 120572 and fixed 1205900= 04 120573 = minus1 119903 = 005 119902 = 0 and

119870 = 40

examples show that the LTM is more efficient than theFDM The methods in this paper are quite different to thepredictor-corrector approach for pricing American optionsunder space-fractional derivatives in [11]The extension of the

methods in this paper to the American option pricing underspace-fractional models will be left for the future work Inaddition it will be also interesting to extend the methods tothe newly established models [21 22]

8 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant nos 15CX141110 andJBK1307012) and National Natural Science Foundation ofChina (Grant no 11471137)

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods World ScientificPublishers London UK 2012

[2] P Carr and L Wu ldquoThe finite moment log stable process andoption pricingrdquo The Journal of Finance vol 58 no 2 pp 597ndash626 2003

[3] A Cartea and D del-Castillo-Negrete ldquoFractional diffusionmodels of option prices in markets with jumpsrdquo Physica AStatistical Mechanics and Its Applications vol 374 no 2 pp749ndash763 2007

[4] W Chen X Xu and S-P Zhu ldquoAnalytically pricing European-style options under the modified Black-Scholes equation with aspatial-fractional derivativerdquoQuarterly of AppliedMathematicsvol 72 no 3 pp 597ndash611 2014

[5] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus amp Applied Analysis vol 3 no 1 pp 51ndash61 2000

[6] S Kumar A Yildirim Y Khan H Jafari K Sayevand and LWei ldquoAnalytical solution of fractional Black-Scholes Europeanoption pricing equation by using Laplace transformrdquo FractionalCalculus and Applied Analysis vol 2 pp 1ndash9 2012

[7] A Cartea ldquoDerivatives pricing with marked point processesusing tick-by-tick datardquo Quantitative Finance vol 13 no 1 pp111ndash123 2013

[8] G Jumarie ldquoStock exchange fractional dynamics defined asfractional exponential growth driven by (usual) Gaussian whitenoise Application to fractional Black-Scholes equationsrdquo Insur-ance Mathematics and Economics vol 42 no 1 pp 271ndash2872008

[9] J-R Liang JWangW-J ZhangW-Y Qiu and F-Y Ren ldquoThesolution to a bifractional Black-Scholes-Merton differentialequationrdquo International Journal of Pure and Applied Mathemat-ics vol 58 no 1 pp 99ndash112 2010

[10] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[11] W Chen X Xu and S-P Zhu ldquoA predictor-corrector approachfor pricing American options under the finite moment log-stable modelrdquo Applied Numerical Mathematics vol 97 pp 15ndash29 2015

[12] H Y Wong and J Zhao ldquoValuing American options underthe CEV model by Laplace-Carson transformsrdquo OperationsResearch Letters vol 38 no 5 pp 474ndash481 2010

[13] K Muthuraman ldquoA moving boundary approach to Americanoption pricingrdquo Journal of Economic Dynamics amp Control vol32 no 11 pp 3520ndash3537 2008

[14] A Chockalingam and K Muthuraman ldquoAn approximate mov-ing boundary method for American option pricingrdquo EuropeanJournal of Operational Research vol 240 no 2 pp 431ndash4382015

[15] J Cox ldquoNotes on option pricing I constant elasticity of variancediffusionsrdquoThe Journal of PortfolioManagement vol 22 pp 15ndash17 1996

[16] I Podlubny Fractional Differential Equations Academic Press1999

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

[18] D Davydov and V Linetsky ldquoPricing and hedging path-dependent options under the CEV processrdquo Management Sci-ence vol 47 no 7 pp 949ndash965 2001

[19] Y Jiang and J Ma ldquoMoving finite element methods for timefractional partial differential equationsrdquo Science China Mathe-matics vol 56 no 6 pp 1287ndash1300 2013

[20] P P Valko and S Vajda ldquoInversion of noise-free Laplacetransforms towards a standardized set of test problemsrdquo InverseProblems in Engineering vol 10 no 5 pp 467ndash483 2002

[21] C Huang X Ma and Q Lan ldquoAn empirical study on listedcompanyrsquos value of cash holdings an information asymmetryperspectiverdquoDiscrete Dynamics in Nature and Society vol 2014Article ID 897278 12 pages 2014

[22] C Huang X Yang X G Yang and H Sheng ldquoAn empiricalstudy of the effect of investor sentiment on returns of differentindustriesrdquo Mathematical Problems in Engineering vol 2014Article ID 545723 11 pages 2014

Submit your manuscripts athttpwwwhindawicom

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MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

where 119872119896119898

(119909) and 119882119896119898

(119909) are the Whittaker functions119868](119909) and 119870](119909) are the modified Bessel functions (see eg[18]) and

120583 = 119903 minus 119902

119909 (119878) =

10038161003816100381610038161205831003816100381610038161003816

12057521003816100381610038161003816120573

1003816100381610038161003816

119878minus2120573

119911 (119878) =1

1205751003816100381610038161003816120573

1003816100381610038161003816

119878minus120573

120598 = sign (120583120573)

119898 =1

41003816100381610038161003816120573

1003816100381610038161003816

119896 (120582) = 120598 (1

2+

1

4120573) minus

120582120572+ 119903

21003816100381610038161003816120583120573

1003816100381610038161003816

] =1

21003816100381610038161003816120573

1003816100381610038161003816

(27)

Moreover it can be calculated that

Λ (119878) = 120585 (119878) 120596120582

120585 (119878) = exp(120583

1205752120573119878minus2120573

)

120596120582=

21003816100381610038161003816119903 minus 119902

1003816100381610038161003816 Γ (2119898 + 1)

1205752Γ (119898 minus 119896 + 12) 119903 minus 119902 = 0

10038161003816100381610038161205731003816100381610038161003816 119903 minus 119902 = 0

(28)

Next we give a particular solution 119906120582(119878) and derive the

nonlinear equation for 119878119891(120582) By matching conditions (20)ndash

(22) we derive that

119906120582(119878) = minus

120582120572119878

120582120572 + 119902+

120582120572119870

120582120572 + 119903

11986212

= 0

11986211

=1

1198865Λ (119870)

[(11988641198865minus 11988631198866) 1198871

+ (11988611198866minus 11988621198865minus Λ (119870)) 119887

2minus Λ (119870)]

11986221

=1

1198865Λ (119870)

[minus119886311988661198871+ (11988611198866minus Λ (119870)) 119887

2

minus Λ (119870)]

11986222

=1

Λ (119870)[11988631198871minus 11988611198872]

(29)

where1198861= 120601120582(119870)

1198862= 120595120582(119870)

1198863=

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

1198864=

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

1198865=

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

1198866=

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

1198871= 119906120582(119870)

1198872=

minus120582120572

120582120572 + 119902

(30)

Furthermore letting 119878 = 119878119891(120582) in formula (23) and using

conditions (20) and (21) we derive the nonlinear equation for119878119891(120582)

1198601Λ(119878119891(120582)) = 119860

2120601120582(119878119891(120582))

+ 1198603119878119891(120582)

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

+ 1198604

119889120595120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119878119891

(120582)

(31)

where

1198601= (

120582120572119870

120582120572 + 119903minus

120582120572119870

120582120572 + 119902)

119889120601120582(119878)

119889119878

10038161003816100381610038161003816100381610038161003816119878=119870

+120582120572120601120582(119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198602= minus

119902Λ (119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198603=

119902Λ (119878)

120582120572 + 119902

10038161003816100381610038161003816100381610038161003816119878=119870

1198604= minus

119903119870Λ (119878)

120582120572 + 119903

10038161003816100381610038161003816100381610038161003816119878=119870

(32)

Using the secant method to solve (31) we obtain 119878119891(120582) for

different values of 120582 Finally the optimal exercise boundaryand put option price can be expressed in terms of the Laplaceinversion

119878119891(120591) = L

minus1[119878119891(120582)

120582]

119881 (119878 120591) = Lminus1

[ (119878 120582)

120582]

(33)

Mathematical Problems in Engineering 5

4 Finite Difference Methods

Suppose 119878max is a large enough positive number such that119881(119878max 120591) asymp 0 for all 120591 isin [0 119879] Define uniform time andspace mesh

Δ120591 =119879

119872

120591119896= 119896Δ120591 119896 = 0 1 119872

Δ119878 =119878max119873

119878119895= 119895Δ119878 119895 = 0 1 119873

(34)

The time-fractional derivative 1198620119863120572

120591119881(119878 120591) at 119878 = 119878

119895 120591 = 120591

119899

can be formulated as (see [19])

119862

0119863120572

120591119881 (119878 120591)

10038161003816100381610038161003816119878=119878119895

120591=120591119899

=

119899

sum

119896=1

119887(119899)

119896

119881(119878119895 120591119896) minus 119881 (119878

119895 120591119896minus1

)

Δ120591+ 119874 ((Δ120591)

2minus120572)

(35)

with

119887(119899)

119896=

1

Γ (1 minus 120572)int

120591119896

120591119896minus1

119889120596

(120591119899minus 120596)120572

=1

Γ (2 minus 120572)[(120591119899minus 120591119896minus1

)1minus120572

minus (120591119899minus 120591119896)1minus120572

]

119896 = 1 2 119899 minus 1

119887(119899)

119899=

(Δ120591)1minus120572

Γ (2 minus 120572)

(36)

We approximate the space derivatives by central difference

1205972119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895minus1

120591119899) minus 2119881 (119878

119895 120591119899) + 119881 (119878

119895+1 120591119899)

(Δ119878)2

+ 119874 ((Δ119878)2)

120597119881

120597119878

10038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895+1

120591119899) minus 119881 (119878

119895minus1 120591119899)

2Δ119878+ 119874 ((Δ119878)

2)

(37)

Inserting (35) and (37) into PDE (10) and omitting the higherterm 119874((Δ120591)

2minus120572) and 119874((Δ119878)

2) we obtain the FDM scheme

(with notation 119881119899

119895asymp 119881(119878

119895 120591119899)) as follows

119888119895minus1

119881119899

119895minus1+ 119888119895119881119899

119895+ 119888119895+1

119881119899

119895+1= 119889119899

119895 (38)

where

119888119895minus1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895+

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119888119895= 119887(119899)

119899+

Δ120591

(Δ119878)212057521198782120573+2

119895

119888119895+1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895minus

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119889119899

119895= 119887(119899)

119899119881119899minus1

119895minus

119899minus1

sum

119896=1

119887(119899)

119896(119881119896

119895minus 119881119896minus1

119895)

(39)

The initial condition is1198810119895= max(119870minus119878

119895 0) and the left-hand

side boundary and the boundary conditions are specified bythe following algorithm

Differently from fixed boundary problems how to deter-mine the moving boundaries 119878

119891(120591) is the key for pricing

American option We design a simple algorithm namelyboundary-searching method for solving time-fractionalfree-boundary problem (10)ndash(14)

Boundary-Searching Algorithm

Step 1 Let 119878119891(0) = 119870 Since 119881

0

119895= max(119870 minus 119878

119895 0) we

determine 119869(1205910) by

119869 (1205910) = min119895

119895 | 1198810

119895= 0 (40)

So we know 119878119869(1205910

)minus1le 119878119891(0) le 119878

119869(1205910

)

Step 2 For 119899 = 1 119872

Step 21 Search backward for 119869(120591119899)

For 119894 = 119869(120591119899minus1

) minus 1 119869(120591119899minus1

) minus 2 1 0

Solve (38) with boundary conditions119881119899119894

= 119870minus119878119894

and 119881119899

119873= 0

If the solutions 119881119899

119894+1and 119881

119899

119894satisfy (119881

119899

119894+1minus

119881119899

119894)Δ119878 le minus1 then119869(120591119899) = 119894 + 1 Break

End If

End For

Then we have that 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899

)] More

accurately searching of 119878119891(120591119899) can be carried out in the next

substep

Step 22 Let 119878120579119891(120591119899) equiv 120579119878

119869(120591119899

)+ (1 minus 120579)119878

119869(120591119899

)minus1 0 le 120579 le 1 Then

redefine the boundary conditions for (38) as

119871minus1

(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1minus

1198710(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1= 119870 minus 119878

120579

119891(120591119899)

(41)

6 Mathematical Problems in Engineering

with Lagrange basis functions

119871ℓ(119878) = prod

119895=minus101

119895 =ℓ

[119878 minus (119869 (120591119899) + 119895) Δ119878] ℓ = minus1 0 1

(42)

Then using the solutions of (38) with the new boundarycondition (41) we calculate the approximation

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp

1198711015840

minus1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1

minus1198711015840

0(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711015840

1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1

(43)

We find the appropriate value of 0 le 120579 le 1 such that

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp minus1 (44)

Step 3 Output early exercise boundaries 119878120579

119891(120591119899) and Ameri-

can put option prices119881119899119895(for 119895 = 0 1 119869(120591

119899)minus1119881119899

119895= 119870minus119878

119895

and for 119895 = 119869(120591119899) 119873119881119899

119895are obtained fromFDM(38)with

the new boundary condition (41))

Remark 1 (a) The boundary-searching method is based onthe fact that the early exercise boundary 119878

119891(120591) is monotoni-

cally decreasing in 120591 and the derivative 120597119881120597119878 is monotoni-cally increasing with respect to 119878 (b) Note that the left-handside of (41) is just the quadratic Lagrange approximation of119881(119878120579

119891(120591119899) 120591119899) The role of Step 22 is to numerically search a

value 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899)] such that

120597

120597119878119881 (119878119891(120591119899) 120591119899) = minus1

119881 (119878119891(120591119899) 120591119899) = 119870 minus 119878

119891(120591119899)

(45)

In fact the boundary-searching method still works withoutStep 22 however Step 22 can lead to more accurate results

5 Numerical Examples

In this section we implement and compare the Laplace trans-form method and the boundary-searching finite differencemethod (FDM)The Laplace inversion is calculated byGaver-Wynn-Rho (GWR) algorithm in [20] The mesh parametersare taken as 119872 = 200 119873 = 800 and 119878max = 200 for FDMThe volatility at time 119905 = 0 is defined by 120590

0equiv 120575119878120573

0

Table 1 lists the computational values of American putoptions Columns entitled ldquoLTMrdquo and ldquoFDMrdquo representthe Laplace transform method and the boundary-searchingFDM respectively For 120572 = 1 the time-fractional derivativereduces to the first-order derivative in time and the problembecomes the classical American option pricing problem

Table 1 Prices of American put option at 120591 = 3

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = 0

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 12189 12362 34116 3479209 11771 11912 32651 3315707 10959 11028 29817 3007104 09778 09793 25770 2582902 09005 09002 23184 23191CPU time (s) 165 23468 173 24134

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = minus1

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 11877 12020 33325 3383409 11485 11604 31918 3229707 10722 10802 29208 2940004 09609 09657 25347 2539702 08877 08922 22876 22898CPU time (s) 153 23921 198 24722

From Table 1 we can see that when 120572 = 1 the results areconsistent with the prices listed in [12] At time to maturity120591 = 3 one could observe that the option prices are decreasingas 120572 is decreasing Moreover the LTM takes much less CPUtime than the FDM as observed from Table 1

In Figure 1 we plot the early exercise boundaries com-puted by LTM and FDM for the cases of 120572 = 07 and 120572 = 04The numbers of mesh nodes are taken as 119872 = 50 119873 = 200

for FDM It can be seen that all the boundaries at differentobtained by LTM are very close time 120591 obtained by LTM arevery close to those by FDM Figure 2 illustrates the effectof different time-fractional derivative order 120572 on the earlyexercise boundaries and option prices It can be seen fromFigure 2(a) that when the value of120572 is decreasing the exerciseregion will be shrunk in the region of time to maturity 120591

close to 0 and will be enlarged in the region of 120591 closeto 119879 From Figure 2(b) we observe that the time-fractionalderivatives have little effect on the option price for the casesof deep-in-the-money (119878

0≪ 119870) and deep-out-the-money

(1198780

≫ 119870) and have significant effect near to on-the-money(1198780asymp 119870)

6 Conclusions

There have been increasing applications of fractional PDEsin the field of option pricing In the history most focuseshave been on the valuation of options without early exercisefeatures In current work we study the American optionpricing with time-fractional PDEs The option pricing prob-lem is formulated into a time-fractional PDE with freeboundary Two methods namely Laplace transform method(LTM) and finite difference method (FDM) are proposed tosolve the time-fractional free-boundary problem Numerical

Mathematical Problems in Engineering 7

28

30

32

34

36

38

40

FDMLTM

0 05 1 15 2 25 330

31

32

33

34

35

36

37

38

39

40

FDMLTM

0 05 1 15 2 25 3

120572 = 07 1205900 = 02 120572 = 04 1205900 = 02

Time to maturity (120591) Time to maturity (120591)

Sf

(120591)

Sf

(120591)

Figure 1 Early exercise boundaries computed by LTM and FDM

0 05 1 15 2 25 310

15

20

25

30

35

40

120572 = 09

120572 = 07

120572 = 05

120572 = 03

Time to maturity (120591)

Sf

(120591)

(a)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

Underlying

Opt

ion

pric

e

50 60 70 800

2

4

6

120572 = 09

120572 = 07

120572 = 05

120572 = 03

(b)

Figure 2 Early exercise boundaries (a) and option prices at 120591 = 119879 = 3 (b) with different 120572 and fixed 1205900= 04 120573 = minus1 119903 = 005 119902 = 0 and

119870 = 40

examples show that the LTM is more efficient than theFDM The methods in this paper are quite different to thepredictor-corrector approach for pricing American optionsunder space-fractional derivatives in [11]The extension of the

methods in this paper to the American option pricing underspace-fractional models will be left for the future work Inaddition it will be also interesting to extend the methods tothe newly established models [21 22]

8 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant nos 15CX141110 andJBK1307012) and National Natural Science Foundation ofChina (Grant no 11471137)

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods World ScientificPublishers London UK 2012

[2] P Carr and L Wu ldquoThe finite moment log stable process andoption pricingrdquo The Journal of Finance vol 58 no 2 pp 597ndash626 2003

[3] A Cartea and D del-Castillo-Negrete ldquoFractional diffusionmodels of option prices in markets with jumpsrdquo Physica AStatistical Mechanics and Its Applications vol 374 no 2 pp749ndash763 2007

[4] W Chen X Xu and S-P Zhu ldquoAnalytically pricing European-style options under the modified Black-Scholes equation with aspatial-fractional derivativerdquoQuarterly of AppliedMathematicsvol 72 no 3 pp 597ndash611 2014

[5] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus amp Applied Analysis vol 3 no 1 pp 51ndash61 2000

[6] S Kumar A Yildirim Y Khan H Jafari K Sayevand and LWei ldquoAnalytical solution of fractional Black-Scholes Europeanoption pricing equation by using Laplace transformrdquo FractionalCalculus and Applied Analysis vol 2 pp 1ndash9 2012

[7] A Cartea ldquoDerivatives pricing with marked point processesusing tick-by-tick datardquo Quantitative Finance vol 13 no 1 pp111ndash123 2013

[8] G Jumarie ldquoStock exchange fractional dynamics defined asfractional exponential growth driven by (usual) Gaussian whitenoise Application to fractional Black-Scholes equationsrdquo Insur-ance Mathematics and Economics vol 42 no 1 pp 271ndash2872008

[9] J-R Liang JWangW-J ZhangW-Y Qiu and F-Y Ren ldquoThesolution to a bifractional Black-Scholes-Merton differentialequationrdquo International Journal of Pure and Applied Mathemat-ics vol 58 no 1 pp 99ndash112 2010

[10] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[11] W Chen X Xu and S-P Zhu ldquoA predictor-corrector approachfor pricing American options under the finite moment log-stable modelrdquo Applied Numerical Mathematics vol 97 pp 15ndash29 2015

[12] H Y Wong and J Zhao ldquoValuing American options underthe CEV model by Laplace-Carson transformsrdquo OperationsResearch Letters vol 38 no 5 pp 474ndash481 2010

[13] K Muthuraman ldquoA moving boundary approach to Americanoption pricingrdquo Journal of Economic Dynamics amp Control vol32 no 11 pp 3520ndash3537 2008

[14] A Chockalingam and K Muthuraman ldquoAn approximate mov-ing boundary method for American option pricingrdquo EuropeanJournal of Operational Research vol 240 no 2 pp 431ndash4382015

[15] J Cox ldquoNotes on option pricing I constant elasticity of variancediffusionsrdquoThe Journal of PortfolioManagement vol 22 pp 15ndash17 1996

[16] I Podlubny Fractional Differential Equations Academic Press1999

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

[18] D Davydov and V Linetsky ldquoPricing and hedging path-dependent options under the CEV processrdquo Management Sci-ence vol 47 no 7 pp 949ndash965 2001

[19] Y Jiang and J Ma ldquoMoving finite element methods for timefractional partial differential equationsrdquo Science China Mathe-matics vol 56 no 6 pp 1287ndash1300 2013

[20] P P Valko and S Vajda ldquoInversion of noise-free Laplacetransforms towards a standardized set of test problemsrdquo InverseProblems in Engineering vol 10 no 5 pp 467ndash483 2002

[21] C Huang X Ma and Q Lan ldquoAn empirical study on listedcompanyrsquos value of cash holdings an information asymmetryperspectiverdquoDiscrete Dynamics in Nature and Society vol 2014Article ID 897278 12 pages 2014

[22] C Huang X Yang X G Yang and H Sheng ldquoAn empiricalstudy of the effect of investor sentiment on returns of differentindustriesrdquo Mathematical Problems in Engineering vol 2014Article ID 545723 11 pages 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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

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

Mathematical Problems in Engineering 5

4 Finite Difference Methods

Suppose 119878max is a large enough positive number such that119881(119878max 120591) asymp 0 for all 120591 isin [0 119879] Define uniform time andspace mesh

Δ120591 =119879

119872

120591119896= 119896Δ120591 119896 = 0 1 119872

Δ119878 =119878max119873

119878119895= 119895Δ119878 119895 = 0 1 119873

(34)

The time-fractional derivative 1198620119863120572

120591119881(119878 120591) at 119878 = 119878

119895 120591 = 120591

119899

can be formulated as (see [19])

119862

0119863120572

120591119881 (119878 120591)

10038161003816100381610038161003816119878=119878119895

120591=120591119899

=

119899

sum

119896=1

119887(119899)

119896

119881(119878119895 120591119896) minus 119881 (119878

119895 120591119896minus1

)

Δ120591+ 119874 ((Δ120591)

2minus120572)

(35)

with

119887(119899)

119896=

1

Γ (1 minus 120572)int

120591119896

120591119896minus1

119889120596

(120591119899minus 120596)120572

=1

Γ (2 minus 120572)[(120591119899minus 120591119896minus1

)1minus120572

minus (120591119899minus 120591119896)1minus120572

]

119896 = 1 2 119899 minus 1

119887(119899)

119899=

(Δ120591)1minus120572

Γ (2 minus 120572)

(36)

We approximate the space derivatives by central difference

1205972119881

1205971198782

100381610038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895minus1

120591119899) minus 2119881 (119878

119895 120591119899) + 119881 (119878

119895+1 120591119899)

(Δ119878)2

+ 119874 ((Δ119878)2)

120597119881

120597119878

10038161003816100381610038161003816100381610038161003816119878=119878119895

120591=120591119899

=119881 (119878119895+1

120591119899) minus 119881 (119878

119895minus1 120591119899)

2Δ119878+ 119874 ((Δ119878)

2)

(37)

Inserting (35) and (37) into PDE (10) and omitting the higherterm 119874((Δ120591)

2minus120572) and 119874((Δ119878)

2) we obtain the FDM scheme

(with notation 119881119899

119895asymp 119881(119878

119895 120591119899)) as follows

119888119895minus1

119881119899

119895minus1+ 119888119895119881119899

119895+ 119888119895+1

119881119899

119895+1= 119889119899

119895 (38)

where

119888119895minus1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895+

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119888119895= 119887(119899)

119899+

Δ120591

(Δ119878)212057521198782120573+2

119895

119888119895+1

= minusΔ120591

2 (Δ119878)212057521198782120573+2

119895minus

Δ120591

2Δ119878(119903 minus 119902) 119878

119895

119889119899

119895= 119887(119899)

119899119881119899minus1

119895minus

119899minus1

sum

119896=1

119887(119899)

119896(119881119896

119895minus 119881119896minus1

119895)

(39)

The initial condition is1198810119895= max(119870minus119878

119895 0) and the left-hand

side boundary and the boundary conditions are specified bythe following algorithm

Differently from fixed boundary problems how to deter-mine the moving boundaries 119878

119891(120591) is the key for pricing

American option We design a simple algorithm namelyboundary-searching method for solving time-fractionalfree-boundary problem (10)ndash(14)

Boundary-Searching Algorithm

Step 1 Let 119878119891(0) = 119870 Since 119881

0

119895= max(119870 minus 119878

119895 0) we

determine 119869(1205910) by

119869 (1205910) = min119895

119895 | 1198810

119895= 0 (40)

So we know 119878119869(1205910

)minus1le 119878119891(0) le 119878

119869(1205910

)

Step 2 For 119899 = 1 119872

Step 21 Search backward for 119869(120591119899)

For 119894 = 119869(120591119899minus1

) minus 1 119869(120591119899minus1

) minus 2 1 0

Solve (38) with boundary conditions119881119899119894

= 119870minus119878119894

and 119881119899

119873= 0

If the solutions 119881119899

119894+1and 119881

119899

119894satisfy (119881

119899

119894+1minus

119881119899

119894)Δ119878 le minus1 then119869(120591119899) = 119894 + 1 Break

End If

End For

Then we have that 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899

)] More

accurately searching of 119878119891(120591119899) can be carried out in the next

substep

Step 22 Let 119878120579119891(120591119899) equiv 120579119878

119869(120591119899

)+ (1 minus 120579)119878

119869(120591119899

)minus1 0 le 120579 le 1 Then

redefine the boundary conditions for (38) as

119871minus1

(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1minus

1198710(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1= 119870 minus 119878

120579

119891(120591119899)

(41)

6 Mathematical Problems in Engineering

with Lagrange basis functions

119871ℓ(119878) = prod

119895=minus101

119895 =ℓ

[119878 minus (119869 (120591119899) + 119895) Δ119878] ℓ = minus1 0 1

(42)

Then using the solutions of (38) with the new boundarycondition (41) we calculate the approximation

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp

1198711015840

minus1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1

minus1198711015840

0(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711015840

1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1

(43)

We find the appropriate value of 0 le 120579 le 1 such that

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp minus1 (44)

Step 3 Output early exercise boundaries 119878120579

119891(120591119899) and Ameri-

can put option prices119881119899119895(for 119895 = 0 1 119869(120591

119899)minus1119881119899

119895= 119870minus119878

119895

and for 119895 = 119869(120591119899) 119873119881119899

119895are obtained fromFDM(38)with

the new boundary condition (41))

Remark 1 (a) The boundary-searching method is based onthe fact that the early exercise boundary 119878

119891(120591) is monotoni-

cally decreasing in 120591 and the derivative 120597119881120597119878 is monotoni-cally increasing with respect to 119878 (b) Note that the left-handside of (41) is just the quadratic Lagrange approximation of119881(119878120579

119891(120591119899) 120591119899) The role of Step 22 is to numerically search a

value 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899)] such that

120597

120597119878119881 (119878119891(120591119899) 120591119899) = minus1

119881 (119878119891(120591119899) 120591119899) = 119870 minus 119878

119891(120591119899)

(45)

In fact the boundary-searching method still works withoutStep 22 however Step 22 can lead to more accurate results

5 Numerical Examples

In this section we implement and compare the Laplace trans-form method and the boundary-searching finite differencemethod (FDM)The Laplace inversion is calculated byGaver-Wynn-Rho (GWR) algorithm in [20] The mesh parametersare taken as 119872 = 200 119873 = 800 and 119878max = 200 for FDMThe volatility at time 119905 = 0 is defined by 120590

0equiv 120575119878120573

0

Table 1 lists the computational values of American putoptions Columns entitled ldquoLTMrdquo and ldquoFDMrdquo representthe Laplace transform method and the boundary-searchingFDM respectively For 120572 = 1 the time-fractional derivativereduces to the first-order derivative in time and the problembecomes the classical American option pricing problem

Table 1 Prices of American put option at 120591 = 3

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = 0

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 12189 12362 34116 3479209 11771 11912 32651 3315707 10959 11028 29817 3007104 09778 09793 25770 2582902 09005 09002 23184 23191CPU time (s) 165 23468 173 24134

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = minus1

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 11877 12020 33325 3383409 11485 11604 31918 3229707 10722 10802 29208 2940004 09609 09657 25347 2539702 08877 08922 22876 22898CPU time (s) 153 23921 198 24722

From Table 1 we can see that when 120572 = 1 the results areconsistent with the prices listed in [12] At time to maturity120591 = 3 one could observe that the option prices are decreasingas 120572 is decreasing Moreover the LTM takes much less CPUtime than the FDM as observed from Table 1

In Figure 1 we plot the early exercise boundaries com-puted by LTM and FDM for the cases of 120572 = 07 and 120572 = 04The numbers of mesh nodes are taken as 119872 = 50 119873 = 200

for FDM It can be seen that all the boundaries at differentobtained by LTM are very close time 120591 obtained by LTM arevery close to those by FDM Figure 2 illustrates the effectof different time-fractional derivative order 120572 on the earlyexercise boundaries and option prices It can be seen fromFigure 2(a) that when the value of120572 is decreasing the exerciseregion will be shrunk in the region of time to maturity 120591

close to 0 and will be enlarged in the region of 120591 closeto 119879 From Figure 2(b) we observe that the time-fractionalderivatives have little effect on the option price for the casesof deep-in-the-money (119878

0≪ 119870) and deep-out-the-money

(1198780

≫ 119870) and have significant effect near to on-the-money(1198780asymp 119870)

6 Conclusions

There have been increasing applications of fractional PDEsin the field of option pricing In the history most focuseshave been on the valuation of options without early exercisefeatures In current work we study the American optionpricing with time-fractional PDEs The option pricing prob-lem is formulated into a time-fractional PDE with freeboundary Two methods namely Laplace transform method(LTM) and finite difference method (FDM) are proposed tosolve the time-fractional free-boundary problem Numerical

Mathematical Problems in Engineering 7

28

30

32

34

36

38

40

FDMLTM

0 05 1 15 2 25 330

31

32

33

34

35

36

37

38

39

40

FDMLTM

0 05 1 15 2 25 3

120572 = 07 1205900 = 02 120572 = 04 1205900 = 02

Time to maturity (120591) Time to maturity (120591)

Sf

(120591)

Sf

(120591)

Figure 1 Early exercise boundaries computed by LTM and FDM

0 05 1 15 2 25 310

15

20

25

30

35

40

120572 = 09

120572 = 07

120572 = 05

120572 = 03

Time to maturity (120591)

Sf

(120591)

(a)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

Underlying

Opt

ion

pric

e

50 60 70 800

2

4

6

120572 = 09

120572 = 07

120572 = 05

120572 = 03

(b)

Figure 2 Early exercise boundaries (a) and option prices at 120591 = 119879 = 3 (b) with different 120572 and fixed 1205900= 04 120573 = minus1 119903 = 005 119902 = 0 and

119870 = 40

examples show that the LTM is more efficient than theFDM The methods in this paper are quite different to thepredictor-corrector approach for pricing American optionsunder space-fractional derivatives in [11]The extension of the

methods in this paper to the American option pricing underspace-fractional models will be left for the future work Inaddition it will be also interesting to extend the methods tothe newly established models [21 22]

8 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant nos 15CX141110 andJBK1307012) and National Natural Science Foundation ofChina (Grant no 11471137)

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods World ScientificPublishers London UK 2012

[2] P Carr and L Wu ldquoThe finite moment log stable process andoption pricingrdquo The Journal of Finance vol 58 no 2 pp 597ndash626 2003

[3] A Cartea and D del-Castillo-Negrete ldquoFractional diffusionmodels of option prices in markets with jumpsrdquo Physica AStatistical Mechanics and Its Applications vol 374 no 2 pp749ndash763 2007

[4] W Chen X Xu and S-P Zhu ldquoAnalytically pricing European-style options under the modified Black-Scholes equation with aspatial-fractional derivativerdquoQuarterly of AppliedMathematicsvol 72 no 3 pp 597ndash611 2014

[5] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus amp Applied Analysis vol 3 no 1 pp 51ndash61 2000

[6] S Kumar A Yildirim Y Khan H Jafari K Sayevand and LWei ldquoAnalytical solution of fractional Black-Scholes Europeanoption pricing equation by using Laplace transformrdquo FractionalCalculus and Applied Analysis vol 2 pp 1ndash9 2012

[7] A Cartea ldquoDerivatives pricing with marked point processesusing tick-by-tick datardquo Quantitative Finance vol 13 no 1 pp111ndash123 2013

[8] G Jumarie ldquoStock exchange fractional dynamics defined asfractional exponential growth driven by (usual) Gaussian whitenoise Application to fractional Black-Scholes equationsrdquo Insur-ance Mathematics and Economics vol 42 no 1 pp 271ndash2872008

[9] J-R Liang JWangW-J ZhangW-Y Qiu and F-Y Ren ldquoThesolution to a bifractional Black-Scholes-Merton differentialequationrdquo International Journal of Pure and Applied Mathemat-ics vol 58 no 1 pp 99ndash112 2010

[10] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[11] W Chen X Xu and S-P Zhu ldquoA predictor-corrector approachfor pricing American options under the finite moment log-stable modelrdquo Applied Numerical Mathematics vol 97 pp 15ndash29 2015

[12] H Y Wong and J Zhao ldquoValuing American options underthe CEV model by Laplace-Carson transformsrdquo OperationsResearch Letters vol 38 no 5 pp 474ndash481 2010

[13] K Muthuraman ldquoA moving boundary approach to Americanoption pricingrdquo Journal of Economic Dynamics amp Control vol32 no 11 pp 3520ndash3537 2008

[14] A Chockalingam and K Muthuraman ldquoAn approximate mov-ing boundary method for American option pricingrdquo EuropeanJournal of Operational Research vol 240 no 2 pp 431ndash4382015

[15] J Cox ldquoNotes on option pricing I constant elasticity of variancediffusionsrdquoThe Journal of PortfolioManagement vol 22 pp 15ndash17 1996

[16] I Podlubny Fractional Differential Equations Academic Press1999

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

[18] D Davydov and V Linetsky ldquoPricing and hedging path-dependent options under the CEV processrdquo Management Sci-ence vol 47 no 7 pp 949ndash965 2001

[19] Y Jiang and J Ma ldquoMoving finite element methods for timefractional partial differential equationsrdquo Science China Mathe-matics vol 56 no 6 pp 1287ndash1300 2013

[20] P P Valko and S Vajda ldquoInversion of noise-free Laplacetransforms towards a standardized set of test problemsrdquo InverseProblems in Engineering vol 10 no 5 pp 467ndash483 2002

[21] C Huang X Ma and Q Lan ldquoAn empirical study on listedcompanyrsquos value of cash holdings an information asymmetryperspectiverdquoDiscrete Dynamics in Nature and Society vol 2014Article ID 897278 12 pages 2014

[22] C Huang X Yang X G Yang and H Sheng ldquoAn empiricalstudy of the effect of investor sentiment on returns of differentindustriesrdquo Mathematical Problems in Engineering vol 2014Article ID 545723 11 pages 2014

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 Mathematical Problems in Engineering

with Lagrange basis functions

119871ℓ(119878) = prod

119895=minus101

119895 =ℓ

[119878 minus (119869 (120591119899) + 119895) Δ119878] ℓ = minus1 0 1

(42)

Then using the solutions of (38) with the new boundarycondition (41) we calculate the approximation

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp

1198711015840

minus1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)minus1

minus1198711015840

0(119878120579

119891(120591119899))

(Δ119878)2

119881119899

119869(120591119899

)

+1198711015840

1(119878120579

119891(120591119899))

2 (Δ119878)2

119881119899

119869(120591119899

)+1

(43)

We find the appropriate value of 0 le 120579 le 1 such that

120597

120597119878119881 (119878120579

119891(120591119899) 120591119899) asymp minus1 (44)

Step 3 Output early exercise boundaries 119878120579

119891(120591119899) and Ameri-

can put option prices119881119899119895(for 119895 = 0 1 119869(120591

119899)minus1119881119899

119895= 119870minus119878

119895

and for 119895 = 119869(120591119899) 119873119881119899

119895are obtained fromFDM(38)with

the new boundary condition (41))

Remark 1 (a) The boundary-searching method is based onthe fact that the early exercise boundary 119878

119891(120591) is monotoni-

cally decreasing in 120591 and the derivative 120597119881120597119878 is monotoni-cally increasing with respect to 119878 (b) Note that the left-handside of (41) is just the quadratic Lagrange approximation of119881(119878120579

119891(120591119899) 120591119899) The role of Step 22 is to numerically search a

value 119878119891(120591119899) isin [119878

119869(120591119899

)minus1 119878119869(120591119899)] such that

120597

120597119878119881 (119878119891(120591119899) 120591119899) = minus1

119881 (119878119891(120591119899) 120591119899) = 119870 minus 119878

119891(120591119899)

(45)

In fact the boundary-searching method still works withoutStep 22 however Step 22 can lead to more accurate results

5 Numerical Examples

In this section we implement and compare the Laplace trans-form method and the boundary-searching finite differencemethod (FDM)The Laplace inversion is calculated byGaver-Wynn-Rho (GWR) algorithm in [20] The mesh parametersare taken as 119872 = 200 119873 = 800 and 119878max = 200 for FDMThe volatility at time 119905 = 0 is defined by 120590

0equiv 120575119878120573

0

Table 1 lists the computational values of American putoptions Columns entitled ldquoLTMrdquo and ldquoFDMrdquo representthe Laplace transform method and the boundary-searchingFDM respectively For 120572 = 1 the time-fractional derivativereduces to the first-order derivative in time and the problembecomes the classical American option pricing problem

Table 1 Prices of American put option at 120591 = 3

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = 0

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 12189 12362 34116 3479209 11771 11912 32651 3315707 10959 11028 29817 3007104 09778 09793 25770 2582902 09005 09002 23184 23191CPU time (s) 165 23468 173 24134

119903 = 005 119902 = 0 119879 = 3 1198780= 40 119870 = 40 120573 = minus1

1205721205900= 01 120590

0= 02

LTM FDM LTM FDM10 11877 12020 33325 3383409 11485 11604 31918 3229707 10722 10802 29208 2940004 09609 09657 25347 2539702 08877 08922 22876 22898CPU time (s) 153 23921 198 24722

From Table 1 we can see that when 120572 = 1 the results areconsistent with the prices listed in [12] At time to maturity120591 = 3 one could observe that the option prices are decreasingas 120572 is decreasing Moreover the LTM takes much less CPUtime than the FDM as observed from Table 1

In Figure 1 we plot the early exercise boundaries com-puted by LTM and FDM for the cases of 120572 = 07 and 120572 = 04The numbers of mesh nodes are taken as 119872 = 50 119873 = 200

for FDM It can be seen that all the boundaries at differentobtained by LTM are very close time 120591 obtained by LTM arevery close to those by FDM Figure 2 illustrates the effectof different time-fractional derivative order 120572 on the earlyexercise boundaries and option prices It can be seen fromFigure 2(a) that when the value of120572 is decreasing the exerciseregion will be shrunk in the region of time to maturity 120591

close to 0 and will be enlarged in the region of 120591 closeto 119879 From Figure 2(b) we observe that the time-fractionalderivatives have little effect on the option price for the casesof deep-in-the-money (119878

0≪ 119870) and deep-out-the-money

(1198780

≫ 119870) and have significant effect near to on-the-money(1198780asymp 119870)

6 Conclusions

There have been increasing applications of fractional PDEsin the field of option pricing In the history most focuseshave been on the valuation of options without early exercisefeatures In current work we study the American optionpricing with time-fractional PDEs The option pricing prob-lem is formulated into a time-fractional PDE with freeboundary Two methods namely Laplace transform method(LTM) and finite difference method (FDM) are proposed tosolve the time-fractional free-boundary problem Numerical

Mathematical Problems in Engineering 7

28

30

32

34

36

38

40

FDMLTM

0 05 1 15 2 25 330

31

32

33

34

35

36

37

38

39

40

FDMLTM

0 05 1 15 2 25 3

120572 = 07 1205900 = 02 120572 = 04 1205900 = 02

Time to maturity (120591) Time to maturity (120591)

Sf

(120591)

Sf

(120591)

Figure 1 Early exercise boundaries computed by LTM and FDM

0 05 1 15 2 25 310

15

20

25

30

35

40

120572 = 09

120572 = 07

120572 = 05

120572 = 03

Time to maturity (120591)

Sf

(120591)

(a)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

Underlying

Opt

ion

pric

e

50 60 70 800

2

4

6

120572 = 09

120572 = 07

120572 = 05

120572 = 03

(b)

Figure 2 Early exercise boundaries (a) and option prices at 120591 = 119879 = 3 (b) with different 120572 and fixed 1205900= 04 120573 = minus1 119903 = 005 119902 = 0 and

119870 = 40

examples show that the LTM is more efficient than theFDM The methods in this paper are quite different to thepredictor-corrector approach for pricing American optionsunder space-fractional derivatives in [11]The extension of the

methods in this paper to the American option pricing underspace-fractional models will be left for the future work Inaddition it will be also interesting to extend the methods tothe newly established models [21 22]

8 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant nos 15CX141110 andJBK1307012) and National Natural Science Foundation ofChina (Grant no 11471137)

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods World ScientificPublishers London UK 2012

[2] P Carr and L Wu ldquoThe finite moment log stable process andoption pricingrdquo The Journal of Finance vol 58 no 2 pp 597ndash626 2003

[3] A Cartea and D del-Castillo-Negrete ldquoFractional diffusionmodels of option prices in markets with jumpsrdquo Physica AStatistical Mechanics and Its Applications vol 374 no 2 pp749ndash763 2007

[4] W Chen X Xu and S-P Zhu ldquoAnalytically pricing European-style options under the modified Black-Scholes equation with aspatial-fractional derivativerdquoQuarterly of AppliedMathematicsvol 72 no 3 pp 597ndash611 2014

[5] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus amp Applied Analysis vol 3 no 1 pp 51ndash61 2000

[6] S Kumar A Yildirim Y Khan H Jafari K Sayevand and LWei ldquoAnalytical solution of fractional Black-Scholes Europeanoption pricing equation by using Laplace transformrdquo FractionalCalculus and Applied Analysis vol 2 pp 1ndash9 2012

[7] A Cartea ldquoDerivatives pricing with marked point processesusing tick-by-tick datardquo Quantitative Finance vol 13 no 1 pp111ndash123 2013

[8] G Jumarie ldquoStock exchange fractional dynamics defined asfractional exponential growth driven by (usual) Gaussian whitenoise Application to fractional Black-Scholes equationsrdquo Insur-ance Mathematics and Economics vol 42 no 1 pp 271ndash2872008

[9] J-R Liang JWangW-J ZhangW-Y Qiu and F-Y Ren ldquoThesolution to a bifractional Black-Scholes-Merton differentialequationrdquo International Journal of Pure and Applied Mathemat-ics vol 58 no 1 pp 99ndash112 2010

[10] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[11] W Chen X Xu and S-P Zhu ldquoA predictor-corrector approachfor pricing American options under the finite moment log-stable modelrdquo Applied Numerical Mathematics vol 97 pp 15ndash29 2015

[12] H Y Wong and J Zhao ldquoValuing American options underthe CEV model by Laplace-Carson transformsrdquo OperationsResearch Letters vol 38 no 5 pp 474ndash481 2010

[13] K Muthuraman ldquoA moving boundary approach to Americanoption pricingrdquo Journal of Economic Dynamics amp Control vol32 no 11 pp 3520ndash3537 2008

[14] A Chockalingam and K Muthuraman ldquoAn approximate mov-ing boundary method for American option pricingrdquo EuropeanJournal of Operational Research vol 240 no 2 pp 431ndash4382015

[15] J Cox ldquoNotes on option pricing I constant elasticity of variancediffusionsrdquoThe Journal of PortfolioManagement vol 22 pp 15ndash17 1996

[16] I Podlubny Fractional Differential Equations Academic Press1999

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

[18] D Davydov and V Linetsky ldquoPricing and hedging path-dependent options under the CEV processrdquo Management Sci-ence vol 47 no 7 pp 949ndash965 2001

[19] Y Jiang and J Ma ldquoMoving finite element methods for timefractional partial differential equationsrdquo Science China Mathe-matics vol 56 no 6 pp 1287ndash1300 2013

[20] P P Valko and S Vajda ldquoInversion of noise-free Laplacetransforms towards a standardized set of test problemsrdquo InverseProblems in Engineering vol 10 no 5 pp 467ndash483 2002

[21] C Huang X Ma and Q Lan ldquoAn empirical study on listedcompanyrsquos value of cash holdings an information asymmetryperspectiverdquoDiscrete Dynamics in Nature and Society vol 2014Article ID 897278 12 pages 2014

[22] C Huang X Yang X G Yang and H Sheng ldquoAn empiricalstudy of the effect of investor sentiment on returns of differentindustriesrdquo Mathematical Problems in Engineering vol 2014Article ID 545723 11 pages 2014

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

Mathematical Problems in Engineering 7

28

30

32

34

36

38

40

FDMLTM

0 05 1 15 2 25 330

31

32

33

34

35

36

37

38

39

40

FDMLTM

0 05 1 15 2 25 3

120572 = 07 1205900 = 02 120572 = 04 1205900 = 02

Time to maturity (120591) Time to maturity (120591)

Sf

(120591)

Sf

(120591)

Figure 1 Early exercise boundaries computed by LTM and FDM

0 05 1 15 2 25 310

15

20

25

30

35

40

120572 = 09

120572 = 07

120572 = 05

120572 = 03

Time to maturity (120591)

Sf

(120591)

(a)

0 20 40 60 80 1000

5

10

15

20

25

30

35

40

45

Underlying

Opt

ion

pric

e

50 60 70 800

2

4

6

120572 = 09

120572 = 07

120572 = 05

120572 = 03

(b)

Figure 2 Early exercise boundaries (a) and option prices at 120591 = 119879 = 3 (b) with different 120572 and fixed 1205900= 04 120573 = minus1 119903 = 005 119902 = 0 and

119870 = 40

examples show that the LTM is more efficient than theFDM The methods in this paper are quite different to thepredictor-corrector approach for pricing American optionsunder space-fractional derivatives in [11]The extension of the

methods in this paper to the American option pricing underspace-fractional models will be left for the future work Inaddition it will be also interesting to extend the methods tothe newly established models [21 22]

8 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant nos 15CX141110 andJBK1307012) and National Natural Science Foundation ofChina (Grant no 11471137)

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods World ScientificPublishers London UK 2012

[2] P Carr and L Wu ldquoThe finite moment log stable process andoption pricingrdquo The Journal of Finance vol 58 no 2 pp 597ndash626 2003

[3] A Cartea and D del-Castillo-Negrete ldquoFractional diffusionmodels of option prices in markets with jumpsrdquo Physica AStatistical Mechanics and Its Applications vol 374 no 2 pp749ndash763 2007

[4] W Chen X Xu and S-P Zhu ldquoAnalytically pricing European-style options under the modified Black-Scholes equation with aspatial-fractional derivativerdquoQuarterly of AppliedMathematicsvol 72 no 3 pp 597ndash611 2014

[5] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus amp Applied Analysis vol 3 no 1 pp 51ndash61 2000

[6] S Kumar A Yildirim Y Khan H Jafari K Sayevand and LWei ldquoAnalytical solution of fractional Black-Scholes Europeanoption pricing equation by using Laplace transformrdquo FractionalCalculus and Applied Analysis vol 2 pp 1ndash9 2012

[7] A Cartea ldquoDerivatives pricing with marked point processesusing tick-by-tick datardquo Quantitative Finance vol 13 no 1 pp111ndash123 2013

[8] G Jumarie ldquoStock exchange fractional dynamics defined asfractional exponential growth driven by (usual) Gaussian whitenoise Application to fractional Black-Scholes equationsrdquo Insur-ance Mathematics and Economics vol 42 no 1 pp 271ndash2872008

[9] J-R Liang JWangW-J ZhangW-Y Qiu and F-Y Ren ldquoThesolution to a bifractional Black-Scholes-Merton differentialequationrdquo International Journal of Pure and Applied Mathemat-ics vol 58 no 1 pp 99ndash112 2010

[10] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[11] W Chen X Xu and S-P Zhu ldquoA predictor-corrector approachfor pricing American options under the finite moment log-stable modelrdquo Applied Numerical Mathematics vol 97 pp 15ndash29 2015

[12] H Y Wong and J Zhao ldquoValuing American options underthe CEV model by Laplace-Carson transformsrdquo OperationsResearch Letters vol 38 no 5 pp 474ndash481 2010

[13] K Muthuraman ldquoA moving boundary approach to Americanoption pricingrdquo Journal of Economic Dynamics amp Control vol32 no 11 pp 3520ndash3537 2008

[14] A Chockalingam and K Muthuraman ldquoAn approximate mov-ing boundary method for American option pricingrdquo EuropeanJournal of Operational Research vol 240 no 2 pp 431ndash4382015

[15] J Cox ldquoNotes on option pricing I constant elasticity of variancediffusionsrdquoThe Journal of PortfolioManagement vol 22 pp 15ndash17 1996

[16] I Podlubny Fractional Differential Equations Academic Press1999

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

[18] D Davydov and V Linetsky ldquoPricing and hedging path-dependent options under the CEV processrdquo Management Sci-ence vol 47 no 7 pp 949ndash965 2001

[19] Y Jiang and J Ma ldquoMoving finite element methods for timefractional partial differential equationsrdquo Science China Mathe-matics vol 56 no 6 pp 1287ndash1300 2013

[20] P P Valko and S Vajda ldquoInversion of noise-free Laplacetransforms towards a standardized set of test problemsrdquo InverseProblems in Engineering vol 10 no 5 pp 467ndash483 2002

[21] C Huang X Ma and Q Lan ldquoAn empirical study on listedcompanyrsquos value of cash holdings an information asymmetryperspectiverdquoDiscrete Dynamics in Nature and Society vol 2014Article ID 897278 12 pages 2014

[22] C Huang X Yang X G Yang and H Sheng ldquoAn empiricalstudy of the effect of investor sentiment on returns of differentindustriesrdquo Mathematical Problems in Engineering vol 2014Article ID 545723 11 pages 2014

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 Mathematical Problems in Engineering

Conflict of Interests

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

Acknowledgments

Theworkwas supported by the Fundamental Research Fundsfor the Central Universities (Grant nos 15CX141110 andJBK1307012) and National Natural Science Foundation ofChina (Grant no 11471137)

References

[1] D Baleanu K Diethelm E Scalas and J J Trujillo FractionalCalculus Models and Numerical Methods World ScientificPublishers London UK 2012

[2] P Carr and L Wu ldquoThe finite moment log stable process andoption pricingrdquo The Journal of Finance vol 58 no 2 pp 597ndash626 2003

[3] A Cartea and D del-Castillo-Negrete ldquoFractional diffusionmodels of option prices in markets with jumpsrdquo Physica AStatistical Mechanics and Its Applications vol 374 no 2 pp749ndash763 2007

[4] W Chen X Xu and S-P Zhu ldquoAnalytically pricing European-style options under the modified Black-Scholes equation with aspatial-fractional derivativerdquoQuarterly of AppliedMathematicsvol 72 no 3 pp 597ndash611 2014

[5] W Wyss ldquoThe fractional Black-Scholes equationrdquo FractionalCalculus amp Applied Analysis vol 3 no 1 pp 51ndash61 2000

[6] S Kumar A Yildirim Y Khan H Jafari K Sayevand and LWei ldquoAnalytical solution of fractional Black-Scholes Europeanoption pricing equation by using Laplace transformrdquo FractionalCalculus and Applied Analysis vol 2 pp 1ndash9 2012

[7] A Cartea ldquoDerivatives pricing with marked point processesusing tick-by-tick datardquo Quantitative Finance vol 13 no 1 pp111ndash123 2013

[8] G Jumarie ldquoStock exchange fractional dynamics defined asfractional exponential growth driven by (usual) Gaussian whitenoise Application to fractional Black-Scholes equationsrdquo Insur-ance Mathematics and Economics vol 42 no 1 pp 271ndash2872008

[9] J-R Liang JWangW-J ZhangW-Y Qiu and F-Y Ren ldquoThesolution to a bifractional Black-Scholes-Merton differentialequationrdquo International Journal of Pure and Applied Mathemat-ics vol 58 no 1 pp 99ndash112 2010

[10] W Chen X Xu and S-P Zhu ldquoAnalytically pricing double bar-rier options based on a time-fractional Black-Scholes equationrdquoComputers amp Mathematics with Applications vol 69 no 12 pp1407ndash1419 2015

[11] W Chen X Xu and S-P Zhu ldquoA predictor-corrector approachfor pricing American options under the finite moment log-stable modelrdquo Applied Numerical Mathematics vol 97 pp 15ndash29 2015

[12] H Y Wong and J Zhao ldquoValuing American options underthe CEV model by Laplace-Carson transformsrdquo OperationsResearch Letters vol 38 no 5 pp 474ndash481 2010

[13] K Muthuraman ldquoA moving boundary approach to Americanoption pricingrdquo Journal of Economic Dynamics amp Control vol32 no 11 pp 3520ndash3537 2008

[14] A Chockalingam and K Muthuraman ldquoAn approximate mov-ing boundary method for American option pricingrdquo EuropeanJournal of Operational Research vol 240 no 2 pp 431ndash4382015

[15] J Cox ldquoNotes on option pricing I constant elasticity of variancediffusionsrdquoThe Journal of PortfolioManagement vol 22 pp 15ndash17 1996

[16] I Podlubny Fractional Differential Equations Academic Press1999

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

[18] D Davydov and V Linetsky ldquoPricing and hedging path-dependent options under the CEV processrdquo Management Sci-ence vol 47 no 7 pp 949ndash965 2001

[19] Y Jiang and J Ma ldquoMoving finite element methods for timefractional partial differential equationsrdquo Science China Mathe-matics vol 56 no 6 pp 1287ndash1300 2013

[20] P P Valko and S Vajda ldquoInversion of noise-free Laplacetransforms towards a standardized set of test problemsrdquo InverseProblems in Engineering vol 10 no 5 pp 467ndash483 2002

[21] C Huang X Ma and Q Lan ldquoAn empirical study on listedcompanyrsquos value of cash holdings an information asymmetryperspectiverdquoDiscrete Dynamics in Nature and Society vol 2014Article ID 897278 12 pages 2014

[22] C Huang X Yang X G Yang and H Sheng ldquoAn empiricalstudy of the effect of investor sentiment on returns of differentindustriesrdquo Mathematical Problems in Engineering vol 2014Article ID 545723 11 pages 2014

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