Research ArticleSolving American Option Pricing Models by the Front FixingMethod Numerical Analysis and Computing

R Company V N Egorova and L Joacutedar

Instituto de Matematica Multidisciplinar Universitat Politecnica de Valencia Camino de Vera sn 46022 Valencia Spain

Correspondence should be addressed to V N Egorova egorovavngmailcom

Received 12 December 2013 Revised 3 April 2014 Accepted 6 April 2014 Published 28 April 2014

Academic Editor Carlos Vazquez

Copyright copy 2014 R Company et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper presents an explicit finite-differencemethod for nonlinear partial differential equation appearing as a transformedBlack-Scholes equation for American put option under logarithmic front fixing transformation Numerical analysis of the method isprovided The method preserves positivity and monotonicity of the numerical solution Consistency and stability properties of thescheme are studied Explicit calculations avoid iterative algorithms for solving nonlinear systemsTheoretical results are confirmedby numerical experiments Comparison with other approaches shows that the proposed method is accurate and competitive

1 Introduction

American options are contracts allowing the holder the rightto sell (buy) an asset at a certain price at any time until aprespecified future date The pricing of American optionsplays an important role both in theory and in real derivativemarkets

The American option pricing problem can be posedeither as a linear complementarity problem (LCP) or a freeboundary value problem These two different formulationshave led to different methods for solving American options

The most algebraic approach of LCPs for Americanoption pricing can be found in [1 2] and the referencestherein

In this paper we follow the free boundary value approachMcKean [3] and van Moerbeke [4] show that the valuationof American options constitutes a free boundary problemlooking for a boundary changing in time to maturity knownas the optimal exercise boundary The problem of findingoptimal exercise boundary can be treated analytically ornumerically With respect to the analytical approach Geskeand Johnson [5] obtained a valuation formula for Americanputs expressed in terms of a series of compound-optionfunctions See also Barone-Adesi andWhaley [6] MacMillan[7] and Ju [8]

Numerical methods were initiated by Brennan andSchwartz [9] and the convergence of their finite difference

method was proved by Jaillet et al [10] Other relevant worksusing finite differencemethods areHull andWhite [11] Duffy[12] Wilmott et al [1] Forsyth and Vetzal [13] Tavella andRandall [14] Tangman et al [15] Zhu andChen [16] andKimet al [17]

Free boundary value problems are challenging becauseone has to find the solution of a partial differential equationthat satisfies auxiliary initial conditions and boundary con-ditions on a fixed boundary as well as on an unknown freeboundary This complexity is reduced by transforming theproblem into a new nonlinear partial differential equationwhere the free boundary appears as a new variable of the PDEproblem

This technique which originated in physics problems isthe so called front fixing method based on Landau transform[18] to fix the optimal exercise boundary on a vertical axisThe front fixing method has been applied successfully toa wide range of problems arising in physics (see [19]) andreferences therein In 1997 Wu and Kwok [20] proposed thefront fixing technique for solving free boundary problemto the field of option pricing Other relevant papers relatedto the fixed domain transformation are Nielsen et al [21]Sev covic [22] and Zhang and Zhu [23] All these papersuse finite difference schemes after the corresponding fixeddomain transformations Nielsen et al [21] use both explicitand implicit schemes while Sev covic [22] uses differenceschemes of operator splitting type

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 146745 9 pageshttpdxdoiorg1011552014146745

2 Abstract and Applied Analysis

Wu and Kwok [20] transform the original problem intoone more manageable equation because the coefficients ofthe transformed equation do not depend on the spatialvariable In [20] one uses a tree-level scheme and performsnumerical tests versus the binomial method The drawbackof the initialization and iteration of the three-level method isovercome by Zhang and Zhu [23] using a predictor-correctorapproach

The front fixing method combined with the use of anexplicit finite difference scheme avoids the drawbacks ofalternative algebraic approaches since it avoids the use ofiterative methods and underlying difficulties such as how toinitiate the algorithm when to stop it and which is the errorafter the stopping

As the best model can be wasted with a disregardednumerical analysis in this paper we focus on the numericalissues of explicit finite difference scheme for American putoption problem after using the fixed domain transforma-tion used in [20] Consistency of the numerical schemewith the PDE problem is stated Conditions on the sizeof discretization steps in order to guarantee stability andpositivity of the numerical solution are given Furthermorethe numerical solution preserves properties of the theoreticalsolution presented in [24] such as the positivity and themonotonicity of the optimal exercise boundary

This paper is organized as follows In Section 2 dimen-sionless fixed domain transformation of the American putoption problem is introduced as well as the discretization ofthe continuous problem Section 3 is addressed to guaranteethe positivity of the numerical solution as well as the non-increasing time behaviour of the optimal exercise boundaryIn Section 4 stability and consistency are treated Finally inSection 5 numerical experiments and comparisonwith othermethods are illustrated

Throughout the paper we will denote for a given 119909 =

[119909

1 119909

2 119909

119873]

119879isin R119873 its supremum norm as 119909

infin=

max|119909119894| 1 le 119894 le 119873

2 Fixed Domain Transformation andDiscretization

It is well-known that the American put option price model isgiven by [1] the moving free boundary PDE as follows

120597119875

120597120591

=

1

2

120590

2119878

2 1205972119875

120597119878

2+ 119903119878

120597119875

120597119878

minus 119903119875 119878 gt 119861 (120591) 0 lt 120591 le 119879

(1)

together with the initial and boundary conditions

119875 (119878 0) = max (119864 minus 119878 0) 119878 ge 0

120597119875

120597119878

(119861 (120591) 120591) = minus1

119875 (119861 (120591) 120591) = 119864 minus 119861 (120591)

lim119878rarrinfin

119875 (119878 120591) = 0

119861 (0) = 119864

119875 (119878 120591) = 119864 minus 119878 0 le 119878 lt 119861 (120591)

(2)

where 120591 = 119879minus119905 denotes the time to maturity 119879 119878 is the assetrsquosprice 119875(119878 120591) is the option price 119861(120591) is the unknown earlyexercise boundary 120590 is a volatility of the asset 119903 is the risk-free interest rate and 119864 is the strike price

Note that if asset price 119878 le 119861(120591) the optimal strategy is toexercise the option while if 119878 gt 119861(120591) the optimal strategy isto hold the option

Let us consider the dimensionless change of the followingvariables [20]

119901 (119909 120591) =

119875 (119878 120591)

119864

119878

119891(120591) =

119861 (120591)

119864

119909 = ln 119878

119878

119891(120591)

(3)

that transforms the problem (1)-(2) into the normalized formas follows

120597119901

120597120591

=

1

2

120590

2 1205972119901

120597119909

2+ (119903 minus

120590

2

2

)

120597119901

120597119909

minus 119903119901 +

119878

1015840

119891

119878

119891

120597119901

120597119909

119909 gt 0 0 lt 120591 le 119879

(4)

where 1198781015840119891denotes the derivative of 119878

119891with respect to 120591 The

new boundary and initial conditions are

119901 (119909 0) = 0 119909 ge 0 (5)

120597119901

120597119909

(0 120591) = minus119878

119891(120591)

(6)

119901 (0 120591) = 1 minus 119878

119891(120591) (7)

lim119909rarrinfin

119901 (119909 120591) = 0 (8)

119878

119891(0) = 1 (9)

The Equation (4) is a nonlinear differential equation onthe domain (0infin) times (0 119879] In order to solve numericallyproblem (4)ndash(9) one has to consider a bounded numericaldomain Let us introduce 119909max large enough to translatethe boundary condition (8) that is 119901(119909max 120591) = 0 Thenthe problem (4)ndash(9) can be studied on the fixed domain[0 119909max] times (0 119879] The value 119909max is chosen following thecriterion pointed out in [25]

Let us introduce the computational grid of 119872 + 1 spacepoints and 119873 time levels with respective stepsizes ℎ and 119896 asfollows

ℎ =

119909max119872+ 1

119896 =

119879

119873

119909

119895= ℎ119895 119895 = 0 119872 + 1

120591

119899= 119896119899 119899 = 0 119873

(10)

Abstract and Applied Analysis 3

The approximate value of 119901(119909 120591) at the point 119909119895and time

120591

119899 is denoted by 119901119899119895asymp 119901(119909

119895 120591

119899) Then a forward-time central-

space explicit scheme is constructed for internal spacial nodesas follows

119901

119899+1

119895minus 119901

119899

119895

119896

=

1

2

120590

2119901

119899

119895minus1minus 2119901

119899

119895+ 119901

119899

119895+1

ℎ

2+ (119903 minus

120590

2

2

)

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

minus 119903119901

119899

119895+

119878

119899+1

119891minus 119878

119899

119891

119896119878

119899

119891

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

1 le 119895 le 119872 0 le 119899 le 119873 minus 1

(11)

By denoting 120583 = 119896ℎ

2 the scheme (11) can be rewritten inthe following form

119901

119899+1

119895= 119886119901

119899

119895minus1+ 119887119901

119899

119895+ 119888119901

119899

119895+1+

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

(119901

119899

119895+1minus 119901

119895minus1)

(12)

where

119886 =

120583

2

(120590

2minus (119903 minus

120590

2

2

) ℎ) 119887 = 1 minus 120590

2120583 minus 119903119896

119888 =

120583

2

(120590

2+ (119903 minus

120590

2

2

) ℎ)

(13)

From the boundary conditions (6) and (7) we can obtain

119901

119899

1minus 119901

119899

minus1

2ℎ

= minus119878

119899

119891 119901

119899

0= 1 minus 119878

119899

119891

(14)

where 119909minus1

= minusℎ is an auxiliary point out of the domain Byconsidering (4) at the point 119909

0= 0 120591 gt 0 (see [26] p 341)

and replacing of the boundary conditions (6) and (7) into (4)at (0+ 120591) one gets the new boundary condition as follows

1

2

120590

2 1205972119901

120597119909

2(0

+ 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(15)

(see [20 23 26]) and its central difference discretization is

120590

2

2

119901

119899

1minus 2119901

119899

0+ 119901

119899

minus1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(16)

From (14) and (16) the value of 119901119899minus1

can be eliminatedobtaining the relationship

119901

119899

1= 120572 minus 120573119878

119899

119891 119899 ge 1 (17)

between the free boundary approximation 119878

119899

119891and 119901

119899

1 where

120572 = 1 +

119903ℎ

2

120590

2 120573 = 1 + ℎ +

1

2

ℎ

2

(18)

By using the scheme (12) for 119895 = 1 and evaluating (17) atthe (119899+1)st time level the free boundary 119878119899+1

119891can be expressed

as

119878

119899+1

119891= 119889

119899119878

119899

119891 0 le 119899 le 119873 minus 1 (19)

where

119889

119899=

120572 minus (119886119901

119899

0+ 119887119901

119899

1+ 119888119901

119899

2minus (119901

119899

2minus 119901

119899

0) 2ℎ)

(119901

119899

2minus 119901

119899

0) 2ℎ + 120573119878

119899

119891

(20)

After expression (19) the value 119878

119899+1

119891can be replaced in

(12) (17) and (14) to obtain values 119901119899+1119895

0 le 119895 le 119872 Then thenumerical scheme for the problem (4)ndash(9) can be rewrittenfor any 119899 = 0 119873 minus 1 in the following algorithmic form

119878

119899+1

119891= 119889

119899119878

119899

119891 (21)

119901

119899+1

0= 1 minus 119878

119899+1

119891 (22)

119901

119899+1

1= 120572 minus 120573119878

119899+1

119891 (23)

119901

119899+1

119895= 119886

119899119901

119899

119895minus1+ 119887119901

119899

119895+ 119888

119899119901

119899

119895+1 119895 = 2 119872 (24)

where

119886

119899= 119886 minus

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

119888

119899= 119888 +

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

119901

119899+1

119872+1= 0

(25)

with the initial conditions

119878

0

119891= 1 119901

0

119895= 0 0 le 119895 le 119872 + 1 (26)

3 Positivity and Monotonicity

In this section we will show the free boundary nonincreasingmonotonicity as well as the positivity and nonincreasingspacial monotonicity of the numerical option price Let usstart this section by showing that constant coefficients 119886119887 and 119888 of scheme (12) for the transformed problem (4)ndash(9) are positive under appropriate conditions on the stepsizediscretization ℎ and 119896

Lemma 1 Assuming that the stepsizes ℎ and 119896 satisfy thefollowing conditions

C1 ℎ le 120590

2|119903 minus (120590

22)| 119903 = 120590

22

C2 119896 le ℎ

2(120590

2+ 119903ℎ

2)

then the coefficients of the scheme 119886 119887 and 119888 are nonnegativeIf 119903 = 120590

22 then under the condition C2 coefficients 119886 119887 and

119888 are nonnegative

Proof From (13) for nonnegativity of 119886 it is necessary that

120590

2minus (119903 minus

120590

2

2

) ℎ ge 0 (27)

If 119903 le 120590

22 from (13) note that 119886 ge 0 for any ℎ gt 0

Otherwise (27) is satisfied under the condition C1From (13) 119887 is nonnegative under the condition C2If 119903 ge 120590

22 nonnegativity of 119888 is guaranteed by (13) for

any ℎ gt 0 Otherwise 119888 is nonnegative under the conditionC1

4 Abstract and Applied Analysis

The following lemma prepares the study of positivity ofthe numerical solution 119901

119899

119895as well as the monotonicity of

the free boundary sequence 119878

119899

119891 that will be established in a

further result

Lemma 2 Let 119901119899119895 119878

119899

119891 be the numerical solution of scheme

(21)ndash(26) for a transformed American put option problem (4)and let 119889119899 be defined by (20) Then under hypothesis of theLemma 1 for small enough ℎ one verifies the following

(1) For each fixed 119899

0 lt 119889

119899le 1 (28)

(2) Values 119901119899+1119895

ge 0 for 119895 = 0 119872 119899 = 0 119873 minus 1

(3) 119901119899+1119895

ge 119901

119899+1

119895+1for 119895 = 0 119872 minus 1 119899 = 0 119873 minus 1

Proof We use the induction principle Note that from (26)we have 1199010

119895ge 0 and 119901

0

119895ge 119901

0

119895+1 Consider the first time level

119899 = 0 From (18) (20) (26) and hypothesis C1 of Lemma 1one gets

0 lt 119889

0=

120572

120573

le 1 (29)

Note that from (21)ndash(23) and (29) one gets

0 lt 119878

1

119891= 119889

0le 1 119901

1

0= 1 minus 119889

0ge 0 119901

1

1= 0 (30)

and from (24) and (26) every 1199011119895= 0 for 119895 = 2 119872

For the sake of clarity let us show firstly that 0 lt 119889

1le 1

From (20) (18) and (30) it follows that

119889

1= 1 minus 119886

120573 minus 120572

120572120573 minus (120573 minus 120572) 2ℎ

= 1 minus 119886

ℎ + ℎ

2(12 minus 119903120590

2)

12 + ℎ (34 + 1199032120590

2) + 119874 (ℎ

2)

(31)

As 119886 is positive by Lemma 1 for small enough values of ℎone gets

0 lt 119889

1le 1 (32)

It is easy to show that 11990122= 0 and for 1198892 analogously as

(31) and (32) 0 lt 119889

2le 1

Let us assume the induction hypothesis that conclusionshold true for index 119899 minus 1 that is

0 lt 119889

119899minus1le 1 119901

119899

119895ge 0 119901

119899

119895ge 119901

119899

119895+1 (33)

Now we are going to prove that conclusions hold true forindex 119899 Let us consider value of 119889119899 for 119899 ge 1 By denoting

119891

119899= 1 +

119903ℎ

2

120590

2minus (119886119901

119899

0+ 119887119901

119899

1+ 119888119901

119899

2minus

119901

119899

2minus 119901

119899

0

2ℎ

) (34)

119892

119899=

119901

119899

2minus 119901

119899

0

2ℎ

+ (

1

2

(1 + (1 + ℎ)

2)) 119878

119899

119891

(35)

then from (20)

119889

119899=

119891

119899

119892

119899 (36)

For 119899 gt 2 using Taylorrsquos expansion the approximationof the involved derivatives and boundary conditions (6) (7)and (15)

119901

119899

2= 1 +

4119903ℎ

2

120590

2minus (1 + 2ℎ + 2ℎ

2) 119878

119899

119891+ 119874 (ℎ

3)

(37)

From (37) and (34) numerator 119891119899 takes the form

119891

119899= 119903ℎ [119896 +

119903119896ℎ + 2 (1 minus 119903119896)

120590

2]

+ (

ℎ

2

2

(1 minus 119903119896) minus 119896ℎ

120590

2

2

) 119878

119899

119891+ 119874 (ℎ

2)

(38)

and verifies 119891119899 gt 0 since 119896 lt ℎ(119903ℎ + 120590

2) under condition C2

of lemma and for ℎ lt 1From (37) and (20) denominator 119892119899 is positive for small

enough values of ℎ since

119892

119899=

119901

119899

2minus 119901

119899

0

2ℎ

+ (1 + ℎ +

ℎ

2

2

) 119878

119899

119891

=

2119903ℎ

120590

2+

ℎ

2

2

119878

119899

119891+ 119874 (ℎ

2) gt 0

(39)

From (36) and previous comments one gets 119889119899 gt 0 Inorder to prove that 119889119899 le 1 let us consider the difference 119891119899 minus119892

119899 By using (38) and (39) under hypothesis of Lemma 2 onecan obtain

119891

119899minus 119892

119899= 119896ℎ(119903

120590

2minus 2119903 + 119903ℎ

120590

2minus

119903ℎ + 120590

2

2

119878

119899

119891) + 119874(ℎ

2)

(40)

Note that if 1205902 lt 2119903 then (40) is nonpositive for smallenough values of ℎ However even if 120590 ge 2119903 the Samuelsonasymptotic limit [27] 119878119899

119891ge 2119903(2119903 + 120590

2) (see [20] page 87)

guarantees the nonpositivity of (40) Therefore 119889119899 le 1In order to prove the positivity of 119901119899+1

119895 it will be useful

to show the nonnegativity of coefficients 119886119899 and 119888

119899 appearingin (21) Coefficient 119886119899 is positive since

119886

119899= 119886 minus

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

= 119886 minus

119889

119899minus 1

2ℎ

ge 119886 ge 0 (41)

In order to show the positivity of the coefficient 119888119899 notethat from (13) and (39) the sign of 119888119899 is the same as the signof (2ℎ119888119892119899 +119891

119899minus119892

119899) and from (13) (38) (39) and 119878

119899

119891le 1 one

gets the following for small enough values of ℎ

2ℎ119888119892

119899+ 119891

119899minus 119892

119899gt 119903119896 + (120590

2120583 + 119903119896)

119903ℎ

2

120590

2minus

119896ℎ

2120590

2

4

gt 0

(42)

Abstract and Applied Analysis 5

Under hypotheses of induction (33) together with positiv-ity of coefficients 119886119899 and 119888

119899 the positivity of 119901119899+1119895

is provedMoreover 119901119899+1

119895 is nonincreasing with respect to index 119895

from (24) since

119901

119899+1

119895minus 119901

119899+1

119895+1= 119886

119899(119901

119899

119895minus1minus 119901

119899

119895) + 119887 (119901

119899

119895minus 119901

119899

119895+1)

+ 119888

119899(119901

119899

119895+1minus 119901

119899

119895+2) ge 0

(43)

Summarizing the following result has been established

Theorem 3 Under assumptions of Lemma 2 the numericalscheme (12) for solving the American option transformedproblem guarantees the following properties of the numericalsolution

(i) nonincreasing monotonicity and positivity of values 119878119899119891

119899 = 0 119873(ii) positivity of the vectors 119901119899 119899 = 0 119873(iii) nonincreasing monotonicity of the vectors 119901

119899=

(119901

119899

0 119901

119899

119872)with respect to space indexes for each fixed

119899 = 0 119873

4 Stability and Consistency

In this section we study the stability and consistency prop-erties of the scheme (21)ndash(26) For the sake of clarity in thepresentation knowing that several different concepts of thestability are used in the literature we begin the section withthe following definition

Definition 4 The numerical scheme (11) is said to be sdot infin-

stable in the domain [0 119909infin]times[0 119879] if for every partitionwith

119896 = Δ120591 ℎ = Δ119909119873119896 = 119879 and (119872 + 1)ℎ = 119909

infin

1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infinle 119862 0 le 119899 le 119873 (44)

where 119862 is independent of ℎ 119896 and 119899 = 0 119873 (see [28])

Theorem 5 Under assumptions of Lemma 2 the numericalscheme (21)ndash(26) for solving transformed problem (4)ndash(9) is sdot

infin-stable

Proof of Theorem 5 Since for each fixed 119899 119901119899119895 is a nonin-

creasing sequence with respect to 119895 then according to theboundary condition (22) and positivity of 119878119899

119891since (28) one

gets1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infin= 119901

119899

0= 1 minus 119878

119899

119891lt 1 0 le 119899 le 119873 (45)

Thus the scheme is sdot infin-stable

Classical consistency of a numerical scheme with respectto a partial differential equation means that the exact theo-retical solution of the PDE approximates well the solutionof the difference scheme as the discretization stepsizes tendto zero [29] However in our case we have to harmonizethe behaviour not only of the PDE (4) with the scheme

(24) but also the boundary conditions (6) (7) and (15) withthe numerical scheme for the free boundaries (21) and (20)With respect to the first part of consistency let us write thenumerical scheme (24) in the following form

119865 (119901

119899

119895 119878

119899

119891)

=

119901

119899+1

119895minus 119901

119899

119895

119896

minus

1

2

120590

2119901

119899

119895minus1minus 2119901

119899

119895+ 119901

119899

119895+1

ℎ

2

minus (119903 minus

120590

2

2

)

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

+ 119903119901

119899

119895minus

119878

119899+1

119891minus 119878

119899

119891

119896119878

119899

119891

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

= 0

(46)

Let us denote by 119901

119899

119895= 119901(119909

119895 120591

119899) the exact theoretical

solution value of the PDE at the mesh point (119909119895 120591

119899) and let

119878

119899

119891= 119878

119891(120591

119899) be the exact solution of the free boundary at time

120591

119899 The scheme (46) is said to be consistent with

119871 (119901 119878

119891)

=

120597119901

120597120591

minus

1

2

120590

2 1205972119901

120597119909

2minus (119903 minus

120590

2

2

)

120597119901

120597119909

+ 119903119901 minus

119878

1015840

119891

119878

119891

120597119901

120597119909

= 0

(47)

if the local truncation error

119879

119899

119895(119901

119878

119891) = 119865 (

119901

119899

119895

119878

119899

119891) minus 119871 (

119901

119899

119895

119878

119899

119891) (48)

satisfies

119879

119899

119895(119901

119878

119891) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (49)

Assuming the existence of the continuous partial deriva-tives up to order two in time and up to order four in spaceusing Taylorrsquos expansion about (119909

119895 120591

119899) one gets

119879

119899

119895(119901

119878

119891)

= 119896119864

119899

119895(3) minus

120590

2

2

ℎ

2119864

119899

119895(2) + (119903 minus

120590

2

2

) ℎ

2119864

119899

119895(1)

minus 119896119864

119899

119895(4)

120597119901

120597119909

(119909

119895 120591

119899) minus ℎ

2119864

119899

119895(1)

1

119878

119899

119891

119889119878

119891

119889120591

(120591

119899)

minus 119896ℎ

2119864

119899

119895(4) 119864

119899

119895(1)

(50)

where

119864

119899

119895(1) =

1

6

120597

3119901

120597119909

3(119909

119895 120591

119899) 119864

119899

119895(2) =

1

12

120597

4119901

120597119909

4(119909

119895 120591

119899)

119864

119899

119895(3) =

1

2

120597

2119901

120597120591

2(119909

119895 120591

119899) 119864

119899

119895(4) =

119896

2

119878

119891

119889

2119878

119891

119889120591

2(120591

119899)

(51)

6 Abstract and Applied Analysis

Equations (50) and (51) show the local truncation error ofthe numerical scheme (24) with respect to the PDE (4) Notethat from (50) and (51)

119879

119899

119895(119901

119878

119891) = 119874 (ℎ

2) + 119874 (119896) (52)

In order to complete the consistency of the solution of the freeboundary problem with the scheme (21)ndash(26) it is necessaryto rewrite the boundary conditions (6) (7) and (15) in thefollowing form

119871

1(119901 119878

119891) = 119901 (0 120591) minus 1 + 119878

119891(120591) = 0

119871

2(119901 119878

119891) =

120597119901

120597119909

(0 120591) + 119878

119891(120591) = 0

119871

3(119901 119878

119891) =

120590

2

2

120597

2119901

120597119909

2(0 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(53)

Furthermore we have finite difference approximation for thefollowing boundary conditions

119865

1(119901 119878

119891) = 119901

119899

0minus 1 + 119878

119899

119891= 0

119865

2(119901 119878

119891) =

119901

119899

1minus 119901

119899

minus1

2ℎ

+ 119878

119899

119891= 0

119865

3(119901 119878

119891) =

120590

2

2

119901

119899

minus1minus 2119901

119899

0+ 119901

119899

1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(54)

It is not difficult to check using Taylorrsquos expansion that thelocal truncation error satisfies

119879

1(119901

119878

119891) = 119865

1(119901

119878

119891) minus 119871

1(119901

119878

119891) = 0

119879

2(119901

119878

119891) = 119865

2(119901

119878

119891) minus 119871

2(119901

119878

119891) = 119874 (ℎ

2)

119879

3(119901

119878

119891) = 119865

3(119901

119878

119891) minus 119871

3(119901

119878

119891) = 119874 (ℎ

2)

(55)

The truncation error for the boundary condition behavesas ℎ2

Theorem 6 Assuming that the solution of the PDE problem(4)ndash(9) admits two times continuous partial derivative withrespect to time and up to order four with respect to spacethe numerical solution computed by the scheme (21)ndash(26) isconsistent with (4) and boundary conditions of order two inspace and order one in time

5 Numerical Experiments

In this section we illustrate the previous theoretical resultswith numerical experiments showing the potential advan-tages of the proposed method such as preserving the quali-tative properties of the theoretical solution [24] such as posi-tivity andmonotonicity A comparisonwith other approachesis also presented in this section

51 Confirmations of Theoretical Results In this experimentwe check that the stability condition can be broken showing

0 02 04 06 08 1086

088

09

092

094

096

098

1

Time to maturity

Free

bou

ndar

y

(a)

088

092

094

096

098

1

102

0 02 04 06 08 1

09

Time to maturity

Free

bou

ndar

y

(b)

Figure 1 Dependence of stability on positivity of coefficient 119887 (120583 =

24 119887 = 00398) for (a) and (120583 = 252 119887 = minus00083) for (b)

that in such case the results could become unreliable Let usconsider an American put option pricing problem as in [21]with the following parameters

119903 = 01 120590 = 02 119879 = 1 119909max = 2 (56)

We consider fixed space step ℎ = 001 that satisfies conditionC1 of Lemma 1 and change time step (Figure 1) Numericaltests show that the condition C2 is critical for positivity ofcoefficient 119887 and as a result for the stability of the schemeThe monotonicity of the free boundary is numerically shownby numerical tests (Figure 1(a)) In Figure 1(b) one can seethat when condition C2 is not satisfied spurious oscillationsappear so the monotonicity is broken

It was theoretically proved in Section 4 that the schemehas order of approximation 119874(ℎ

2) + 119874(119896) The results of the

Abstract and Applied Analysis 7

Table 1 Convergence in space for fixed time step 119896 = 2 sdot10

minus3 for theproblem with parameters (56)

Asset price True value 004 002 00190 116974 117283 117054 116991100 69320 69308 69309 69312110 41550 41694 41564 41531120 25102 25219 25151 25114

RMSE 18037-2 47857-3 14623-3CPU-time (sec) 0052 0075 0105

0

0001

0002

0003

0004

0005

0 50 100 150 200

RMSE

CPU-time (s)

Figure 2 RMSE against the computational time

numerical experiments for the problem with the followingparameters

119903 = 008 120590 = 02 119879 = 3 119909max = 2 (57)

are presented in Table 1 For ldquoTruerdquo values we use thereference values offered in [30] We consider the differentspace steps for fixed time step 119896 = 0002 to guaranteethe stability The root-mean-square error (RMSE) is used tomeasure the accuracy of the scheme The last row presentsthe CPU-time in seconds for each experimentThe plot of theRMSE against computational time is presented in Figure 2

To check the order of approximation in space we intro-duce the convergence rate

120574 (ℎ

1 ℎ

2) =

ln RMSEℎ1

minus ln RMSEℎ2

ln ℎ

1minus ln ℎ

2

(58)

From Table 1 and (58) one obtains 120574(4 sdot 10

minus2 2 sdot 10

minus2) =

191 that is close to 2To check the order of approximation in time the space

step ℎ is fixed (ℎ = 10

minus2) and time step 119896 is variableFrom Table 2 by analogous to (58) formula one gets 120574(5 sdot

10

minus4 10

minus3) = 080 that is close to 1 It proves the second order

of approximation in space and the first order in time

52 Comparison with Other Approaches The front fixingmethod is used in [21] for American put option pricing prob-lemwith parameters (56) but with another transformation asfollows

119909 =

119878

119878

119891(120591)

119901 (119909 120591) = 119875 (119878 120591) = 119875 (119909119878

119891(119905) 120591)

(59)

Table 2 Convergence in time for fixed space step ℎ = 001 for theproblem with parameters (56)

Asset price True value 00005 0001 000290 116974 116984 116989 116991100 69320 69319 69318 69312110 41550 41555 41544 41531120 25102 25103 25091 25114

RMSE 56347-4 98234-4 14623-3CPU-time (sec) 0142 0113 0105

Table 3 Comparison with front-fixing method under transforma-tion (59)

Method 119878

119891(119879)

Implicit (in [21]) 08615Explicit (in [21]) 08622Explicit (proposed) 08628

By the numerical tests it is shown that the consideredscheme is stable for 120583 le 24 while the condition for a stabilityof the scheme in [21] is 120583 le 6 The results are representedin Table 3 The front fixing method with the logarithmictransformation has a good advantage weaker condition forthe stability It reduces the computational costs

Let us compare front fixing method with anotherapproach [31] based on Mellinrsquos transform The parametersof the problem are 119903 = 00488 120590 = 03 and 119879 = 05833

To compare results of the explicit front fixing methodwith Mellinrsquos transform [31] we have to multiply our dimen-sionless value on119864 = 45Then forMellinrsquos transformmethod119878

119891(119879) = 3277 while for front fixing method 119878

119891(119879) =

327655Close to maturity exercise boundary for the American

put can be analytically approximated [32] as follows

119878

119891(120591) sim 119864[1 minus

radic2120590120591 log( 120590

2

6119903radic120587120591120590

22

)] (60)

This approximation can be used only near expiry date InFigure 3 the value of the free boundary for both approachesis presented Near initial point front fixing method andanalytical approximation (60) give close results

We compare explicit front fixing method (FF) with ℎ

1=

0001 and ℎ

2= 0002 and 120583 = 5 for American put with other

numerical methods shown in [15] in Table 4 for the followingproblem parameters

119903 = 005 120590 = 02 119879 = 3

119864 = 100 119909max = 2

(61)

There are several considered methods as follows

(i) penalty method (PM) is considered in [21 33](ii) Ikonen and Toivanen [34] proposed an operator

splitting technique (OS) for solving the linear com-plementarity problem

8 Abstract and Applied Analysis

Table 4 Comparison with other methods put option value for different asset prices

Asset price True value PM OS HW WK FF(ℎ1) FF(ℎ

2)

80 202797 202793 202795 202803 202825 202795 20279390 133075 133071 133074 133075 133117 133075 133074100 87106 87100 87104 87103 87135 87106 87104110 56825 56820 56824 56823 56867 56825 56823120 36964 36960 36963 36965 37001 36963 36963RMSE 46690-4 89442-6 31623-4 36116-3 10229-4 22966-4CPU-time 2567 1191 472 423 2599 4617

0 002 004 006 008 01 012 014075

08

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(a)

0 02 04 06 08

08

1

065

07

075

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(b)

Figure 3 Comparison with the analytical approximation of Kuskeclose to 120591 = 0 (a) and for [0 119879] (b)

(iii) Han-Wu algorithm (HW) transforms the Black-Scholes equation into a heat equation on an infinitedomain [35]

(iv) the front fixing method proposed by Wu and Kwok(WK) [20]

Apart from the previous comparisons showing that themethod is competitive we remark that the proposed schemehere guarantees the positivity of the solution as well as themonotonicity of both solution and the free boundary

6 Conclusion

The front fixing method for American put option is consid-ered We found that the proposed explicit numerical schemeis stable under appropriate conditions (see Lemma 1) andconsistent with the differential equation with the secondorder in space and first order in time Moreover the mono-tonicity and positivity of the solution and the free boundaryunder that condition are proved in agreement with Kimrsquosresults in [24] about the properties of the theoretical solutionThe theoretical study is confirmed by numerical tests It isshown that the condition (27) is critical that is violation ofthe condition leads to destabilization of the scheme

By the comparison with other approaches advantages ofthe method are discovered The logarithmic transformationrequires weaker stability condition than Landaursquos transfor-mation presented in [21] The explicit calculations withquite weak stability conditions avoid iterations to solve thenonlinear partial differential equation It makes the methodsimpler and reduces computational costs

Conflict of Interests

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

Acknowledgment

This paper has been partially supported by the Euro-pean Union in the FP7-PEOPLE-2012-ITN Program under

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

Submit your manuscripts athttpwwwhindawicom

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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

2 Abstract and Applied Analysis

Wu and Kwok [20] transform the original problem intoone more manageable equation because the coefficients ofthe transformed equation do not depend on the spatialvariable In [20] one uses a tree-level scheme and performsnumerical tests versus the binomial method The drawbackof the initialization and iteration of the three-level method isovercome by Zhang and Zhu [23] using a predictor-correctorapproach

The front fixing method combined with the use of anexplicit finite difference scheme avoids the drawbacks ofalternative algebraic approaches since it avoids the use ofiterative methods and underlying difficulties such as how toinitiate the algorithm when to stop it and which is the errorafter the stopping

As the best model can be wasted with a disregardednumerical analysis in this paper we focus on the numericalissues of explicit finite difference scheme for American putoption problem after using the fixed domain transforma-tion used in [20] Consistency of the numerical schemewith the PDE problem is stated Conditions on the sizeof discretization steps in order to guarantee stability andpositivity of the numerical solution are given Furthermorethe numerical solution preserves properties of the theoreticalsolution presented in [24] such as the positivity and themonotonicity of the optimal exercise boundary

This paper is organized as follows In Section 2 dimen-sionless fixed domain transformation of the American putoption problem is introduced as well as the discretization ofthe continuous problem Section 3 is addressed to guaranteethe positivity of the numerical solution as well as the non-increasing time behaviour of the optimal exercise boundaryIn Section 4 stability and consistency are treated Finally inSection 5 numerical experiments and comparisonwith othermethods are illustrated

Throughout the paper we will denote for a given 119909 =

[119909

1 119909

2 119909

119873]

119879isin R119873 its supremum norm as 119909

infin=

max|119909119894| 1 le 119894 le 119873

2 Fixed Domain Transformation andDiscretization

It is well-known that the American put option price model isgiven by [1] the moving free boundary PDE as follows

120597119875

120597120591

=

1

2

120590

2119878

2 1205972119875

120597119878

2+ 119903119878

120597119875

120597119878

minus 119903119875 119878 gt 119861 (120591) 0 lt 120591 le 119879

(1)

together with the initial and boundary conditions

119875 (119878 0) = max (119864 minus 119878 0) 119878 ge 0

120597119875

120597119878

(119861 (120591) 120591) = minus1

119875 (119861 (120591) 120591) = 119864 minus 119861 (120591)

lim119878rarrinfin

119875 (119878 120591) = 0

119861 (0) = 119864

119875 (119878 120591) = 119864 minus 119878 0 le 119878 lt 119861 (120591)

(2)

where 120591 = 119879minus119905 denotes the time to maturity 119879 119878 is the assetrsquosprice 119875(119878 120591) is the option price 119861(120591) is the unknown earlyexercise boundary 120590 is a volatility of the asset 119903 is the risk-free interest rate and 119864 is the strike price

Note that if asset price 119878 le 119861(120591) the optimal strategy is toexercise the option while if 119878 gt 119861(120591) the optimal strategy isto hold the option

Let us consider the dimensionless change of the followingvariables [20]

119901 (119909 120591) =

119875 (119878 120591)

119864

119878

119891(120591) =

119861 (120591)

119864

119909 = ln 119878

119878

119891(120591)

(3)

that transforms the problem (1)-(2) into the normalized formas follows

120597119901

120597120591

=

1

2

120590

2 1205972119901

120597119909

2+ (119903 minus

120590

2

2

)

120597119901

120597119909

minus 119903119901 +

119878

1015840

119891

119878

119891

120597119901

120597119909

119909 gt 0 0 lt 120591 le 119879

(4)

where 1198781015840119891denotes the derivative of 119878

119891with respect to 120591 The

new boundary and initial conditions are

119901 (119909 0) = 0 119909 ge 0 (5)

120597119901

120597119909

(0 120591) = minus119878

119891(120591)

(6)

119901 (0 120591) = 1 minus 119878

119891(120591) (7)

lim119909rarrinfin

119901 (119909 120591) = 0 (8)

119878

119891(0) = 1 (9)

The Equation (4) is a nonlinear differential equation onthe domain (0infin) times (0 119879] In order to solve numericallyproblem (4)ndash(9) one has to consider a bounded numericaldomain Let us introduce 119909max large enough to translatethe boundary condition (8) that is 119901(119909max 120591) = 0 Thenthe problem (4)ndash(9) can be studied on the fixed domain[0 119909max] times (0 119879] The value 119909max is chosen following thecriterion pointed out in [25]

Let us introduce the computational grid of 119872 + 1 spacepoints and 119873 time levels with respective stepsizes ℎ and 119896 asfollows

ℎ =

119909max119872+ 1

119896 =

119879

119873

119909

119895= ℎ119895 119895 = 0 119872 + 1

120591

119899= 119896119899 119899 = 0 119873

(10)

Abstract and Applied Analysis 3

The approximate value of 119901(119909 120591) at the point 119909119895and time

120591

119899 is denoted by 119901119899119895asymp 119901(119909

119895 120591

119899) Then a forward-time central-

space explicit scheme is constructed for internal spacial nodesas follows

119901

119899+1

119895minus 119901

119899

119895

119896

=

1

2

120590

2119901

119899

119895minus1minus 2119901

119899

119895+ 119901

119899

119895+1

ℎ

2+ (119903 minus

120590

2

2

)

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

minus 119903119901

119899

119895+

119878

119899+1

119891minus 119878

119899

119891

119896119878

119899

119891

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

1 le 119895 le 119872 0 le 119899 le 119873 minus 1

(11)

By denoting 120583 = 119896ℎ

2 the scheme (11) can be rewritten inthe following form

119901

119899+1

119895= 119886119901

119899

119895minus1+ 119887119901

119899

119895+ 119888119901

119899

119895+1+

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

(119901

119899

119895+1minus 119901

119895minus1)

(12)

where

119886 =

120583

2

(120590

2minus (119903 minus

120590

2

2

) ℎ) 119887 = 1 minus 120590

2120583 minus 119903119896

119888 =

120583

2

(120590

2+ (119903 minus

120590

2

2

) ℎ)

(13)

From the boundary conditions (6) and (7) we can obtain

119901

119899

1minus 119901

119899

minus1

2ℎ

= minus119878

119899

119891 119901

119899

0= 1 minus 119878

119899

119891

(14)

where 119909minus1

= minusℎ is an auxiliary point out of the domain Byconsidering (4) at the point 119909

0= 0 120591 gt 0 (see [26] p 341)

and replacing of the boundary conditions (6) and (7) into (4)at (0+ 120591) one gets the new boundary condition as follows

1

2

120590

2 1205972119901

120597119909

2(0

+ 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(15)

(see [20 23 26]) and its central difference discretization is

120590

2

2

119901

119899

1minus 2119901

119899

0+ 119901

119899

minus1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(16)

From (14) and (16) the value of 119901119899minus1

can be eliminatedobtaining the relationship

119901

119899

1= 120572 minus 120573119878

119899

119891 119899 ge 1 (17)

between the free boundary approximation 119878

119899

119891and 119901

119899

1 where

120572 = 1 +

119903ℎ

2

120590

2 120573 = 1 + ℎ +

1

2

ℎ

2

(18)

By using the scheme (12) for 119895 = 1 and evaluating (17) atthe (119899+1)st time level the free boundary 119878119899+1

119891can be expressed

as

119878

119899+1

119891= 119889

119899119878

119899

119891 0 le 119899 le 119873 minus 1 (19)

where

119889

119899=

120572 minus (119886119901

119899

0+ 119887119901

119899

1+ 119888119901

119899

2minus (119901

119899

2minus 119901

119899

0) 2ℎ)

(119901

119899

2minus 119901

119899

0) 2ℎ + 120573119878

119899

119891

(20)

After expression (19) the value 119878

119899+1

119891can be replaced in

(12) (17) and (14) to obtain values 119901119899+1119895

0 le 119895 le 119872 Then thenumerical scheme for the problem (4)ndash(9) can be rewrittenfor any 119899 = 0 119873 minus 1 in the following algorithmic form

119878

119899+1

119891= 119889

119899119878

119899

119891 (21)

119901

119899+1

0= 1 minus 119878

119899+1

119891 (22)

119901

119899+1

1= 120572 minus 120573119878

119899+1

119891 (23)

119901

119899+1

119895= 119886

119899119901

119899

119895minus1+ 119887119901

119899

119895+ 119888

119899119901

119899

119895+1 119895 = 2 119872 (24)

where

119886

119899= 119886 minus

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

119888

119899= 119888 +

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

119901

119899+1

119872+1= 0

(25)

with the initial conditions

119878

0

119891= 1 119901

0

119895= 0 0 le 119895 le 119872 + 1 (26)

3 Positivity and Monotonicity

In this section we will show the free boundary nonincreasingmonotonicity as well as the positivity and nonincreasingspacial monotonicity of the numerical option price Let usstart this section by showing that constant coefficients 119886119887 and 119888 of scheme (12) for the transformed problem (4)ndash(9) are positive under appropriate conditions on the stepsizediscretization ℎ and 119896

Lemma 1 Assuming that the stepsizes ℎ and 119896 satisfy thefollowing conditions

C1 ℎ le 120590

2|119903 minus (120590

22)| 119903 = 120590

22

C2 119896 le ℎ

2(120590

2+ 119903ℎ

2)

then the coefficients of the scheme 119886 119887 and 119888 are nonnegativeIf 119903 = 120590

22 then under the condition C2 coefficients 119886 119887 and

119888 are nonnegative

Proof From (13) for nonnegativity of 119886 it is necessary that

120590

2minus (119903 minus

120590

2

2

) ℎ ge 0 (27)

If 119903 le 120590

22 from (13) note that 119886 ge 0 for any ℎ gt 0

Otherwise (27) is satisfied under the condition C1From (13) 119887 is nonnegative under the condition C2If 119903 ge 120590

22 nonnegativity of 119888 is guaranteed by (13) for

any ℎ gt 0 Otherwise 119888 is nonnegative under the conditionC1

4 Abstract and Applied Analysis

The following lemma prepares the study of positivity ofthe numerical solution 119901

119899

119895as well as the monotonicity of

the free boundary sequence 119878

119899

119891 that will be established in a

further result

Lemma 2 Let 119901119899119895 119878

119899

119891 be the numerical solution of scheme

(21)ndash(26) for a transformed American put option problem (4)and let 119889119899 be defined by (20) Then under hypothesis of theLemma 1 for small enough ℎ one verifies the following

(1) For each fixed 119899

0 lt 119889

119899le 1 (28)

(2) Values 119901119899+1119895

ge 0 for 119895 = 0 119872 119899 = 0 119873 minus 1

(3) 119901119899+1119895

ge 119901

119899+1

119895+1for 119895 = 0 119872 minus 1 119899 = 0 119873 minus 1

Proof We use the induction principle Note that from (26)we have 1199010

119895ge 0 and 119901

0

119895ge 119901

0

119895+1 Consider the first time level

119899 = 0 From (18) (20) (26) and hypothesis C1 of Lemma 1one gets

0 lt 119889

0=

120572

120573

le 1 (29)

Note that from (21)ndash(23) and (29) one gets

0 lt 119878

1

119891= 119889

0le 1 119901

1

0= 1 minus 119889

0ge 0 119901

1

1= 0 (30)

and from (24) and (26) every 1199011119895= 0 for 119895 = 2 119872

For the sake of clarity let us show firstly that 0 lt 119889

1le 1

From (20) (18) and (30) it follows that

119889

1= 1 minus 119886

120573 minus 120572

120572120573 minus (120573 minus 120572) 2ℎ

= 1 minus 119886

ℎ + ℎ

2(12 minus 119903120590

2)

12 + ℎ (34 + 1199032120590

2) + 119874 (ℎ

2)

(31)

As 119886 is positive by Lemma 1 for small enough values of ℎone gets

0 lt 119889

1le 1 (32)

It is easy to show that 11990122= 0 and for 1198892 analogously as

(31) and (32) 0 lt 119889

2le 1

Let us assume the induction hypothesis that conclusionshold true for index 119899 minus 1 that is

0 lt 119889

119899minus1le 1 119901

119899

119895ge 0 119901

119899

119895ge 119901

119899

119895+1 (33)

Now we are going to prove that conclusions hold true forindex 119899 Let us consider value of 119889119899 for 119899 ge 1 By denoting

119891

119899= 1 +

119903ℎ

2

120590

2minus (119886119901

119899

0+ 119887119901

119899

1+ 119888119901

119899

2minus

119901

119899

2minus 119901

119899

0

2ℎ

) (34)

119892

119899=

119901

119899

2minus 119901

119899

0

2ℎ

+ (

1

2

(1 + (1 + ℎ)

2)) 119878

119899

119891

(35)

then from (20)

119889

119899=

119891

119899

119892

119899 (36)

For 119899 gt 2 using Taylorrsquos expansion the approximationof the involved derivatives and boundary conditions (6) (7)and (15)

119901

119899

2= 1 +

4119903ℎ

2

120590

2minus (1 + 2ℎ + 2ℎ

2) 119878

119899

119891+ 119874 (ℎ

3)

(37)

From (37) and (34) numerator 119891119899 takes the form

119891

119899= 119903ℎ [119896 +

119903119896ℎ + 2 (1 minus 119903119896)

120590

2]

+ (

ℎ

2

2

(1 minus 119903119896) minus 119896ℎ

120590

2

2

) 119878

119899

119891+ 119874 (ℎ

2)

(38)

and verifies 119891119899 gt 0 since 119896 lt ℎ(119903ℎ + 120590

2) under condition C2

of lemma and for ℎ lt 1From (37) and (20) denominator 119892119899 is positive for small

enough values of ℎ since

119892

119899=

119901

119899

2minus 119901

119899

0

2ℎ

+ (1 + ℎ +

ℎ

2

2

) 119878

119899

119891

=

2119903ℎ

120590

2+

ℎ

2

2

119878

119899

119891+ 119874 (ℎ

2) gt 0

(39)

From (36) and previous comments one gets 119889119899 gt 0 Inorder to prove that 119889119899 le 1 let us consider the difference 119891119899 minus119892

119899 By using (38) and (39) under hypothesis of Lemma 2 onecan obtain

119891

119899minus 119892

119899= 119896ℎ(119903

120590

2minus 2119903 + 119903ℎ

120590

2minus

119903ℎ + 120590

2

2

119878

119899

119891) + 119874(ℎ

2)

(40)

Note that if 1205902 lt 2119903 then (40) is nonpositive for smallenough values of ℎ However even if 120590 ge 2119903 the Samuelsonasymptotic limit [27] 119878119899

119891ge 2119903(2119903 + 120590

2) (see [20] page 87)

guarantees the nonpositivity of (40) Therefore 119889119899 le 1In order to prove the positivity of 119901119899+1

119895 it will be useful

to show the nonnegativity of coefficients 119886119899 and 119888

119899 appearingin (21) Coefficient 119886119899 is positive since

119886

119899= 119886 minus

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

= 119886 minus

119889

119899minus 1

2ℎ

ge 119886 ge 0 (41)

In order to show the positivity of the coefficient 119888119899 notethat from (13) and (39) the sign of 119888119899 is the same as the signof (2ℎ119888119892119899 +119891

119899minus119892

119899) and from (13) (38) (39) and 119878

119899

119891le 1 one

gets the following for small enough values of ℎ

2ℎ119888119892

119899+ 119891

119899minus 119892

119899gt 119903119896 + (120590

2120583 + 119903119896)

119903ℎ

2

120590

2minus

119896ℎ

2120590

2

4

gt 0

(42)

Abstract and Applied Analysis 5

Under hypotheses of induction (33) together with positiv-ity of coefficients 119886119899 and 119888

119899 the positivity of 119901119899+1119895

is provedMoreover 119901119899+1

119895 is nonincreasing with respect to index 119895

from (24) since

119901

119899+1

119895minus 119901

119899+1

119895+1= 119886

119899(119901

119899

119895minus1minus 119901

119899

119895) + 119887 (119901

119899

119895minus 119901

119899

119895+1)

+ 119888

119899(119901

119899

119895+1minus 119901

119899

119895+2) ge 0

(43)

Summarizing the following result has been established

Theorem 3 Under assumptions of Lemma 2 the numericalscheme (12) for solving the American option transformedproblem guarantees the following properties of the numericalsolution

(i) nonincreasing monotonicity and positivity of values 119878119899119891

119899 = 0 119873(ii) positivity of the vectors 119901119899 119899 = 0 119873(iii) nonincreasing monotonicity of the vectors 119901

119899=

(119901

119899

0 119901

119899

119872)with respect to space indexes for each fixed

119899 = 0 119873

4 Stability and Consistency

In this section we study the stability and consistency prop-erties of the scheme (21)ndash(26) For the sake of clarity in thepresentation knowing that several different concepts of thestability are used in the literature we begin the section withthe following definition

Definition 4 The numerical scheme (11) is said to be sdot infin-

stable in the domain [0 119909infin]times[0 119879] if for every partitionwith

119896 = Δ120591 ℎ = Δ119909119873119896 = 119879 and (119872 + 1)ℎ = 119909

infin

1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infinle 119862 0 le 119899 le 119873 (44)

where 119862 is independent of ℎ 119896 and 119899 = 0 119873 (see [28])

Theorem 5 Under assumptions of Lemma 2 the numericalscheme (21)ndash(26) for solving transformed problem (4)ndash(9) is sdot

infin-stable

Proof of Theorem 5 Since for each fixed 119899 119901119899119895 is a nonin-

creasing sequence with respect to 119895 then according to theboundary condition (22) and positivity of 119878119899

119891since (28) one

gets1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infin= 119901

119899

0= 1 minus 119878

119899

119891lt 1 0 le 119899 le 119873 (45)

Thus the scheme is sdot infin-stable

Classical consistency of a numerical scheme with respectto a partial differential equation means that the exact theo-retical solution of the PDE approximates well the solutionof the difference scheme as the discretization stepsizes tendto zero [29] However in our case we have to harmonizethe behaviour not only of the PDE (4) with the scheme

(24) but also the boundary conditions (6) (7) and (15) withthe numerical scheme for the free boundaries (21) and (20)With respect to the first part of consistency let us write thenumerical scheme (24) in the following form

119865 (119901

119899

119895 119878

119899

119891)

=

119901

119899+1

119895minus 119901

119899

119895

119896

minus

1

2

120590

2119901

119899

119895minus1minus 2119901

119899

119895+ 119901

119899

119895+1

ℎ

2

minus (119903 minus

120590

2

2

)

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

+ 119903119901

119899

119895minus

119878

119899+1

119891minus 119878

119899

119891

119896119878

119899

119891

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

= 0

(46)

Let us denote by 119901

119899

119895= 119901(119909

119895 120591

119899) the exact theoretical

solution value of the PDE at the mesh point (119909119895 120591

119899) and let

119878

119899

119891= 119878

119891(120591

119899) be the exact solution of the free boundary at time

120591

119899 The scheme (46) is said to be consistent with

119871 (119901 119878

119891)

=

120597119901

120597120591

minus

1

2

120590

2 1205972119901

120597119909

2minus (119903 minus

120590

2

2

)

120597119901

120597119909

+ 119903119901 minus

119878

1015840

119891

119878

119891

120597119901

120597119909

= 0

(47)

if the local truncation error

119879

119899

119895(119901

119878

119891) = 119865 (

119901

119899

119895

119878

119899

119891) minus 119871 (

119901

119899

119895

119878

119899

119891) (48)

satisfies

119879

119899

119895(119901

119878

119891) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (49)

Assuming the existence of the continuous partial deriva-tives up to order two in time and up to order four in spaceusing Taylorrsquos expansion about (119909

119895 120591

119899) one gets

119879

119899

119895(119901

119878

119891)

= 119896119864

119899

119895(3) minus

120590

2

2

ℎ

2119864

119899

119895(2) + (119903 minus

120590

2

2

) ℎ

2119864

119899

119895(1)

minus 119896119864

119899

119895(4)

120597119901

120597119909

(119909

119895 120591

119899) minus ℎ

2119864

119899

119895(1)

1

119878

119899

119891

119889119878

119891

119889120591

(120591

119899)

minus 119896ℎ

2119864

119899

119895(4) 119864

119899

119895(1)

(50)

where

119864

119899

119895(1) =

1

6

120597

3119901

120597119909

3(119909

119895 120591

119899) 119864

119899

119895(2) =

1

12

120597

4119901

120597119909

4(119909

119895 120591

119899)

119864

119899

119895(3) =

1

2

120597

2119901

120597120591

2(119909

119895 120591

119899) 119864

119899

119895(4) =

119896

2

119878

119891

119889

2119878

119891

119889120591

2(120591

119899)

(51)

6 Abstract and Applied Analysis

Equations (50) and (51) show the local truncation error ofthe numerical scheme (24) with respect to the PDE (4) Notethat from (50) and (51)

119879

119899

119895(119901

119878

119891) = 119874 (ℎ

2) + 119874 (119896) (52)

In order to complete the consistency of the solution of the freeboundary problem with the scheme (21)ndash(26) it is necessaryto rewrite the boundary conditions (6) (7) and (15) in thefollowing form

119871

1(119901 119878

119891) = 119901 (0 120591) minus 1 + 119878

119891(120591) = 0

119871

2(119901 119878

119891) =

120597119901

120597119909

(0 120591) + 119878

119891(120591) = 0

119871

3(119901 119878

119891) =

120590

2

2

120597

2119901

120597119909

2(0 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(53)

Furthermore we have finite difference approximation for thefollowing boundary conditions

119865

1(119901 119878

119891) = 119901

119899

0minus 1 + 119878

119899

119891= 0

119865

2(119901 119878

119891) =

119901

119899

1minus 119901

119899

minus1

2ℎ

+ 119878

119899

119891= 0

119865

3(119901 119878

119891) =

120590

2

2

119901

119899

minus1minus 2119901

119899

0+ 119901

119899

1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(54)

It is not difficult to check using Taylorrsquos expansion that thelocal truncation error satisfies

119879

1(119901

119878

119891) = 119865

1(119901

119878

119891) minus 119871

1(119901

119878

119891) = 0

119879

2(119901

119878

119891) = 119865

2(119901

119878

119891) minus 119871

2(119901

119878

119891) = 119874 (ℎ

2)

119879

3(119901

119878

119891) = 119865

3(119901

119878

119891) minus 119871

3(119901

119878

119891) = 119874 (ℎ

2)

(55)

The truncation error for the boundary condition behavesas ℎ2

Theorem 6 Assuming that the solution of the PDE problem(4)ndash(9) admits two times continuous partial derivative withrespect to time and up to order four with respect to spacethe numerical solution computed by the scheme (21)ndash(26) isconsistent with (4) and boundary conditions of order two inspace and order one in time

5 Numerical Experiments

In this section we illustrate the previous theoretical resultswith numerical experiments showing the potential advan-tages of the proposed method such as preserving the quali-tative properties of the theoretical solution [24] such as posi-tivity andmonotonicity A comparisonwith other approachesis also presented in this section

51 Confirmations of Theoretical Results In this experimentwe check that the stability condition can be broken showing

0 02 04 06 08 1086

088

09

092

094

096

098

1

Time to maturity

Free

bou

ndar

y

(a)

088

092

094

096

098

1

102

0 02 04 06 08 1

09

Time to maturity

Free

bou

ndar

y

(b)

Figure 1 Dependence of stability on positivity of coefficient 119887 (120583 =

24 119887 = 00398) for (a) and (120583 = 252 119887 = minus00083) for (b)

that in such case the results could become unreliable Let usconsider an American put option pricing problem as in [21]with the following parameters

119903 = 01 120590 = 02 119879 = 1 119909max = 2 (56)

We consider fixed space step ℎ = 001 that satisfies conditionC1 of Lemma 1 and change time step (Figure 1) Numericaltests show that the condition C2 is critical for positivity ofcoefficient 119887 and as a result for the stability of the schemeThe monotonicity of the free boundary is numerically shownby numerical tests (Figure 1(a)) In Figure 1(b) one can seethat when condition C2 is not satisfied spurious oscillationsappear so the monotonicity is broken

It was theoretically proved in Section 4 that the schemehas order of approximation 119874(ℎ

2) + 119874(119896) The results of the

Abstract and Applied Analysis 7

Table 1 Convergence in space for fixed time step 119896 = 2 sdot10

minus3 for theproblem with parameters (56)

Asset price True value 004 002 00190 116974 117283 117054 116991100 69320 69308 69309 69312110 41550 41694 41564 41531120 25102 25219 25151 25114

RMSE 18037-2 47857-3 14623-3CPU-time (sec) 0052 0075 0105

0

0001

0002

0003

0004

0005

0 50 100 150 200

RMSE

CPU-time (s)

Figure 2 RMSE against the computational time

numerical experiments for the problem with the followingparameters

119903 = 008 120590 = 02 119879 = 3 119909max = 2 (57)

are presented in Table 1 For ldquoTruerdquo values we use thereference values offered in [30] We consider the differentspace steps for fixed time step 119896 = 0002 to guaranteethe stability The root-mean-square error (RMSE) is used tomeasure the accuracy of the scheme The last row presentsthe CPU-time in seconds for each experimentThe plot of theRMSE against computational time is presented in Figure 2

To check the order of approximation in space we intro-duce the convergence rate

120574 (ℎ

1 ℎ

2) =

ln RMSEℎ1

minus ln RMSEℎ2

ln ℎ

1minus ln ℎ

2

(58)

From Table 1 and (58) one obtains 120574(4 sdot 10

minus2 2 sdot 10

minus2) =

191 that is close to 2To check the order of approximation in time the space

step ℎ is fixed (ℎ = 10

minus2) and time step 119896 is variableFrom Table 2 by analogous to (58) formula one gets 120574(5 sdot

10

minus4 10

minus3) = 080 that is close to 1 It proves the second order

of approximation in space and the first order in time

52 Comparison with Other Approaches The front fixingmethod is used in [21] for American put option pricing prob-lemwith parameters (56) but with another transformation asfollows

119909 =

119878

119878

119891(120591)

119901 (119909 120591) = 119875 (119878 120591) = 119875 (119909119878

119891(119905) 120591)

(59)

Table 2 Convergence in time for fixed space step ℎ = 001 for theproblem with parameters (56)

Asset price True value 00005 0001 000290 116974 116984 116989 116991100 69320 69319 69318 69312110 41550 41555 41544 41531120 25102 25103 25091 25114

RMSE 56347-4 98234-4 14623-3CPU-time (sec) 0142 0113 0105

Table 3 Comparison with front-fixing method under transforma-tion (59)

Method 119878

119891(119879)

Implicit (in [21]) 08615Explicit (in [21]) 08622Explicit (proposed) 08628

By the numerical tests it is shown that the consideredscheme is stable for 120583 le 24 while the condition for a stabilityof the scheme in [21] is 120583 le 6 The results are representedin Table 3 The front fixing method with the logarithmictransformation has a good advantage weaker condition forthe stability It reduces the computational costs

Let us compare front fixing method with anotherapproach [31] based on Mellinrsquos transform The parametersof the problem are 119903 = 00488 120590 = 03 and 119879 = 05833

To compare results of the explicit front fixing methodwith Mellinrsquos transform [31] we have to multiply our dimen-sionless value on119864 = 45Then forMellinrsquos transformmethod119878

119891(119879) = 3277 while for front fixing method 119878

119891(119879) =

327655Close to maturity exercise boundary for the American

put can be analytically approximated [32] as follows

119878

119891(120591) sim 119864[1 minus

radic2120590120591 log( 120590

2

6119903radic120587120591120590

22

)] (60)

This approximation can be used only near expiry date InFigure 3 the value of the free boundary for both approachesis presented Near initial point front fixing method andanalytical approximation (60) give close results

We compare explicit front fixing method (FF) with ℎ

1=

0001 and ℎ

2= 0002 and 120583 = 5 for American put with other

numerical methods shown in [15] in Table 4 for the followingproblem parameters

119903 = 005 120590 = 02 119879 = 3

119864 = 100 119909max = 2

(61)

There are several considered methods as follows

(i) penalty method (PM) is considered in [21 33](ii) Ikonen and Toivanen [34] proposed an operator

splitting technique (OS) for solving the linear com-plementarity problem

8 Abstract and Applied Analysis

Table 4 Comparison with other methods put option value for different asset prices

Asset price True value PM OS HW WK FF(ℎ1) FF(ℎ

2)

80 202797 202793 202795 202803 202825 202795 20279390 133075 133071 133074 133075 133117 133075 133074100 87106 87100 87104 87103 87135 87106 87104110 56825 56820 56824 56823 56867 56825 56823120 36964 36960 36963 36965 37001 36963 36963RMSE 46690-4 89442-6 31623-4 36116-3 10229-4 22966-4CPU-time 2567 1191 472 423 2599 4617

0 002 004 006 008 01 012 014075

08

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(a)

0 02 04 06 08

08

1

065

07

075

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(b)

Figure 3 Comparison with the analytical approximation of Kuskeclose to 120591 = 0 (a) and for [0 119879] (b)

(iii) Han-Wu algorithm (HW) transforms the Black-Scholes equation into a heat equation on an infinitedomain [35]

(iv) the front fixing method proposed by Wu and Kwok(WK) [20]

Apart from the previous comparisons showing that themethod is competitive we remark that the proposed schemehere guarantees the positivity of the solution as well as themonotonicity of both solution and the free boundary

6 Conclusion

The front fixing method for American put option is consid-ered We found that the proposed explicit numerical schemeis stable under appropriate conditions (see Lemma 1) andconsistent with the differential equation with the secondorder in space and first order in time Moreover the mono-tonicity and positivity of the solution and the free boundaryunder that condition are proved in agreement with Kimrsquosresults in [24] about the properties of the theoretical solutionThe theoretical study is confirmed by numerical tests It isshown that the condition (27) is critical that is violation ofthe condition leads to destabilization of the scheme

By the comparison with other approaches advantages ofthe method are discovered The logarithmic transformationrequires weaker stability condition than Landaursquos transfor-mation presented in [21] The explicit calculations withquite weak stability conditions avoid iterations to solve thenonlinear partial differential equation It makes the methodsimpler and reduces computational costs

Conflict of Interests

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

Acknowledgment

This paper has been partially supported by the Euro-pean Union in the FP7-PEOPLE-2012-ITN Program under

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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

Abstract and Applied Analysis 3

The approximate value of 119901(119909 120591) at the point 119909119895and time

120591

119899 is denoted by 119901119899119895asymp 119901(119909

119895 120591

119899) Then a forward-time central-

space explicit scheme is constructed for internal spacial nodesas follows

119901

119899+1

119895minus 119901

119899

119895

119896

=

1

2

120590

2119901

119899

119895minus1minus 2119901

119899

119895+ 119901

119899

119895+1

ℎ

2+ (119903 minus

120590

2

2

)

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

minus 119903119901

119899

119895+

119878

119899+1

119891minus 119878

119899

119891

119896119878

119899

119891

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

1 le 119895 le 119872 0 le 119899 le 119873 minus 1

(11)

By denoting 120583 = 119896ℎ

2 the scheme (11) can be rewritten inthe following form

119901

119899+1

119895= 119886119901

119899

119895minus1+ 119887119901

119899

119895+ 119888119901

119899

119895+1+

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

(119901

119899

119895+1minus 119901

119895minus1)

(12)

where

119886 =

120583

2

(120590

2minus (119903 minus

120590

2

2

) ℎ) 119887 = 1 minus 120590

2120583 minus 119903119896

119888 =

120583

2

(120590

2+ (119903 minus

120590

2

2

) ℎ)

(13)

From the boundary conditions (6) and (7) we can obtain

119901

119899

1minus 119901

119899

minus1

2ℎ

= minus119878

119899

119891 119901

119899

0= 1 minus 119878

119899

119891

(14)

where 119909minus1

= minusℎ is an auxiliary point out of the domain Byconsidering (4) at the point 119909

0= 0 120591 gt 0 (see [26] p 341)

and replacing of the boundary conditions (6) and (7) into (4)at (0+ 120591) one gets the new boundary condition as follows

1

2

120590

2 1205972119901

120597119909

2(0

+ 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(15)

(see [20 23 26]) and its central difference discretization is

120590

2

2

119901

119899

1minus 2119901

119899

0+ 119901

119899

minus1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(16)

From (14) and (16) the value of 119901119899minus1

can be eliminatedobtaining the relationship

119901

119899

1= 120572 minus 120573119878

119899

119891 119899 ge 1 (17)

between the free boundary approximation 119878

119899

119891and 119901

119899

1 where

120572 = 1 +

119903ℎ

2

120590

2 120573 = 1 + ℎ +

1

2

ℎ

2

(18)

By using the scheme (12) for 119895 = 1 and evaluating (17) atthe (119899+1)st time level the free boundary 119878119899+1

119891can be expressed

as

119878

119899+1

119891= 119889

119899119878

119899

119891 0 le 119899 le 119873 minus 1 (19)

where

119889

119899=

120572 minus (119886119901

119899

0+ 119887119901

119899

1+ 119888119901

119899

2minus (119901

119899

2minus 119901

119899

0) 2ℎ)

(119901

119899

2minus 119901

119899

0) 2ℎ + 120573119878

119899

119891

(20)

After expression (19) the value 119878

119899+1

119891can be replaced in

(12) (17) and (14) to obtain values 119901119899+1119895

0 le 119895 le 119872 Then thenumerical scheme for the problem (4)ndash(9) can be rewrittenfor any 119899 = 0 119873 minus 1 in the following algorithmic form

119878

119899+1

119891= 119889

119899119878

119899

119891 (21)

119901

119899+1

0= 1 minus 119878

119899+1

119891 (22)

119901

119899+1

1= 120572 minus 120573119878

119899+1

119891 (23)

119901

119899+1

119895= 119886

119899119901

119899

119895minus1+ 119887119901

119899

119895+ 119888

119899119901

119899

119895+1 119895 = 2 119872 (24)

where

119886

119899= 119886 minus

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

119888

119899= 119888 +

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

119901

119899+1

119872+1= 0

(25)

with the initial conditions

119878

0

119891= 1 119901

0

119895= 0 0 le 119895 le 119872 + 1 (26)

3 Positivity and Monotonicity

In this section we will show the free boundary nonincreasingmonotonicity as well as the positivity and nonincreasingspacial monotonicity of the numerical option price Let usstart this section by showing that constant coefficients 119886119887 and 119888 of scheme (12) for the transformed problem (4)ndash(9) are positive under appropriate conditions on the stepsizediscretization ℎ and 119896

Lemma 1 Assuming that the stepsizes ℎ and 119896 satisfy thefollowing conditions

C1 ℎ le 120590

2|119903 minus (120590

22)| 119903 = 120590

22

C2 119896 le ℎ

2(120590

2+ 119903ℎ

2)

then the coefficients of the scheme 119886 119887 and 119888 are nonnegativeIf 119903 = 120590

22 then under the condition C2 coefficients 119886 119887 and

119888 are nonnegative

Proof From (13) for nonnegativity of 119886 it is necessary that

120590

2minus (119903 minus

120590

2

2

) ℎ ge 0 (27)

If 119903 le 120590

22 from (13) note that 119886 ge 0 for any ℎ gt 0

Otherwise (27) is satisfied under the condition C1From (13) 119887 is nonnegative under the condition C2If 119903 ge 120590

22 nonnegativity of 119888 is guaranteed by (13) for

any ℎ gt 0 Otherwise 119888 is nonnegative under the conditionC1

4 Abstract and Applied Analysis

The following lemma prepares the study of positivity ofthe numerical solution 119901

119899

119895as well as the monotonicity of

the free boundary sequence 119878

119899

119891 that will be established in a

further result

Lemma 2 Let 119901119899119895 119878

119899

119891 be the numerical solution of scheme

(21)ndash(26) for a transformed American put option problem (4)and let 119889119899 be defined by (20) Then under hypothesis of theLemma 1 for small enough ℎ one verifies the following

(1) For each fixed 119899

0 lt 119889

119899le 1 (28)

(2) Values 119901119899+1119895

ge 0 for 119895 = 0 119872 119899 = 0 119873 minus 1

(3) 119901119899+1119895

ge 119901

119899+1

119895+1for 119895 = 0 119872 minus 1 119899 = 0 119873 minus 1

Proof We use the induction principle Note that from (26)we have 1199010

119895ge 0 and 119901

0

119895ge 119901

0

119895+1 Consider the first time level

119899 = 0 From (18) (20) (26) and hypothesis C1 of Lemma 1one gets

0 lt 119889

0=

120572

120573

le 1 (29)

Note that from (21)ndash(23) and (29) one gets

0 lt 119878

1

119891= 119889

0le 1 119901

1

0= 1 minus 119889

0ge 0 119901

1

1= 0 (30)

and from (24) and (26) every 1199011119895= 0 for 119895 = 2 119872

For the sake of clarity let us show firstly that 0 lt 119889

1le 1

From (20) (18) and (30) it follows that

119889

1= 1 minus 119886

120573 minus 120572

120572120573 minus (120573 minus 120572) 2ℎ

= 1 minus 119886

ℎ + ℎ

2(12 minus 119903120590

2)

12 + ℎ (34 + 1199032120590

2) + 119874 (ℎ

2)

(31)

As 119886 is positive by Lemma 1 for small enough values of ℎone gets

0 lt 119889

1le 1 (32)

It is easy to show that 11990122= 0 and for 1198892 analogously as

(31) and (32) 0 lt 119889

2le 1

Let us assume the induction hypothesis that conclusionshold true for index 119899 minus 1 that is

0 lt 119889

119899minus1le 1 119901

119899

119895ge 0 119901

119899

119895ge 119901

119899

119895+1 (33)

Now we are going to prove that conclusions hold true forindex 119899 Let us consider value of 119889119899 for 119899 ge 1 By denoting

119891

119899= 1 +

119903ℎ

2

120590

2minus (119886119901

119899

0+ 119887119901

119899

1+ 119888119901

119899

2minus

119901

119899

2minus 119901

119899

0

2ℎ

) (34)

119892

119899=

119901

119899

2minus 119901

119899

0

2ℎ

+ (

1

2

(1 + (1 + ℎ)

2)) 119878

119899

119891

(35)

then from (20)

119889

119899=

119891

119899

119892

119899 (36)

For 119899 gt 2 using Taylorrsquos expansion the approximationof the involved derivatives and boundary conditions (6) (7)and (15)

119901

119899

2= 1 +

4119903ℎ

2

120590

2minus (1 + 2ℎ + 2ℎ

2) 119878

119899

119891+ 119874 (ℎ

3)

(37)

From (37) and (34) numerator 119891119899 takes the form

119891

119899= 119903ℎ [119896 +

119903119896ℎ + 2 (1 minus 119903119896)

120590

2]

+ (

ℎ

2

2

(1 minus 119903119896) minus 119896ℎ

120590

2

2

) 119878

119899

119891+ 119874 (ℎ

2)

(38)

and verifies 119891119899 gt 0 since 119896 lt ℎ(119903ℎ + 120590

2) under condition C2

of lemma and for ℎ lt 1From (37) and (20) denominator 119892119899 is positive for small

enough values of ℎ since

119892

119899=

119901

119899

2minus 119901

119899

0

2ℎ

+ (1 + ℎ +

ℎ

2

2

) 119878

119899

119891

=

2119903ℎ

120590

2+

ℎ

2

2

119878

119899

119891+ 119874 (ℎ

2) gt 0

(39)

From (36) and previous comments one gets 119889119899 gt 0 Inorder to prove that 119889119899 le 1 let us consider the difference 119891119899 minus119892

119899 By using (38) and (39) under hypothesis of Lemma 2 onecan obtain

119891

119899minus 119892

119899= 119896ℎ(119903

120590

2minus 2119903 + 119903ℎ

120590

2minus

119903ℎ + 120590

2

2

119878

119899

119891) + 119874(ℎ

2)

(40)

Note that if 1205902 lt 2119903 then (40) is nonpositive for smallenough values of ℎ However even if 120590 ge 2119903 the Samuelsonasymptotic limit [27] 119878119899

119891ge 2119903(2119903 + 120590

2) (see [20] page 87)

guarantees the nonpositivity of (40) Therefore 119889119899 le 1In order to prove the positivity of 119901119899+1

119895 it will be useful

to show the nonnegativity of coefficients 119886119899 and 119888

119899 appearingin (21) Coefficient 119886119899 is positive since

119886

119899= 119886 minus

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

= 119886 minus

119889

119899minus 1

2ℎ

ge 119886 ge 0 (41)

In order to show the positivity of the coefficient 119888119899 notethat from (13) and (39) the sign of 119888119899 is the same as the signof (2ℎ119888119892119899 +119891

119899minus119892

119899) and from (13) (38) (39) and 119878

119899

119891le 1 one

gets the following for small enough values of ℎ

2ℎ119888119892

119899+ 119891

119899minus 119892

119899gt 119903119896 + (120590

2120583 + 119903119896)

119903ℎ

2

120590

2minus

119896ℎ

2120590

2

4

gt 0

(42)

Abstract and Applied Analysis 5

Under hypotheses of induction (33) together with positiv-ity of coefficients 119886119899 and 119888

119899 the positivity of 119901119899+1119895

is provedMoreover 119901119899+1

119895 is nonincreasing with respect to index 119895

from (24) since

119901

119899+1

119895minus 119901

119899+1

119895+1= 119886

119899(119901

119899

119895minus1minus 119901

119899

119895) + 119887 (119901

119899

119895minus 119901

119899

119895+1)

+ 119888

119899(119901

119899

119895+1minus 119901

119899

119895+2) ge 0

(43)

Summarizing the following result has been established

Theorem 3 Under assumptions of Lemma 2 the numericalscheme (12) for solving the American option transformedproblem guarantees the following properties of the numericalsolution

(i) nonincreasing monotonicity and positivity of values 119878119899119891

119899 = 0 119873(ii) positivity of the vectors 119901119899 119899 = 0 119873(iii) nonincreasing monotonicity of the vectors 119901

119899=

(119901

119899

0 119901

119899

119872)with respect to space indexes for each fixed

119899 = 0 119873

4 Stability and Consistency

In this section we study the stability and consistency prop-erties of the scheme (21)ndash(26) For the sake of clarity in thepresentation knowing that several different concepts of thestability are used in the literature we begin the section withthe following definition

Definition 4 The numerical scheme (11) is said to be sdot infin-

stable in the domain [0 119909infin]times[0 119879] if for every partitionwith

119896 = Δ120591 ℎ = Δ119909119873119896 = 119879 and (119872 + 1)ℎ = 119909

infin

1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infinle 119862 0 le 119899 le 119873 (44)

where 119862 is independent of ℎ 119896 and 119899 = 0 119873 (see [28])

Theorem 5 Under assumptions of Lemma 2 the numericalscheme (21)ndash(26) for solving transformed problem (4)ndash(9) is sdot

infin-stable

Proof of Theorem 5 Since for each fixed 119899 119901119899119895 is a nonin-

creasing sequence with respect to 119895 then according to theboundary condition (22) and positivity of 119878119899

119891since (28) one

gets1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infin= 119901

119899

0= 1 minus 119878

119899

119891lt 1 0 le 119899 le 119873 (45)

Thus the scheme is sdot infin-stable

Classical consistency of a numerical scheme with respectto a partial differential equation means that the exact theo-retical solution of the PDE approximates well the solutionof the difference scheme as the discretization stepsizes tendto zero [29] However in our case we have to harmonizethe behaviour not only of the PDE (4) with the scheme

(24) but also the boundary conditions (6) (7) and (15) withthe numerical scheme for the free boundaries (21) and (20)With respect to the first part of consistency let us write thenumerical scheme (24) in the following form

119865 (119901

119899

119895 119878

119899

119891)

=

119901

119899+1

119895minus 119901

119899

119895

119896

minus

1

2

120590

2119901

119899

119895minus1minus 2119901

119899

119895+ 119901

119899

119895+1

ℎ

2

minus (119903 minus

120590

2

2

)

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

+ 119903119901

119899

119895minus

119878

119899+1

119891minus 119878

119899

119891

119896119878

119899

119891

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

= 0

(46)

Let us denote by 119901

119899

119895= 119901(119909

119895 120591

119899) the exact theoretical

solution value of the PDE at the mesh point (119909119895 120591

119899) and let

119878

119899

119891= 119878

119891(120591

119899) be the exact solution of the free boundary at time

120591

119899 The scheme (46) is said to be consistent with

119871 (119901 119878

119891)

=

120597119901

120597120591

minus

1

2

120590

2 1205972119901

120597119909

2minus (119903 minus

120590

2

2

)

120597119901

120597119909

+ 119903119901 minus

119878

1015840

119891

119878

119891

120597119901

120597119909

= 0

(47)

if the local truncation error

119879

119899

119895(119901

119878

119891) = 119865 (

119901

119899

119895

119878

119899

119891) minus 119871 (

119901

119899

119895

119878

119899

119891) (48)

satisfies

119879

119899

119895(119901

119878

119891) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (49)

Assuming the existence of the continuous partial deriva-tives up to order two in time and up to order four in spaceusing Taylorrsquos expansion about (119909

119895 120591

119899) one gets

119879

119899

119895(119901

119878

119891)

= 119896119864

119899

119895(3) minus

120590

2

2

ℎ

2119864

119899

119895(2) + (119903 minus

120590

2

2

) ℎ

2119864

119899

119895(1)

minus 119896119864

119899

119895(4)

120597119901

120597119909

(119909

119895 120591

119899) minus ℎ

2119864

119899

119895(1)

1

119878

119899

119891

119889119878

119891

119889120591

(120591

119899)

minus 119896ℎ

2119864

119899

119895(4) 119864

119899

119895(1)

(50)

where

119864

119899

119895(1) =

1

6

120597

3119901

120597119909

3(119909

119895 120591

119899) 119864

119899

119895(2) =

1

12

120597

4119901

120597119909

4(119909

119895 120591

119899)

119864

119899

119895(3) =

1

2

120597

2119901

120597120591

2(119909

119895 120591

119899) 119864

119899

119895(4) =

119896

2

119878

119891

119889

2119878

119891

119889120591

2(120591

119899)

(51)

6 Abstract and Applied Analysis

Equations (50) and (51) show the local truncation error ofthe numerical scheme (24) with respect to the PDE (4) Notethat from (50) and (51)

119879

119899

119895(119901

119878

119891) = 119874 (ℎ

2) + 119874 (119896) (52)

In order to complete the consistency of the solution of the freeboundary problem with the scheme (21)ndash(26) it is necessaryto rewrite the boundary conditions (6) (7) and (15) in thefollowing form

119871

1(119901 119878

119891) = 119901 (0 120591) minus 1 + 119878

119891(120591) = 0

119871

2(119901 119878

119891) =

120597119901

120597119909

(0 120591) + 119878

119891(120591) = 0

119871

3(119901 119878

119891) =

120590

2

2

120597

2119901

120597119909

2(0 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(53)

Furthermore we have finite difference approximation for thefollowing boundary conditions

119865

1(119901 119878

119891) = 119901

119899

0minus 1 + 119878

119899

119891= 0

119865

2(119901 119878

119891) =

119901

119899

1minus 119901

119899

minus1

2ℎ

+ 119878

119899

119891= 0

119865

3(119901 119878

119891) =

120590

2

2

119901

119899

minus1minus 2119901

119899

0+ 119901

119899

1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(54)

It is not difficult to check using Taylorrsquos expansion that thelocal truncation error satisfies

119879

1(119901

119878

119891) = 119865

1(119901

119878

119891) minus 119871

1(119901

119878

119891) = 0

119879

2(119901

119878

119891) = 119865

2(119901

119878

119891) minus 119871

2(119901

119878

119891) = 119874 (ℎ

2)

119879

3(119901

119878

119891) = 119865

3(119901

119878

119891) minus 119871

3(119901

119878

119891) = 119874 (ℎ

2)

(55)

The truncation error for the boundary condition behavesas ℎ2

Theorem 6 Assuming that the solution of the PDE problem(4)ndash(9) admits two times continuous partial derivative withrespect to time and up to order four with respect to spacethe numerical solution computed by the scheme (21)ndash(26) isconsistent with (4) and boundary conditions of order two inspace and order one in time

5 Numerical Experiments

In this section we illustrate the previous theoretical resultswith numerical experiments showing the potential advan-tages of the proposed method such as preserving the quali-tative properties of the theoretical solution [24] such as posi-tivity andmonotonicity A comparisonwith other approachesis also presented in this section

51 Confirmations of Theoretical Results In this experimentwe check that the stability condition can be broken showing

0 02 04 06 08 1086

088

09

092

094

096

098

1

Time to maturity

Free

bou

ndar

y

(a)

088

092

094

096

098

1

102

0 02 04 06 08 1

09

Time to maturity

Free

bou

ndar

y

(b)

Figure 1 Dependence of stability on positivity of coefficient 119887 (120583 =

24 119887 = 00398) for (a) and (120583 = 252 119887 = minus00083) for (b)

that in such case the results could become unreliable Let usconsider an American put option pricing problem as in [21]with the following parameters

119903 = 01 120590 = 02 119879 = 1 119909max = 2 (56)

We consider fixed space step ℎ = 001 that satisfies conditionC1 of Lemma 1 and change time step (Figure 1) Numericaltests show that the condition C2 is critical for positivity ofcoefficient 119887 and as a result for the stability of the schemeThe monotonicity of the free boundary is numerically shownby numerical tests (Figure 1(a)) In Figure 1(b) one can seethat when condition C2 is not satisfied spurious oscillationsappear so the monotonicity is broken

It was theoretically proved in Section 4 that the schemehas order of approximation 119874(ℎ

2) + 119874(119896) The results of the

Abstract and Applied Analysis 7

Table 1 Convergence in space for fixed time step 119896 = 2 sdot10

minus3 for theproblem with parameters (56)

Asset price True value 004 002 00190 116974 117283 117054 116991100 69320 69308 69309 69312110 41550 41694 41564 41531120 25102 25219 25151 25114

RMSE 18037-2 47857-3 14623-3CPU-time (sec) 0052 0075 0105

0

0001

0002

0003

0004

0005

0 50 100 150 200

RMSE

CPU-time (s)

Figure 2 RMSE against the computational time

numerical experiments for the problem with the followingparameters

119903 = 008 120590 = 02 119879 = 3 119909max = 2 (57)

are presented in Table 1 For ldquoTruerdquo values we use thereference values offered in [30] We consider the differentspace steps for fixed time step 119896 = 0002 to guaranteethe stability The root-mean-square error (RMSE) is used tomeasure the accuracy of the scheme The last row presentsthe CPU-time in seconds for each experimentThe plot of theRMSE against computational time is presented in Figure 2

To check the order of approximation in space we intro-duce the convergence rate

120574 (ℎ

1 ℎ

2) =

ln RMSEℎ1

minus ln RMSEℎ2

ln ℎ

1minus ln ℎ

2

(58)

From Table 1 and (58) one obtains 120574(4 sdot 10

minus2 2 sdot 10

minus2) =

191 that is close to 2To check the order of approximation in time the space

step ℎ is fixed (ℎ = 10

minus2) and time step 119896 is variableFrom Table 2 by analogous to (58) formula one gets 120574(5 sdot

10

minus4 10

minus3) = 080 that is close to 1 It proves the second order

of approximation in space and the first order in time

52 Comparison with Other Approaches The front fixingmethod is used in [21] for American put option pricing prob-lemwith parameters (56) but with another transformation asfollows

119909 =

119878

119878

119891(120591)

119901 (119909 120591) = 119875 (119878 120591) = 119875 (119909119878

119891(119905) 120591)

(59)

Table 2 Convergence in time for fixed space step ℎ = 001 for theproblem with parameters (56)

Asset price True value 00005 0001 000290 116974 116984 116989 116991100 69320 69319 69318 69312110 41550 41555 41544 41531120 25102 25103 25091 25114

RMSE 56347-4 98234-4 14623-3CPU-time (sec) 0142 0113 0105

Table 3 Comparison with front-fixing method under transforma-tion (59)

Method 119878

119891(119879)

Implicit (in [21]) 08615Explicit (in [21]) 08622Explicit (proposed) 08628

By the numerical tests it is shown that the consideredscheme is stable for 120583 le 24 while the condition for a stabilityof the scheme in [21] is 120583 le 6 The results are representedin Table 3 The front fixing method with the logarithmictransformation has a good advantage weaker condition forthe stability It reduces the computational costs

Let us compare front fixing method with anotherapproach [31] based on Mellinrsquos transform The parametersof the problem are 119903 = 00488 120590 = 03 and 119879 = 05833

To compare results of the explicit front fixing methodwith Mellinrsquos transform [31] we have to multiply our dimen-sionless value on119864 = 45Then forMellinrsquos transformmethod119878

119891(119879) = 3277 while for front fixing method 119878

119891(119879) =

327655Close to maturity exercise boundary for the American

put can be analytically approximated [32] as follows

119878

119891(120591) sim 119864[1 minus

radic2120590120591 log( 120590

2

6119903radic120587120591120590

22

)] (60)

This approximation can be used only near expiry date InFigure 3 the value of the free boundary for both approachesis presented Near initial point front fixing method andanalytical approximation (60) give close results

We compare explicit front fixing method (FF) with ℎ

1=

0001 and ℎ

2= 0002 and 120583 = 5 for American put with other

numerical methods shown in [15] in Table 4 for the followingproblem parameters

119903 = 005 120590 = 02 119879 = 3

119864 = 100 119909max = 2

(61)

There are several considered methods as follows

(i) penalty method (PM) is considered in [21 33](ii) Ikonen and Toivanen [34] proposed an operator

splitting technique (OS) for solving the linear com-plementarity problem

8 Abstract and Applied Analysis

Table 4 Comparison with other methods put option value for different asset prices

Asset price True value PM OS HW WK FF(ℎ1) FF(ℎ

2)

80 202797 202793 202795 202803 202825 202795 20279390 133075 133071 133074 133075 133117 133075 133074100 87106 87100 87104 87103 87135 87106 87104110 56825 56820 56824 56823 56867 56825 56823120 36964 36960 36963 36965 37001 36963 36963RMSE 46690-4 89442-6 31623-4 36116-3 10229-4 22966-4CPU-time 2567 1191 472 423 2599 4617

0 002 004 006 008 01 012 014075

08

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(a)

0 02 04 06 08

08

1

065

07

075

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(b)

Figure 3 Comparison with the analytical approximation of Kuskeclose to 120591 = 0 (a) and for [0 119879] (b)

(iii) Han-Wu algorithm (HW) transforms the Black-Scholes equation into a heat equation on an infinitedomain [35]

(iv) the front fixing method proposed by Wu and Kwok(WK) [20]

Apart from the previous comparisons showing that themethod is competitive we remark that the proposed schemehere guarantees the positivity of the solution as well as themonotonicity of both solution and the free boundary

6 Conclusion

The front fixing method for American put option is consid-ered We found that the proposed explicit numerical schemeis stable under appropriate conditions (see Lemma 1) andconsistent with the differential equation with the secondorder in space and first order in time Moreover the mono-tonicity and positivity of the solution and the free boundaryunder that condition are proved in agreement with Kimrsquosresults in [24] about the properties of the theoretical solutionThe theoretical study is confirmed by numerical tests It isshown that the condition (27) is critical that is violation ofthe condition leads to destabilization of the scheme

By the comparison with other approaches advantages ofthe method are discovered The logarithmic transformationrequires weaker stability condition than Landaursquos transfor-mation presented in [21] The explicit calculations withquite weak stability conditions avoid iterations to solve thenonlinear partial differential equation It makes the methodsimpler and reduces computational costs

Conflict of Interests

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

Acknowledgment

This paper has been partially supported by the Euro-pean Union in the FP7-PEOPLE-2012-ITN Program under

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

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OptimizationJournal of

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CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

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Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Abstract and Applied Analysis

The following lemma prepares the study of positivity ofthe numerical solution 119901

119899

119895as well as the monotonicity of

the free boundary sequence 119878

119899

119891 that will be established in a

further result

Lemma 2 Let 119901119899119895 119878

119899

119891 be the numerical solution of scheme

(21)ndash(26) for a transformed American put option problem (4)and let 119889119899 be defined by (20) Then under hypothesis of theLemma 1 for small enough ℎ one verifies the following

(1) For each fixed 119899

0 lt 119889

119899le 1 (28)

(2) Values 119901119899+1119895

ge 0 for 119895 = 0 119872 119899 = 0 119873 minus 1

(3) 119901119899+1119895

ge 119901

119899+1

119895+1for 119895 = 0 119872 minus 1 119899 = 0 119873 minus 1

Proof We use the induction principle Note that from (26)we have 1199010

119895ge 0 and 119901

0

119895ge 119901

0

119895+1 Consider the first time level

119899 = 0 From (18) (20) (26) and hypothesis C1 of Lemma 1one gets

0 lt 119889

0=

120572

120573

le 1 (29)

Note that from (21)ndash(23) and (29) one gets

0 lt 119878

1

119891= 119889

0le 1 119901

1

0= 1 minus 119889

0ge 0 119901

1

1= 0 (30)

and from (24) and (26) every 1199011119895= 0 for 119895 = 2 119872

For the sake of clarity let us show firstly that 0 lt 119889

1le 1

From (20) (18) and (30) it follows that

119889

1= 1 minus 119886

120573 minus 120572

120572120573 minus (120573 minus 120572) 2ℎ

= 1 minus 119886

ℎ + ℎ

2(12 minus 119903120590

2)

12 + ℎ (34 + 1199032120590

2) + 119874 (ℎ

2)

(31)

As 119886 is positive by Lemma 1 for small enough values of ℎone gets

0 lt 119889

1le 1 (32)

It is easy to show that 11990122= 0 and for 1198892 analogously as

(31) and (32) 0 lt 119889

2le 1

Let us assume the induction hypothesis that conclusionshold true for index 119899 minus 1 that is

0 lt 119889

119899minus1le 1 119901

119899

119895ge 0 119901

119899

119895ge 119901

119899

119895+1 (33)

Now we are going to prove that conclusions hold true forindex 119899 Let us consider value of 119889119899 for 119899 ge 1 By denoting

119891

119899= 1 +

119903ℎ

2

120590

2minus (119886119901

119899

0+ 119887119901

119899

1+ 119888119901

119899

2minus

119901

119899

2minus 119901

119899

0

2ℎ

) (34)

119892

119899=

119901

119899

2minus 119901

119899

0

2ℎ

+ (

1

2

(1 + (1 + ℎ)

2)) 119878

119899

119891

(35)

then from (20)

119889

119899=

119891

119899

119892

119899 (36)

For 119899 gt 2 using Taylorrsquos expansion the approximationof the involved derivatives and boundary conditions (6) (7)and (15)

119901

119899

2= 1 +

4119903ℎ

2

120590

2minus (1 + 2ℎ + 2ℎ

2) 119878

119899

119891+ 119874 (ℎ

3)

(37)

From (37) and (34) numerator 119891119899 takes the form

119891

119899= 119903ℎ [119896 +

119903119896ℎ + 2 (1 minus 119903119896)

120590

2]

+ (

ℎ

2

2

(1 minus 119903119896) minus 119896ℎ

120590

2

2

) 119878

119899

119891+ 119874 (ℎ

2)

(38)

and verifies 119891119899 gt 0 since 119896 lt ℎ(119903ℎ + 120590

2) under condition C2

of lemma and for ℎ lt 1From (37) and (20) denominator 119892119899 is positive for small

enough values of ℎ since

119892

119899=

119901

119899

2minus 119901

119899

0

2ℎ

+ (1 + ℎ +

ℎ

2

2

) 119878

119899

119891

=

2119903ℎ

120590

2+

ℎ

2

2

119878

119899

119891+ 119874 (ℎ

2) gt 0

(39)

From (36) and previous comments one gets 119889119899 gt 0 Inorder to prove that 119889119899 le 1 let us consider the difference 119891119899 minus119892

119899 By using (38) and (39) under hypothesis of Lemma 2 onecan obtain

119891

119899minus 119892

119899= 119896ℎ(119903

120590

2minus 2119903 + 119903ℎ

120590

2minus

119903ℎ + 120590

2

2

119878

119899

119891) + 119874(ℎ

2)

(40)

Note that if 1205902 lt 2119903 then (40) is nonpositive for smallenough values of ℎ However even if 120590 ge 2119903 the Samuelsonasymptotic limit [27] 119878119899

119891ge 2119903(2119903 + 120590

2) (see [20] page 87)

guarantees the nonpositivity of (40) Therefore 119889119899 le 1In order to prove the positivity of 119901119899+1

119895 it will be useful

to show the nonnegativity of coefficients 119886119899 and 119888

119899 appearingin (21) Coefficient 119886119899 is positive since

119886

119899= 119886 minus

119878

119899+1

119891minus 119878

119899

119891

2ℎ119878

119899

119891

= 119886 minus

119889

119899minus 1

2ℎ

ge 119886 ge 0 (41)

In order to show the positivity of the coefficient 119888119899 notethat from (13) and (39) the sign of 119888119899 is the same as the signof (2ℎ119888119892119899 +119891

119899minus119892

119899) and from (13) (38) (39) and 119878

119899

119891le 1 one

gets the following for small enough values of ℎ

2ℎ119888119892

119899+ 119891

119899minus 119892

119899gt 119903119896 + (120590

2120583 + 119903119896)

119903ℎ

2

120590

2minus

119896ℎ

2120590

2

4

gt 0

(42)

Abstract and Applied Analysis 5

Under hypotheses of induction (33) together with positiv-ity of coefficients 119886119899 and 119888

119899 the positivity of 119901119899+1119895

is provedMoreover 119901119899+1

119895 is nonincreasing with respect to index 119895

from (24) since

119901

119899+1

119895minus 119901

119899+1

119895+1= 119886

119899(119901

119899

119895minus1minus 119901

119899

119895) + 119887 (119901

119899

119895minus 119901

119899

119895+1)

+ 119888

119899(119901

119899

119895+1minus 119901

119899

119895+2) ge 0

(43)

Summarizing the following result has been established

Theorem 3 Under assumptions of Lemma 2 the numericalscheme (12) for solving the American option transformedproblem guarantees the following properties of the numericalsolution

(i) nonincreasing monotonicity and positivity of values 119878119899119891

119899 = 0 119873(ii) positivity of the vectors 119901119899 119899 = 0 119873(iii) nonincreasing monotonicity of the vectors 119901

119899=

(119901

119899

0 119901

119899

119872)with respect to space indexes for each fixed

119899 = 0 119873

4 Stability and Consistency

In this section we study the stability and consistency prop-erties of the scheme (21)ndash(26) For the sake of clarity in thepresentation knowing that several different concepts of thestability are used in the literature we begin the section withthe following definition

Definition 4 The numerical scheme (11) is said to be sdot infin-

stable in the domain [0 119909infin]times[0 119879] if for every partitionwith

119896 = Δ120591 ℎ = Δ119909119873119896 = 119879 and (119872 + 1)ℎ = 119909

infin

1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infinle 119862 0 le 119899 le 119873 (44)

where 119862 is independent of ℎ 119896 and 119899 = 0 119873 (see [28])

Theorem 5 Under assumptions of Lemma 2 the numericalscheme (21)ndash(26) for solving transformed problem (4)ndash(9) is sdot

infin-stable

Proof of Theorem 5 Since for each fixed 119899 119901119899119895 is a nonin-

creasing sequence with respect to 119895 then according to theboundary condition (22) and positivity of 119878119899

119891since (28) one

gets1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infin= 119901

119899

0= 1 minus 119878

119899

119891lt 1 0 le 119899 le 119873 (45)

Thus the scheme is sdot infin-stable

Classical consistency of a numerical scheme with respectto a partial differential equation means that the exact theo-retical solution of the PDE approximates well the solutionof the difference scheme as the discretization stepsizes tendto zero [29] However in our case we have to harmonizethe behaviour not only of the PDE (4) with the scheme

(24) but also the boundary conditions (6) (7) and (15) withthe numerical scheme for the free boundaries (21) and (20)With respect to the first part of consistency let us write thenumerical scheme (24) in the following form

119865 (119901

119899

119895 119878

119899

119891)

=

119901

119899+1

119895minus 119901

119899

119895

119896

minus

1

2

120590

2119901

119899

119895minus1minus 2119901

119899

119895+ 119901

119899

119895+1

ℎ

2

minus (119903 minus

120590

2

2

)

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

+ 119903119901

119899

119895minus

119878

119899+1

119891minus 119878

119899

119891

119896119878

119899

119891

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

= 0

(46)

Let us denote by 119901

119899

119895= 119901(119909

119895 120591

119899) the exact theoretical

solution value of the PDE at the mesh point (119909119895 120591

119899) and let

119878

119899

119891= 119878

119891(120591

119899) be the exact solution of the free boundary at time

120591

119899 The scheme (46) is said to be consistent with

119871 (119901 119878

119891)

=

120597119901

120597120591

minus

1

2

120590

2 1205972119901

120597119909

2minus (119903 minus

120590

2

2

)

120597119901

120597119909

+ 119903119901 minus

119878

1015840

119891

119878

119891

120597119901

120597119909

= 0

(47)

if the local truncation error

119879

119899

119895(119901

119878

119891) = 119865 (

119901

119899

119895

119878

119899

119891) minus 119871 (

119901

119899

119895

119878

119899

119891) (48)

satisfies

119879

119899

119895(119901

119878

119891) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (49)

Assuming the existence of the continuous partial deriva-tives up to order two in time and up to order four in spaceusing Taylorrsquos expansion about (119909

119895 120591

119899) one gets

119879

119899

119895(119901

119878

119891)

= 119896119864

119899

119895(3) minus

120590

2

2

ℎ

2119864

119899

119895(2) + (119903 minus

120590

2

2

) ℎ

2119864

119899

119895(1)

minus 119896119864

119899

119895(4)

120597119901

120597119909

(119909

119895 120591

119899) minus ℎ

2119864

119899

119895(1)

1

119878

119899

119891

119889119878

119891

119889120591

(120591

119899)

minus 119896ℎ

2119864

119899

119895(4) 119864

119899

119895(1)

(50)

where

119864

119899

119895(1) =

1

6

120597

3119901

120597119909

3(119909

119895 120591

119899) 119864

119899

119895(2) =

1

12

120597

4119901

120597119909

4(119909

119895 120591

119899)

119864

119899

119895(3) =

1

2

120597

2119901

120597120591

2(119909

119895 120591

119899) 119864

119899

119895(4) =

119896

2

119878

119891

119889

2119878

119891

119889120591

2(120591

119899)

(51)

6 Abstract and Applied Analysis

Equations (50) and (51) show the local truncation error ofthe numerical scheme (24) with respect to the PDE (4) Notethat from (50) and (51)

119879

119899

119895(119901

119878

119891) = 119874 (ℎ

2) + 119874 (119896) (52)

In order to complete the consistency of the solution of the freeboundary problem with the scheme (21)ndash(26) it is necessaryto rewrite the boundary conditions (6) (7) and (15) in thefollowing form

119871

1(119901 119878

119891) = 119901 (0 120591) minus 1 + 119878

119891(120591) = 0

119871

2(119901 119878

119891) =

120597119901

120597119909

(0 120591) + 119878

119891(120591) = 0

119871

3(119901 119878

119891) =

120590

2

2

120597

2119901

120597119909

2(0 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(53)

Furthermore we have finite difference approximation for thefollowing boundary conditions

119865

1(119901 119878

119891) = 119901

119899

0minus 1 + 119878

119899

119891= 0

119865

2(119901 119878

119891) =

119901

119899

1minus 119901

119899

minus1

2ℎ

+ 119878

119899

119891= 0

119865

3(119901 119878

119891) =

120590

2

2

119901

119899

minus1minus 2119901

119899

0+ 119901

119899

1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(54)

It is not difficult to check using Taylorrsquos expansion that thelocal truncation error satisfies

119879

1(119901

119878

119891) = 119865

1(119901

119878

119891) minus 119871

1(119901

119878

119891) = 0

119879

2(119901

119878

119891) = 119865

2(119901

119878

119891) minus 119871

2(119901

119878

119891) = 119874 (ℎ

2)

119879

3(119901

119878

119891) = 119865

3(119901

119878

119891) minus 119871

3(119901

119878

119891) = 119874 (ℎ

2)

(55)

The truncation error for the boundary condition behavesas ℎ2

Theorem 6 Assuming that the solution of the PDE problem(4)ndash(9) admits two times continuous partial derivative withrespect to time and up to order four with respect to spacethe numerical solution computed by the scheme (21)ndash(26) isconsistent with (4) and boundary conditions of order two inspace and order one in time

5 Numerical Experiments

In this section we illustrate the previous theoretical resultswith numerical experiments showing the potential advan-tages of the proposed method such as preserving the quali-tative properties of the theoretical solution [24] such as posi-tivity andmonotonicity A comparisonwith other approachesis also presented in this section

51 Confirmations of Theoretical Results In this experimentwe check that the stability condition can be broken showing

0 02 04 06 08 1086

088

09

092

094

096

098

1

Time to maturity

Free

bou

ndar

y

(a)

088

092

094

096

098

1

102

0 02 04 06 08 1

09

Time to maturity

Free

bou

ndar

y

(b)

Figure 1 Dependence of stability on positivity of coefficient 119887 (120583 =

24 119887 = 00398) for (a) and (120583 = 252 119887 = minus00083) for (b)

that in such case the results could become unreliable Let usconsider an American put option pricing problem as in [21]with the following parameters

119903 = 01 120590 = 02 119879 = 1 119909max = 2 (56)

We consider fixed space step ℎ = 001 that satisfies conditionC1 of Lemma 1 and change time step (Figure 1) Numericaltests show that the condition C2 is critical for positivity ofcoefficient 119887 and as a result for the stability of the schemeThe monotonicity of the free boundary is numerically shownby numerical tests (Figure 1(a)) In Figure 1(b) one can seethat when condition C2 is not satisfied spurious oscillationsappear so the monotonicity is broken

It was theoretically proved in Section 4 that the schemehas order of approximation 119874(ℎ

2) + 119874(119896) The results of the

Abstract and Applied Analysis 7

Table 1 Convergence in space for fixed time step 119896 = 2 sdot10

minus3 for theproblem with parameters (56)

Asset price True value 004 002 00190 116974 117283 117054 116991100 69320 69308 69309 69312110 41550 41694 41564 41531120 25102 25219 25151 25114

RMSE 18037-2 47857-3 14623-3CPU-time (sec) 0052 0075 0105

0

0001

0002

0003

0004

0005

0 50 100 150 200

RMSE

CPU-time (s)

Figure 2 RMSE against the computational time

numerical experiments for the problem with the followingparameters

119903 = 008 120590 = 02 119879 = 3 119909max = 2 (57)

are presented in Table 1 For ldquoTruerdquo values we use thereference values offered in [30] We consider the differentspace steps for fixed time step 119896 = 0002 to guaranteethe stability The root-mean-square error (RMSE) is used tomeasure the accuracy of the scheme The last row presentsthe CPU-time in seconds for each experimentThe plot of theRMSE against computational time is presented in Figure 2

To check the order of approximation in space we intro-duce the convergence rate

120574 (ℎ

1 ℎ

2) =

ln RMSEℎ1

minus ln RMSEℎ2

ln ℎ

1minus ln ℎ

2

(58)

From Table 1 and (58) one obtains 120574(4 sdot 10

minus2 2 sdot 10

minus2) =

191 that is close to 2To check the order of approximation in time the space

step ℎ is fixed (ℎ = 10

minus2) and time step 119896 is variableFrom Table 2 by analogous to (58) formula one gets 120574(5 sdot

10

minus4 10

minus3) = 080 that is close to 1 It proves the second order

of approximation in space and the first order in time

52 Comparison with Other Approaches The front fixingmethod is used in [21] for American put option pricing prob-lemwith parameters (56) but with another transformation asfollows

119909 =

119878

119878

119891(120591)

119901 (119909 120591) = 119875 (119878 120591) = 119875 (119909119878

119891(119905) 120591)

(59)

Table 2 Convergence in time for fixed space step ℎ = 001 for theproblem with parameters (56)

Asset price True value 00005 0001 000290 116974 116984 116989 116991100 69320 69319 69318 69312110 41550 41555 41544 41531120 25102 25103 25091 25114

RMSE 56347-4 98234-4 14623-3CPU-time (sec) 0142 0113 0105

Table 3 Comparison with front-fixing method under transforma-tion (59)

Method 119878

119891(119879)

Implicit (in [21]) 08615Explicit (in [21]) 08622Explicit (proposed) 08628

By the numerical tests it is shown that the consideredscheme is stable for 120583 le 24 while the condition for a stabilityof the scheme in [21] is 120583 le 6 The results are representedin Table 3 The front fixing method with the logarithmictransformation has a good advantage weaker condition forthe stability It reduces the computational costs

Let us compare front fixing method with anotherapproach [31] based on Mellinrsquos transform The parametersof the problem are 119903 = 00488 120590 = 03 and 119879 = 05833

To compare results of the explicit front fixing methodwith Mellinrsquos transform [31] we have to multiply our dimen-sionless value on119864 = 45Then forMellinrsquos transformmethod119878

119891(119879) = 3277 while for front fixing method 119878

119891(119879) =

327655Close to maturity exercise boundary for the American

put can be analytically approximated [32] as follows

119878

119891(120591) sim 119864[1 minus

radic2120590120591 log( 120590

2

6119903radic120587120591120590

22

)] (60)

This approximation can be used only near expiry date InFigure 3 the value of the free boundary for both approachesis presented Near initial point front fixing method andanalytical approximation (60) give close results

We compare explicit front fixing method (FF) with ℎ

1=

0001 and ℎ

2= 0002 and 120583 = 5 for American put with other

numerical methods shown in [15] in Table 4 for the followingproblem parameters

119903 = 005 120590 = 02 119879 = 3

119864 = 100 119909max = 2

(61)

There are several considered methods as follows

(i) penalty method (PM) is considered in [21 33](ii) Ikonen and Toivanen [34] proposed an operator

splitting technique (OS) for solving the linear com-plementarity problem

8 Abstract and Applied Analysis

Table 4 Comparison with other methods put option value for different asset prices

Asset price True value PM OS HW WK FF(ℎ1) FF(ℎ

2)

80 202797 202793 202795 202803 202825 202795 20279390 133075 133071 133074 133075 133117 133075 133074100 87106 87100 87104 87103 87135 87106 87104110 56825 56820 56824 56823 56867 56825 56823120 36964 36960 36963 36965 37001 36963 36963RMSE 46690-4 89442-6 31623-4 36116-3 10229-4 22966-4CPU-time 2567 1191 472 423 2599 4617

0 002 004 006 008 01 012 014075

08

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(a)

0 02 04 06 08

08

1

065

07

075

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(b)

Figure 3 Comparison with the analytical approximation of Kuskeclose to 120591 = 0 (a) and for [0 119879] (b)

(iii) Han-Wu algorithm (HW) transforms the Black-Scholes equation into a heat equation on an infinitedomain [35]

(iv) the front fixing method proposed by Wu and Kwok(WK) [20]

Apart from the previous comparisons showing that themethod is competitive we remark that the proposed schemehere guarantees the positivity of the solution as well as themonotonicity of both solution and the free boundary

6 Conclusion

The front fixing method for American put option is consid-ered We found that the proposed explicit numerical schemeis stable under appropriate conditions (see Lemma 1) andconsistent with the differential equation with the secondorder in space and first order in time Moreover the mono-tonicity and positivity of the solution and the free boundaryunder that condition are proved in agreement with Kimrsquosresults in [24] about the properties of the theoretical solutionThe theoretical study is confirmed by numerical tests It isshown that the condition (27) is critical that is violation ofthe condition leads to destabilization of the scheme

By the comparison with other approaches advantages ofthe method are discovered The logarithmic transformationrequires weaker stability condition than Landaursquos transfor-mation presented in [21] The explicit calculations withquite weak stability conditions avoid iterations to solve thenonlinear partial differential equation It makes the methodsimpler and reduces computational costs

Conflict of Interests

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

Acknowledgment

This paper has been partially supported by the Euro-pean Union in the FP7-PEOPLE-2012-ITN Program under

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical PhysicsAdvances in

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OptimizationJournal of

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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

Abstract and Applied Analysis 5

Under hypotheses of induction (33) together with positiv-ity of coefficients 119886119899 and 119888

119899 the positivity of 119901119899+1119895

is provedMoreover 119901119899+1

119895 is nonincreasing with respect to index 119895

from (24) since

119901

119899+1

119895minus 119901

119899+1

119895+1= 119886

119899(119901

119899

119895minus1minus 119901

119899

119895) + 119887 (119901

119899

119895minus 119901

119899

119895+1)

+ 119888

119899(119901

119899

119895+1minus 119901

119899

119895+2) ge 0

(43)

Summarizing the following result has been established

Theorem 3 Under assumptions of Lemma 2 the numericalscheme (12) for solving the American option transformedproblem guarantees the following properties of the numericalsolution

(i) nonincreasing monotonicity and positivity of values 119878119899119891

119899 = 0 119873(ii) positivity of the vectors 119901119899 119899 = 0 119873(iii) nonincreasing monotonicity of the vectors 119901

119899=

(119901

119899

0 119901

119899

119872)with respect to space indexes for each fixed

119899 = 0 119873

4 Stability and Consistency

In this section we study the stability and consistency prop-erties of the scheme (21)ndash(26) For the sake of clarity in thepresentation knowing that several different concepts of thestability are used in the literature we begin the section withthe following definition

Definition 4 The numerical scheme (11) is said to be sdot infin-

stable in the domain [0 119909infin]times[0 119879] if for every partitionwith

119896 = Δ120591 ℎ = Δ119909119873119896 = 119879 and (119872 + 1)ℎ = 119909

infin

1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infinle 119862 0 le 119899 le 119873 (44)

where 119862 is independent of ℎ 119896 and 119899 = 0 119873 (see [28])

Theorem 5 Under assumptions of Lemma 2 the numericalscheme (21)ndash(26) for solving transformed problem (4)ndash(9) is sdot

infin-stable

Proof of Theorem 5 Since for each fixed 119899 119901119899119895 is a nonin-

creasing sequence with respect to 119895 then according to theboundary condition (22) and positivity of 119878119899

119891since (28) one

gets1003817

1003817

1003817

1003817

119875

11989910038171003817

1003817

1003817infin= 119901

119899

0= 1 minus 119878

119899

119891lt 1 0 le 119899 le 119873 (45)

Thus the scheme is sdot infin-stable

Classical consistency of a numerical scheme with respectto a partial differential equation means that the exact theo-retical solution of the PDE approximates well the solutionof the difference scheme as the discretization stepsizes tendto zero [29] However in our case we have to harmonizethe behaviour not only of the PDE (4) with the scheme

(24) but also the boundary conditions (6) (7) and (15) withthe numerical scheme for the free boundaries (21) and (20)With respect to the first part of consistency let us write thenumerical scheme (24) in the following form

119865 (119901

119899

119895 119878

119899

119891)

=

119901

119899+1

119895minus 119901

119899

119895

119896

minus

1

2

120590

2119901

119899

119895minus1minus 2119901

119899

119895+ 119901

119899

119895+1

ℎ

2

minus (119903 minus

120590

2

2

)

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

+ 119903119901

119899

119895minus

119878

119899+1

119891minus 119878

119899

119891

119896119878

119899

119891

119901

119899

119895+1minus 119901

119899

119895minus1

2ℎ

= 0

(46)

Let us denote by 119901

119899

119895= 119901(119909

119895 120591

119899) the exact theoretical

solution value of the PDE at the mesh point (119909119895 120591

119899) and let

119878

119899

119891= 119878

119891(120591

119899) be the exact solution of the free boundary at time

120591

119899 The scheme (46) is said to be consistent with

119871 (119901 119878

119891)

=

120597119901

120597120591

minus

1

2

120590

2 1205972119901

120597119909

2minus (119903 minus

120590

2

2

)

120597119901

120597119909

+ 119903119901 minus

119878

1015840

119891

119878

119891

120597119901

120597119909

= 0

(47)

if the local truncation error

119879

119899

119895(119901

119878

119891) = 119865 (

119901

119899

119895

119878

119899

119891) minus 119871 (

119901

119899

119895

119878

119899

119891) (48)

satisfies

119879

119899

119895(119901

119878

119891) 997888rarr 0 as ℎ 997888rarr 0 119896 997888rarr 0 (49)

Assuming the existence of the continuous partial deriva-tives up to order two in time and up to order four in spaceusing Taylorrsquos expansion about (119909

119895 120591

119899) one gets

119879

119899

119895(119901

119878

119891)

= 119896119864

119899

119895(3) minus

120590

2

2

ℎ

2119864

119899

119895(2) + (119903 minus

120590

2

2

) ℎ

2119864

119899

119895(1)

minus 119896119864

119899

119895(4)

120597119901

120597119909

(119909

119895 120591

119899) minus ℎ

2119864

119899

119895(1)

1

119878

119899

119891

119889119878

119891

119889120591

(120591

119899)

minus 119896ℎ

2119864

119899

119895(4) 119864

119899

119895(1)

(50)

where

119864

119899

119895(1) =

1

6

120597

3119901

120597119909

3(119909

119895 120591

119899) 119864

119899

119895(2) =

1

12

120597

4119901

120597119909

4(119909

119895 120591

119899)

119864

119899

119895(3) =

1

2

120597

2119901

120597120591

2(119909

119895 120591

119899) 119864

119899

119895(4) =

119896

2

119878

119891

119889

2119878

119891

119889120591

2(120591

119899)

(51)

6 Abstract and Applied Analysis

Equations (50) and (51) show the local truncation error ofthe numerical scheme (24) with respect to the PDE (4) Notethat from (50) and (51)

119879

119899

119895(119901

119878

119891) = 119874 (ℎ

2) + 119874 (119896) (52)

In order to complete the consistency of the solution of the freeboundary problem with the scheme (21)ndash(26) it is necessaryto rewrite the boundary conditions (6) (7) and (15) in thefollowing form

119871

1(119901 119878

119891) = 119901 (0 120591) minus 1 + 119878

119891(120591) = 0

119871

2(119901 119878

119891) =

120597119901

120597119909

(0 120591) + 119878

119891(120591) = 0

119871

3(119901 119878

119891) =

120590

2

2

120597

2119901

120597119909

2(0 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(53)

Furthermore we have finite difference approximation for thefollowing boundary conditions

119865

1(119901 119878

119891) = 119901

119899

0minus 1 + 119878

119899

119891= 0

119865

2(119901 119878

119891) =

119901

119899

1minus 119901

119899

minus1

2ℎ

+ 119878

119899

119891= 0

119865

3(119901 119878

119891) =

120590

2

2

119901

119899

minus1minus 2119901

119899

0+ 119901

119899

1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(54)

It is not difficult to check using Taylorrsquos expansion that thelocal truncation error satisfies

119879

1(119901

119878

119891) = 119865

1(119901

119878

119891) minus 119871

1(119901

119878

119891) = 0

119879

2(119901

119878

119891) = 119865

2(119901

119878

119891) minus 119871

2(119901

119878

119891) = 119874 (ℎ

2)

119879

3(119901

119878

119891) = 119865

3(119901

119878

119891) minus 119871

3(119901

119878

119891) = 119874 (ℎ

2)

(55)

The truncation error for the boundary condition behavesas ℎ2

Theorem 6 Assuming that the solution of the PDE problem(4)ndash(9) admits two times continuous partial derivative withrespect to time and up to order four with respect to spacethe numerical solution computed by the scheme (21)ndash(26) isconsistent with (4) and boundary conditions of order two inspace and order one in time

5 Numerical Experiments

In this section we illustrate the previous theoretical resultswith numerical experiments showing the potential advan-tages of the proposed method such as preserving the quali-tative properties of the theoretical solution [24] such as posi-tivity andmonotonicity A comparisonwith other approachesis also presented in this section

51 Confirmations of Theoretical Results In this experimentwe check that the stability condition can be broken showing

0 02 04 06 08 1086

088

09

092

094

096

098

1

Time to maturity

Free

bou

ndar

y

(a)

088

092

094

096

098

1

102

0 02 04 06 08 1

09

Time to maturity

Free

bou

ndar

y

(b)

Figure 1 Dependence of stability on positivity of coefficient 119887 (120583 =

24 119887 = 00398) for (a) and (120583 = 252 119887 = minus00083) for (b)

that in such case the results could become unreliable Let usconsider an American put option pricing problem as in [21]with the following parameters

119903 = 01 120590 = 02 119879 = 1 119909max = 2 (56)

We consider fixed space step ℎ = 001 that satisfies conditionC1 of Lemma 1 and change time step (Figure 1) Numericaltests show that the condition C2 is critical for positivity ofcoefficient 119887 and as a result for the stability of the schemeThe monotonicity of the free boundary is numerically shownby numerical tests (Figure 1(a)) In Figure 1(b) one can seethat when condition C2 is not satisfied spurious oscillationsappear so the monotonicity is broken

It was theoretically proved in Section 4 that the schemehas order of approximation 119874(ℎ

2) + 119874(119896) The results of the

Abstract and Applied Analysis 7

Table 1 Convergence in space for fixed time step 119896 = 2 sdot10

minus3 for theproblem with parameters (56)

Asset price True value 004 002 00190 116974 117283 117054 116991100 69320 69308 69309 69312110 41550 41694 41564 41531120 25102 25219 25151 25114

RMSE 18037-2 47857-3 14623-3CPU-time (sec) 0052 0075 0105

0

0001

0002

0003

0004

0005

0 50 100 150 200

RMSE

CPU-time (s)

Figure 2 RMSE against the computational time

numerical experiments for the problem with the followingparameters

119903 = 008 120590 = 02 119879 = 3 119909max = 2 (57)

are presented in Table 1 For ldquoTruerdquo values we use thereference values offered in [30] We consider the differentspace steps for fixed time step 119896 = 0002 to guaranteethe stability The root-mean-square error (RMSE) is used tomeasure the accuracy of the scheme The last row presentsthe CPU-time in seconds for each experimentThe plot of theRMSE against computational time is presented in Figure 2

To check the order of approximation in space we intro-duce the convergence rate

120574 (ℎ

1 ℎ

2) =

ln RMSEℎ1

minus ln RMSEℎ2

ln ℎ

1minus ln ℎ

2

(58)

From Table 1 and (58) one obtains 120574(4 sdot 10

minus2 2 sdot 10

minus2) =

191 that is close to 2To check the order of approximation in time the space

step ℎ is fixed (ℎ = 10

minus2) and time step 119896 is variableFrom Table 2 by analogous to (58) formula one gets 120574(5 sdot

10

minus4 10

minus3) = 080 that is close to 1 It proves the second order

of approximation in space and the first order in time

52 Comparison with Other Approaches The front fixingmethod is used in [21] for American put option pricing prob-lemwith parameters (56) but with another transformation asfollows

119909 =

119878

119878

119891(120591)

119901 (119909 120591) = 119875 (119878 120591) = 119875 (119909119878

119891(119905) 120591)

(59)

Table 2 Convergence in time for fixed space step ℎ = 001 for theproblem with parameters (56)

Asset price True value 00005 0001 000290 116974 116984 116989 116991100 69320 69319 69318 69312110 41550 41555 41544 41531120 25102 25103 25091 25114

RMSE 56347-4 98234-4 14623-3CPU-time (sec) 0142 0113 0105

Table 3 Comparison with front-fixing method under transforma-tion (59)

Method 119878

119891(119879)

Implicit (in [21]) 08615Explicit (in [21]) 08622Explicit (proposed) 08628

By the numerical tests it is shown that the consideredscheme is stable for 120583 le 24 while the condition for a stabilityof the scheme in [21] is 120583 le 6 The results are representedin Table 3 The front fixing method with the logarithmictransformation has a good advantage weaker condition forthe stability It reduces the computational costs

Let us compare front fixing method with anotherapproach [31] based on Mellinrsquos transform The parametersof the problem are 119903 = 00488 120590 = 03 and 119879 = 05833

To compare results of the explicit front fixing methodwith Mellinrsquos transform [31] we have to multiply our dimen-sionless value on119864 = 45Then forMellinrsquos transformmethod119878

119891(119879) = 3277 while for front fixing method 119878

119891(119879) =

327655Close to maturity exercise boundary for the American

put can be analytically approximated [32] as follows

119878

119891(120591) sim 119864[1 minus

radic2120590120591 log( 120590

2

6119903radic120587120591120590

22

)] (60)

This approximation can be used only near expiry date InFigure 3 the value of the free boundary for both approachesis presented Near initial point front fixing method andanalytical approximation (60) give close results

We compare explicit front fixing method (FF) with ℎ

1=

0001 and ℎ

2= 0002 and 120583 = 5 for American put with other

numerical methods shown in [15] in Table 4 for the followingproblem parameters

119903 = 005 120590 = 02 119879 = 3

119864 = 100 119909max = 2

(61)

There are several considered methods as follows

(i) penalty method (PM) is considered in [21 33](ii) Ikonen and Toivanen [34] proposed an operator

splitting technique (OS) for solving the linear com-plementarity problem

8 Abstract and Applied Analysis

Table 4 Comparison with other methods put option value for different asset prices

Asset price True value PM OS HW WK FF(ℎ1) FF(ℎ

2)

80 202797 202793 202795 202803 202825 202795 20279390 133075 133071 133074 133075 133117 133075 133074100 87106 87100 87104 87103 87135 87106 87104110 56825 56820 56824 56823 56867 56825 56823120 36964 36960 36963 36965 37001 36963 36963RMSE 46690-4 89442-6 31623-4 36116-3 10229-4 22966-4CPU-time 2567 1191 472 423 2599 4617

0 002 004 006 008 01 012 014075

08

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(a)

0 02 04 06 08

08

1

065

07

075

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(b)

Figure 3 Comparison with the analytical approximation of Kuskeclose to 120591 = 0 (a) and for [0 119879] (b)

(iii) Han-Wu algorithm (HW) transforms the Black-Scholes equation into a heat equation on an infinitedomain [35]

(iv) the front fixing method proposed by Wu and Kwok(WK) [20]

Apart from the previous comparisons showing that themethod is competitive we remark that the proposed schemehere guarantees the positivity of the solution as well as themonotonicity of both solution and the free boundary

6 Conclusion

The front fixing method for American put option is consid-ered We found that the proposed explicit numerical schemeis stable under appropriate conditions (see Lemma 1) andconsistent with the differential equation with the secondorder in space and first order in time Moreover the mono-tonicity and positivity of the solution and the free boundaryunder that condition are proved in agreement with Kimrsquosresults in [24] about the properties of the theoretical solutionThe theoretical study is confirmed by numerical tests It isshown that the condition (27) is critical that is violation ofthe condition leads to destabilization of the scheme

By the comparison with other approaches advantages ofthe method are discovered The logarithmic transformationrequires weaker stability condition than Landaursquos transfor-mation presented in [21] The explicit calculations withquite weak stability conditions avoid iterations to solve thenonlinear partial differential equation It makes the methodsimpler and reduces computational costs

Conflict of Interests

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

Acknowledgment

This paper has been partially supported by the Euro-pean Union in the FP7-PEOPLE-2012-ITN Program under

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

Volume 2014

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

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Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

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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 Abstract and Applied Analysis

Equations (50) and (51) show the local truncation error ofthe numerical scheme (24) with respect to the PDE (4) Notethat from (50) and (51)

119879

119899

119895(119901

119878

119891) = 119874 (ℎ

2) + 119874 (119896) (52)

In order to complete the consistency of the solution of the freeboundary problem with the scheme (21)ndash(26) it is necessaryto rewrite the boundary conditions (6) (7) and (15) in thefollowing form

119871

1(119901 119878

119891) = 119901 (0 120591) minus 1 + 119878

119891(120591) = 0

119871

2(119901 119878

119891) =

120597119901

120597119909

(0 120591) + 119878

119891(120591) = 0

119871

3(119901 119878

119891) =

120590

2

2

120597

2119901

120597119909

2(0 120591) +

120590

2

2

119878

119891(120591) minus 119903 = 0

(53)

Furthermore we have finite difference approximation for thefollowing boundary conditions

119865

1(119901 119878

119891) = 119901

119899

0minus 1 + 119878

119899

119891= 0

119865

2(119901 119878

119891) =

119901

119899

1minus 119901

119899

minus1

2ℎ

+ 119878

119899

119891= 0

119865

3(119901 119878

119891) =

120590

2

2

119901

119899

minus1minus 2119901

119899

0+ 119901

119899

1

ℎ

2+

120590

2

2

119878

119899

119891minus 119903 = 0

(54)

It is not difficult to check using Taylorrsquos expansion that thelocal truncation error satisfies

119879

1(119901

119878

119891) = 119865

1(119901

119878

119891) minus 119871

1(119901

119878

119891) = 0

119879

2(119901

119878

119891) = 119865

2(119901

119878

119891) minus 119871

2(119901

119878

119891) = 119874 (ℎ

2)

119879

3(119901

119878

119891) = 119865

3(119901

119878

119891) minus 119871

3(119901

119878

119891) = 119874 (ℎ

2)

(55)

The truncation error for the boundary condition behavesas ℎ2

Theorem 6 Assuming that the solution of the PDE problem(4)ndash(9) admits two times continuous partial derivative withrespect to time and up to order four with respect to spacethe numerical solution computed by the scheme (21)ndash(26) isconsistent with (4) and boundary conditions of order two inspace and order one in time

5 Numerical Experiments

In this section we illustrate the previous theoretical resultswith numerical experiments showing the potential advan-tages of the proposed method such as preserving the quali-tative properties of the theoretical solution [24] such as posi-tivity andmonotonicity A comparisonwith other approachesis also presented in this section

51 Confirmations of Theoretical Results In this experimentwe check that the stability condition can be broken showing

0 02 04 06 08 1086

088

09

092

094

096

098

1

Time to maturity

Free

bou

ndar

y

(a)

088

092

094

096

098

1

102

0 02 04 06 08 1

09

Time to maturity

Free

bou

ndar

y

(b)

Figure 1 Dependence of stability on positivity of coefficient 119887 (120583 =

24 119887 = 00398) for (a) and (120583 = 252 119887 = minus00083) for (b)

that in such case the results could become unreliable Let usconsider an American put option pricing problem as in [21]with the following parameters

119903 = 01 120590 = 02 119879 = 1 119909max = 2 (56)

We consider fixed space step ℎ = 001 that satisfies conditionC1 of Lemma 1 and change time step (Figure 1) Numericaltests show that the condition C2 is critical for positivity ofcoefficient 119887 and as a result for the stability of the schemeThe monotonicity of the free boundary is numerically shownby numerical tests (Figure 1(a)) In Figure 1(b) one can seethat when condition C2 is not satisfied spurious oscillationsappear so the monotonicity is broken

It was theoretically proved in Section 4 that the schemehas order of approximation 119874(ℎ

2) + 119874(119896) The results of the

Abstract and Applied Analysis 7

Table 1 Convergence in space for fixed time step 119896 = 2 sdot10

minus3 for theproblem with parameters (56)

Asset price True value 004 002 00190 116974 117283 117054 116991100 69320 69308 69309 69312110 41550 41694 41564 41531120 25102 25219 25151 25114

RMSE 18037-2 47857-3 14623-3CPU-time (sec) 0052 0075 0105

0

0001

0002

0003

0004

0005

0 50 100 150 200

RMSE

CPU-time (s)

Figure 2 RMSE against the computational time

numerical experiments for the problem with the followingparameters

119903 = 008 120590 = 02 119879 = 3 119909max = 2 (57)

are presented in Table 1 For ldquoTruerdquo values we use thereference values offered in [30] We consider the differentspace steps for fixed time step 119896 = 0002 to guaranteethe stability The root-mean-square error (RMSE) is used tomeasure the accuracy of the scheme The last row presentsthe CPU-time in seconds for each experimentThe plot of theRMSE against computational time is presented in Figure 2

To check the order of approximation in space we intro-duce the convergence rate

120574 (ℎ

1 ℎ

2) =

ln RMSEℎ1

minus ln RMSEℎ2

ln ℎ

1minus ln ℎ

2

(58)

From Table 1 and (58) one obtains 120574(4 sdot 10

minus2 2 sdot 10

minus2) =

191 that is close to 2To check the order of approximation in time the space

step ℎ is fixed (ℎ = 10

minus2) and time step 119896 is variableFrom Table 2 by analogous to (58) formula one gets 120574(5 sdot

10

minus4 10

minus3) = 080 that is close to 1 It proves the second order

of approximation in space and the first order in time

52 Comparison with Other Approaches The front fixingmethod is used in [21] for American put option pricing prob-lemwith parameters (56) but with another transformation asfollows

119909 =

119878

119878

119891(120591)

119901 (119909 120591) = 119875 (119878 120591) = 119875 (119909119878

119891(119905) 120591)

(59)

Table 2 Convergence in time for fixed space step ℎ = 001 for theproblem with parameters (56)

Asset price True value 00005 0001 000290 116974 116984 116989 116991100 69320 69319 69318 69312110 41550 41555 41544 41531120 25102 25103 25091 25114

RMSE 56347-4 98234-4 14623-3CPU-time (sec) 0142 0113 0105

Table 3 Comparison with front-fixing method under transforma-tion (59)

Method 119878

119891(119879)

Implicit (in [21]) 08615Explicit (in [21]) 08622Explicit (proposed) 08628

By the numerical tests it is shown that the consideredscheme is stable for 120583 le 24 while the condition for a stabilityof the scheme in [21] is 120583 le 6 The results are representedin Table 3 The front fixing method with the logarithmictransformation has a good advantage weaker condition forthe stability It reduces the computational costs

Let us compare front fixing method with anotherapproach [31] based on Mellinrsquos transform The parametersof the problem are 119903 = 00488 120590 = 03 and 119879 = 05833

To compare results of the explicit front fixing methodwith Mellinrsquos transform [31] we have to multiply our dimen-sionless value on119864 = 45Then forMellinrsquos transformmethod119878

119891(119879) = 3277 while for front fixing method 119878

119891(119879) =

327655Close to maturity exercise boundary for the American

put can be analytically approximated [32] as follows

119878

119891(120591) sim 119864[1 minus

radic2120590120591 log( 120590

2

6119903radic120587120591120590

22

)] (60)

This approximation can be used only near expiry date InFigure 3 the value of the free boundary for both approachesis presented Near initial point front fixing method andanalytical approximation (60) give close results

We compare explicit front fixing method (FF) with ℎ

1=

0001 and ℎ

2= 0002 and 120583 = 5 for American put with other

numerical methods shown in [15] in Table 4 for the followingproblem parameters

119903 = 005 120590 = 02 119879 = 3

119864 = 100 119909max = 2

(61)

There are several considered methods as follows

(i) penalty method (PM) is considered in [21 33](ii) Ikonen and Toivanen [34] proposed an operator

splitting technique (OS) for solving the linear com-plementarity problem

8 Abstract and Applied Analysis

Table 4 Comparison with other methods put option value for different asset prices

Asset price True value PM OS HW WK FF(ℎ1) FF(ℎ

2)

80 202797 202793 202795 202803 202825 202795 20279390 133075 133071 133074 133075 133117 133075 133074100 87106 87100 87104 87103 87135 87106 87104110 56825 56820 56824 56823 56867 56825 56823120 36964 36960 36963 36965 37001 36963 36963RMSE 46690-4 89442-6 31623-4 36116-3 10229-4 22966-4CPU-time 2567 1191 472 423 2599 4617

0 002 004 006 008 01 012 014075

08

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(a)

0 02 04 06 08

08

1

065

07

075

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(b)

Figure 3 Comparison with the analytical approximation of Kuskeclose to 120591 = 0 (a) and for [0 119879] (b)

(iii) Han-Wu algorithm (HW) transforms the Black-Scholes equation into a heat equation on an infinitedomain [35]

(iv) the front fixing method proposed by Wu and Kwok(WK) [20]

Apart from the previous comparisons showing that themethod is competitive we remark that the proposed schemehere guarantees the positivity of the solution as well as themonotonicity of both solution and the free boundary

6 Conclusion

The front fixing method for American put option is consid-ered We found that the proposed explicit numerical schemeis stable under appropriate conditions (see Lemma 1) andconsistent with the differential equation with the secondorder in space and first order in time Moreover the mono-tonicity and positivity of the solution and the free boundaryunder that condition are proved in agreement with Kimrsquosresults in [24] about the properties of the theoretical solutionThe theoretical study is confirmed by numerical tests It isshown that the condition (27) is critical that is violation ofthe condition leads to destabilization of the scheme

By the comparison with other approaches advantages ofthe method are discovered The logarithmic transformationrequires weaker stability condition than Landaursquos transfor-mation presented in [21] The explicit calculations withquite weak stability conditions avoid iterations to solve thenonlinear partial differential equation It makes the methodsimpler and reduces computational costs

Conflict of Interests

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

Acknowledgment

This paper has been partially supported by the Euro-pean Union in the FP7-PEOPLE-2012-ITN Program under

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

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

Abstract and Applied Analysis 7

Table 1 Convergence in space for fixed time step 119896 = 2 sdot10

minus3 for theproblem with parameters (56)

Asset price True value 004 002 00190 116974 117283 117054 116991100 69320 69308 69309 69312110 41550 41694 41564 41531120 25102 25219 25151 25114

RMSE 18037-2 47857-3 14623-3CPU-time (sec) 0052 0075 0105

0

0001

0002

0003

0004

0005

0 50 100 150 200

RMSE

CPU-time (s)

Figure 2 RMSE against the computational time

numerical experiments for the problem with the followingparameters

119903 = 008 120590 = 02 119879 = 3 119909max = 2 (57)

are presented in Table 1 For ldquoTruerdquo values we use thereference values offered in [30] We consider the differentspace steps for fixed time step 119896 = 0002 to guaranteethe stability The root-mean-square error (RMSE) is used tomeasure the accuracy of the scheme The last row presentsthe CPU-time in seconds for each experimentThe plot of theRMSE against computational time is presented in Figure 2

To check the order of approximation in space we intro-duce the convergence rate

120574 (ℎ

1 ℎ

2) =

ln RMSEℎ1

minus ln RMSEℎ2

ln ℎ

1minus ln ℎ

2

(58)

From Table 1 and (58) one obtains 120574(4 sdot 10

minus2 2 sdot 10

minus2) =

191 that is close to 2To check the order of approximation in time the space

step ℎ is fixed (ℎ = 10

minus2) and time step 119896 is variableFrom Table 2 by analogous to (58) formula one gets 120574(5 sdot

10

minus4 10

minus3) = 080 that is close to 1 It proves the second order

of approximation in space and the first order in time

52 Comparison with Other Approaches The front fixingmethod is used in [21] for American put option pricing prob-lemwith parameters (56) but with another transformation asfollows

119909 =

119878

119878

119891(120591)

119901 (119909 120591) = 119875 (119878 120591) = 119875 (119909119878

119891(119905) 120591)

(59)

Table 2 Convergence in time for fixed space step ℎ = 001 for theproblem with parameters (56)

Asset price True value 00005 0001 000290 116974 116984 116989 116991100 69320 69319 69318 69312110 41550 41555 41544 41531120 25102 25103 25091 25114

RMSE 56347-4 98234-4 14623-3CPU-time (sec) 0142 0113 0105

Table 3 Comparison with front-fixing method under transforma-tion (59)

Method 119878

119891(119879)

Implicit (in [21]) 08615Explicit (in [21]) 08622Explicit (proposed) 08628

By the numerical tests it is shown that the consideredscheme is stable for 120583 le 24 while the condition for a stabilityof the scheme in [21] is 120583 le 6 The results are representedin Table 3 The front fixing method with the logarithmictransformation has a good advantage weaker condition forthe stability It reduces the computational costs

Let us compare front fixing method with anotherapproach [31] based on Mellinrsquos transform The parametersof the problem are 119903 = 00488 120590 = 03 and 119879 = 05833

To compare results of the explicit front fixing methodwith Mellinrsquos transform [31] we have to multiply our dimen-sionless value on119864 = 45Then forMellinrsquos transformmethod119878

119891(119879) = 3277 while for front fixing method 119878

119891(119879) =

327655Close to maturity exercise boundary for the American

put can be analytically approximated [32] as follows

119878

119891(120591) sim 119864[1 minus

radic2120590120591 log( 120590

2

6119903radic120587120591120590

22

)] (60)

This approximation can be used only near expiry date InFigure 3 the value of the free boundary for both approachesis presented Near initial point front fixing method andanalytical approximation (60) give close results

We compare explicit front fixing method (FF) with ℎ

1=

0001 and ℎ

2= 0002 and 120583 = 5 for American put with other

numerical methods shown in [15] in Table 4 for the followingproblem parameters

119903 = 005 120590 = 02 119879 = 3

119864 = 100 119909max = 2

(61)

There are several considered methods as follows

(i) penalty method (PM) is considered in [21 33](ii) Ikonen and Toivanen [34] proposed an operator

splitting technique (OS) for solving the linear com-plementarity problem

8 Abstract and Applied Analysis

Table 4 Comparison with other methods put option value for different asset prices

Asset price True value PM OS HW WK FF(ℎ1) FF(ℎ

2)

80 202797 202793 202795 202803 202825 202795 20279390 133075 133071 133074 133075 133117 133075 133074100 87106 87100 87104 87103 87135 87106 87104110 56825 56820 56824 56823 56867 56825 56823120 36964 36960 36963 36965 37001 36963 36963RMSE 46690-4 89442-6 31623-4 36116-3 10229-4 22966-4CPU-time 2567 1191 472 423 2599 4617

0 002 004 006 008 01 012 014075

08

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(a)

0 02 04 06 08

08

1

065

07

075

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(b)

Figure 3 Comparison with the analytical approximation of Kuskeclose to 120591 = 0 (a) and for [0 119879] (b)

(iii) Han-Wu algorithm (HW) transforms the Black-Scholes equation into a heat equation on an infinitedomain [35]

(iv) the front fixing method proposed by Wu and Kwok(WK) [20]

Apart from the previous comparisons showing that themethod is competitive we remark that the proposed schemehere guarantees the positivity of the solution as well as themonotonicity of both solution and the free boundary

6 Conclusion

The front fixing method for American put option is consid-ered We found that the proposed explicit numerical schemeis stable under appropriate conditions (see Lemma 1) andconsistent with the differential equation with the secondorder in space and first order in time Moreover the mono-tonicity and positivity of the solution and the free boundaryunder that condition are proved in agreement with Kimrsquosresults in [24] about the properties of the theoretical solutionThe theoretical study is confirmed by numerical tests It isshown that the condition (27) is critical that is violation ofthe condition leads to destabilization of the scheme

By the comparison with other approaches advantages ofthe method are discovered The logarithmic transformationrequires weaker stability condition than Landaursquos transfor-mation presented in [21] The explicit calculations withquite weak stability conditions avoid iterations to solve thenonlinear partial differential equation It makes the methodsimpler and reduces computational costs

Conflict of Interests

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

Acknowledgment

This paper has been partially supported by the Euro-pean Union in the FP7-PEOPLE-2012-ITN Program under

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

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 Abstract and Applied Analysis

Table 4 Comparison with other methods put option value for different asset prices

Asset price True value PM OS HW WK FF(ℎ1) FF(ℎ

2)

80 202797 202793 202795 202803 202825 202795 20279390 133075 133071 133074 133075 133117 133075 133074100 87106 87100 87104 87103 87135 87106 87104110 56825 56820 56824 56823 56867 56825 56823120 36964 36960 36963 36965 37001 36963 36963RMSE 46690-4 89442-6 31623-4 36116-3 10229-4 22966-4CPU-time 2567 1191 472 423 2599 4617

0 002 004 006 008 01 012 014075

08

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(a)

0 02 04 06 08

08

1

065

07

075

085

09

095

1

Time to maturity

Free

bou

ndar

y

Front fixing methodAnalytical

(b)

Figure 3 Comparison with the analytical approximation of Kuskeclose to 120591 = 0 (a) and for [0 119879] (b)

(iii) Han-Wu algorithm (HW) transforms the Black-Scholes equation into a heat equation on an infinitedomain [35]

(iv) the front fixing method proposed by Wu and Kwok(WK) [20]

Apart from the previous comparisons showing that themethod is competitive we remark that the proposed schemehere guarantees the positivity of the solution as well as themonotonicity of both solution and the free boundary

6 Conclusion

The front fixing method for American put option is consid-ered We found that the proposed explicit numerical schemeis stable under appropriate conditions (see Lemma 1) andconsistent with the differential equation with the secondorder in space and first order in time Moreover the mono-tonicity and positivity of the solution and the free boundaryunder that condition are proved in agreement with Kimrsquosresults in [24] about the properties of the theoretical solutionThe theoretical study is confirmed by numerical tests It isshown that the condition (27) is critical that is violation ofthe condition leads to destabilization of the scheme

By the comparison with other approaches advantages ofthe method are discovered The logarithmic transformationrequires weaker stability condition than Landaursquos transfor-mation presented in [21] The explicit calculations withquite weak stability conditions avoid iterations to solve thenonlinear partial differential equation It makes the methodsimpler and reduces computational costs

Conflict of Interests

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

Acknowledgment

This paper has been partially supported by the Euro-pean Union in the FP7-PEOPLE-2012-ITN Program under

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

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

Abstract and Applied Analysis 9

Grant Agreement no 304617 (FP7 Marie Curie ActionProject Multi-ITN STRIKE-Novel Methods in Computa-tional Finance)

References

[1] P Wilmott S Howison and J Dewynne The Mathematics ofFinancial Derivatives Cambridge University Press CambridgeUK 1995

[2] L Feng V Linetsky J L Morales and J Nocedal ldquoOn thesolution of complementarity problems arising in Americanoptions pricingrdquo Optimization Methods amp Software vol 26 no4-5 pp 813ndash825 2011

[3] H P McKean ldquoAppendix a free boundary problem for the heatequation arising from a problem in mathematical economicsrdquoIndustrial Management Review vol 6 pp 32ndash39 1965

[4] P van Moerbeke ldquoOn optimal stopping and free boundaryproblemsrdquo Archive for Rational Mechanics and Analysis vol 60no 2 pp 101ndash148 1976

[5] R Geske and H Johnson ldquoThe American put option valuedanalyticallyrdquo Journal of Finance vol 39 no 5 pp 1511ndash15241984

[6] G Barone-Adesi and R Whaley ldquoEfficient analytic approxima-tion of American option valuesrdquo Journal of Finance vol 42 no2 pp 301ndash320 1987

[7] L W MacMillan ldquoAn analytical approximation for the Ameri-can put pricesrdquo Advances in Futures and Options Research vol1 pp 119ndash139 1986

[8] N Ju ldquoPricing an American option by approximating its earlyexercise boundary as amultipiece exponential functionrdquoReviewof Financial Studies vol 11 no 3 pp 627ndash646 1998

[9] M Brennan and E Schwartz ldquoFinite difference methods andjump processes arising in the pricing of contingent claims asynthesisrdquo Journal of Financial and Quantitative Analysis vol13 no 3461 474 pages 1978

[10] P Jaillet D Lamberton and B Lapeyre ldquoVariational inequal-ities and the pricing of American optionsrdquo Acta ApplicandaeMathematicae vol 21 no 3 pp 263ndash289 1990

[11] J Hull and A White ldquoValuing derivative securities using theexplicit finite difference methodrdquo Journal of Financial andQuantitative Analysis vol 25 no 1 pp 87ndash100 1990

[12] D J Duffy Finite Difference Methods in Financial EngineeringA Partial Differential Equation Approach John Wiley amp Sons2006

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

[14] D Tavella and C Randall Pricing Financial Instruments TheFinite Difference Method John Wiley and Sons New York NYUSA 2000

[15] D Y Tangman A Gopaul and M Bhuruth ldquoA fast high-order finite difference algorithm for pricing American optionsrdquoJournal of Computational andAppliedMathematics vol 222 no1 pp 17ndash29 2008

[16] S-P Zhu andW-T Chen ldquoA predictor-corrector scheme basedon the ADI method for pricing American puts with stochasticvolatilityrdquo Computers amp Mathematics with Applications vol 62no 1 pp 1ndash26 2011

[17] B J Kim Y-K Ma and H J Choe ldquoA simple numericalmethod for pricing an American put optionrdquo Journal of AppliedMathematics vol 2013 Article ID 128025 7 pages 2013

[18] H G Landau ldquoHeat conduction in a melting solidrdquo Quarterlyof Applied Mathematics vol 8 pp 81ndash95 1950

[19] J Crank Free and Moving Boundary Problems Oxford SciencePublications 1984

[20] LWu and Y-K Kwok ldquoA front-fixing method for the valuationof American optionrdquo The Journal of Financial Engineering vol6 no 2 pp 83ndash97 1997

[21] B F Nielsen O Skavhaug and A Tvelto ldquoPenalty and front-fixing methods for the numerical solution of American optionproblemsrdquo Journal of Computational Finance vol 5 2002

[22] D Sevcovic ldquoAn iterative algorithm for evaluating approxima-tions to the optimal exercise boundary for a nonlinear Black-Scholes equationrdquo Canadian Applied Mathematics Quarterlyvol 15 no 1 pp 77ndash97 2007

[23] J Zhang and S P Zhu ldquoA Hybrid Finite Difference Method forValuing American Putsrdquo in Proceedings of theWorld Congress ofEngineering vol 2 London UK July 2009

[24] I J Kim ldquoThe analytic valuation of American optionsrdquo TheReview of Financial Studies vol 3 no 4 pp 547ndash572 1990

[25] R Kangro and R Nicolaides ldquoFar field boundary conditions forBlack-Scholes equationsrdquo SIAM Journal on Numerical Analysisvol 38 no 4 pp 1357ndash1368 2000

[26] Y-K Kwok Mathematical Models of Financial DerivativesSpringer Berlin 2nd edition 2008

[27] P A Samuelson Rational Theory of Warrant Pricing vol 6Industrial Management Review 1979

[28] R Company L Jodar and J-R Pintos ldquoA consistent stablenumerical scheme for a nonlinear option pricing model inilliquid marketsrdquo Mathematics and Computers in Simulationvol 82 no 10 pp 1972ndash1985 2012

[29] G D Smith Numerical Solution of Partial Differential Equa-tions Finite Difference Methods The Clarendon Press OxfordUK 3d edition 1985

[30] F A A E Saib Y D Tangman N Thakoor and M BhuruthldquoOn some finite difference algorithms for pricing Americanoptions and their implementation in mathematicardquo in Proceed-ings of the 11th International Conference on Computational andMathematicalMethods in Science and Engineering (CMMSE rsquo11)June 2011

[31] R Panini and R P Srivastav ldquoOption pricing with MellintransformsrdquoMathematical and ComputerModelling vol 40 no1-2 pp 43ndash56 2004

[32] R A Kuske and J B Keller ldquoOptimal exercise boundary for anAmerican putrdquo Applied Mathematical Finance vol 5 pp 107ndash116 1998

[33] J Toivanen Finite DifferenceMethods for Early Exercise OptionsEncyclopedia of Quantitative Finance 2010

[34] S Ikonen and J Toivanen ldquoOperator splitting methods forAmerican option pricingrdquo Applied Mathematics Letters vol 17no 7 pp 809ndash814 2004

[35] H Han and X Wu ldquoA fast numerical method for the Black-Scholes equation of American optionsrdquo SIAM Journal onNumerical Analysis vol 41 no 6 pp 2081ndash2095 2003

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