Applied Mathematics and Computation 213 (2009) 121–136

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Applied Mathematics and Computation

journal homepage: www.elsevier .com/ locate/amc

Efficient L-stable method for parabolic problems with application to pricingAmerican options under stochastic volatility

M. YousufDepartment of Mathematics and Statistics, King Fahd University of Petroleum and Minerals, Dhahran 31261, Saudi Arabia

a r t i c l e i n f o a b s t r a c t

Keywords:L-stablePadé approximationsParabolic problemAmerican optionsHeston’s stochastic volatility model

0096-3003/$ - see front matter � 2009 Elsevier Incdoi:10.1016/j.amc.2009.02.060

E-mail address: myousuf@kfupm.edu.sa

Efficient L-stable numerical method for semilinear parabolic problems with nonsmoothinitial data is proposed and implemented to solve Heston’s stochastic volatility modelbased PDE for pricing American options under stochastic volatility. The proposed newmethod is also used to solve two asset American options pricing problem. Cox andMatthews [S.M. Cox, P.C. Matthews, Exponential time differencing for stiff systems, Journalof Computational Physics 176 (2002) 430–455] developed a class of exponential timedifferencing Runge–Kutta schemes (ETDRK) for nonlinear parabolic problems. Kassamand Trefethen [A.K. Kassam, L.N. Trefethen, Fourth-order time stepping for stiff PDEs, SIAMJournal on Scientific Computing 26 (4) (2005) 1214–1233] showed that while computingcertain functions involved in the Cox–Matthews schemes, severe cancelation errors canoccur which affect the accuracy and stability of the schemes. Kassam and Trefethenproposed complex contour integration technique to implement these schemes in a waythat avoids these cancelation errors. But this approach creates new difficulties in choosingand evaluating the contour integrals for larger problems. We modify the ETDRK schemesusing positivity preserving Padé approximations of the matrix exponential functions andconstruct computationally efficient parallel version using splitting technique. As a resultof this approach it is required only to solve several backward Euler linear problems in serialor parallel.

� 2009 Elsevier Inc. All rights reserved.

1. Introduction

Problems having irregular initial data or mismatched initial and boundary conditions occur in various applications,including mechanical engineering, computational chemistry, and financial engineering. Nonsmooth payoffs cause disconti-nuities in the solution (or its derivatives) and standard A-stable methods (e.g., Crank–Nicolson) are prone to produce largeand spurious oscillations in the numerical solutions which would mislead to estimating options accurately if one does nottreat the problem carefully.

Cox and Matthews [2] developed a class of exponential time differencing schemes (ETD) for nonlinear stiff systems ofODEs and extended the results to solve nonlinear parabolic problems. This approach reduces the spatially discretized PDEusing Duhamel’s principle on one time step to an integral equation followed by approximation of the integral involvingnonlinear function. The nonlinear function is approximated by a polynomial. In the analysis, Cox and Matthews treatedscalar examples (ODEs) and systems of two PDEs with special form. Kassam and Trefethen [9] addressed the limitedgenerality of the Cox–Matthews schemes and showed that these schemes can suffer from severe cancelation errors whencomputing certain functions involved in the schemes. A new strategy based on contour integral was introduced to improve

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122 M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136

the general applicability of these schemes. Use of complex contour integration to implement these schemes avoids invertingmatrix polynomials, but this approach creates new difficulties in choosing and evaluating the contour integrals for largerproblems.

Cox–Matthews schemes as well as Kassam–Trefethen modifications require calculating matrix exponentials. Even if theoriginal matrix is sparse, the matrix exponential will not itself be sparse, which can be a significant amount of work and af-fect the computational efficiency of the scheme. For one-dimensional problems the calculation is not very expensive, how-ever, the cost goes up as the dimension increases. Also, neither Cox–Matthews development nor Kassam–Trefethenmodification addresses problems with nonsmooth data.

In the Kassam–Trefethen approach, contour integrals are evaluated by means of some numerical technique and must con-tain spectrum of the discretization matrix A. Generally the eigenvalues of A lie in or near the left half of the complex planeand they may cover a wide range which grows with spatial mesh size N. The spectrum of the discretization matrix A is noteasily known and it is typically unbounded as the spatial step goes to zero. This is a primary limitation of the Kassam–Trefe-then modification because the contour varies from problem to problem, with dependence on the spatial mesh. This limita-tion makes the technique problem dependent. For example, eigenvalues for diffusive problems are close to the negative realaxis and for dispersive problems they are close to the imaginary axis.

Another recent development in this area is implementation of the second order exponential time differencing scheme tothe magnetohydrodynamic equations in a spherical shell. A variety of different methods including direct computation, con-tour integration, spectral expansions and recurrence relations are discussed and implemented, see [16].

Main contribution in this article is to develop an alternate solution to these computational difficulties. A second orderETDRK scheme requires inverting a second degree matrix polynomial where as third and fourth order schemes requireinverting cubic matrix polynomials, which can cause serious numerical instability and computational difficulties becauseof the ill-conditioning (see [5, Section 6.2]). We modify Cox–Matthews schemes to the general nondiagonal problems usingPadé approximations of the matrix exponential functions and use splitting technique to construct parallel versions of theschemes. This approach transforms the matrix polynomial inversion problem into a sum of well-conditioned linear problemsthat can be solved in parallel. Our formulation of the modified schemes is generally more accurate for problems with irreg-ular data and computationally more efficient as compared to the aforementioned ETDRK schemes.

A fourth order L-stable method is constructed using positivity preserving sub-diagonal (0,4)-Padé approximation. Toshow the advantage of L-stable method, we have constructed an A-stable method using diagonal (2,2)-Padé approximation.An algorithm based on the modified schemes is developed and implemented to solve two important problems from financialmathematics. Heston’s stochastic volatility model with a small nonlinear penalty term is used for pricing American put op-tions under stochastic volatility. Penalty method was first introduced by Zvan et al. [24] for American options under stochas-tic volatility. Forsyth and Vetzal [3] proposed an implicit finite difference scheme for valuing American options using thepenalty method. Nielsen et al. [17] presented a refinement of Zvan’s work and illustrated the performance of various numer-ical schemes using explicit, semi-implicit, and fully implicit methods. Khaliq et al. [12] used linearly implicit predictor–cor-rector schemes for pricing American options.

Organization of this paper is as follows. In Section 2 we consider an abstract PDE and write its solution using Duhamel’sprinciple. Basic time stepping schemes are given in Section 3 and Padé approximations as well as modified schemes are men-tioned in Section 4. Parallel implementation of these schemes with a parallel algorithm is given in Section 5. Models for pricingAmerican options and penalty method approach is described in Section 6. Section 7 contains an efficient spatial discretizationapproach. Numerical results are discussed in Section 8. Finally we provide some concluding remarks in Section 9.

2. The abstract PDE

We consider the following semilinear initial-boundary value problem:

ut þ Au ¼ Fðu; tÞ in X; t 2 0; t� �

¼ J;

u ¼ v on oX; t 2 J; uð�;0Þ ¼ u0 in X;ð2:1Þ

where X is a bounded domain in Rd with Lipschitz boundary, F is a smooth nonlinear function defined on Rd and A denotes auniformly elliptic operator,

Au :¼ �Xd

j;k¼1

o

oxjaj;k

ouoxk

� �þXd

j¼1

bjouoxjþ b0u: ð2:2Þ

The coefficients aj;k and bj are C1 (or sufficiently smooth) functions on X; aj;k ¼ ak;j; b0 P 0, and for some c0 > 0

Xd

j;k¼1

aj;knjnk P c0jnj2 on X; for all n 2 Rd: ð2:3Þ

The initial-value problem (2.1) is considered in a Hilbert space H, with A being a linear, self-adjoint, positive definite closedoperator with a compact inverse T, defined on a dense domain DðAÞ �H, see [21] for more details. The operator A couldrepresent any of fAhg0<h6h0

, obtained from a spatial discretization and H could be an appropriate finite-dimensionalsubspace of L2ðXÞ, cf. [19,21].

M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136 123

We assume the resolvent set qðAÞ satisfies, for some a 2 ð0;p=2Þ,

qðAÞ � Ra; Ra :¼ fz 2 C : a < j argðzÞj 6 p; z–0g: ð2:4Þ

We also assume there exists M P 1 such that

kðzI � AÞ�1k 6 Mjzj�1; z 2 Ra: ð2:5Þ

It follows that �A is the infinitesimal generator of an analytic semigroup fe�tAgtP0 which is the solution operator for (2.1).There is a standard representation,

e�tA ¼ 12pi

ZK

e�tzðzI � AÞ�1dz; ð2:6Þ

where K :¼ fz 2 C : j argðzÞj ¼ hg, oriented so that ImðzÞ decreases, for any h 2 a; p2� �

.By Duhamel’s principle the exact solution of (2.1) can be written as,

uðtÞ ¼ EðtÞv þZ t

0Eðt � sÞFðuðsÞ; sÞds; ð2:7Þ

where EðtÞ ¼ e�tA is the solution operator of the corresponding homogeneous problem with F � 0.Let 0 < k 6 k0, for some k0, be the fix time step and tn ¼ nk. Replacing t by t þ k in (2.7), we obtain,

uðt þ kÞ ¼ Eðt þ kÞv þZ tþk

0Eðt þ k� sÞFðuðsÞ; sÞds;

¼ EðkÞEðtÞv þ EðkÞZ t

0Eðt � sÞFðuðsÞ; sÞdsþ

Z tþk

tEðt þ k� sÞFðuðsÞ; sÞds

and using the change of variable s� t ¼ ks we can write,

uðt þ kÞ ¼ EðkÞuðtÞ þ kZ 1

0Eðk� ksÞFðuðt þ ksÞ; t þ ksÞds; ð2:8Þ

which satisfies the recurrence formula,

uðtnþ1Þ ¼ EðkÞuðtnÞ þ kZ 1

0Eðk� ksÞFðuðtn þ skÞ; tn þ skÞds: ð2:9Þ

3. Cox–Matthews time stepping scheme

Recurrence formula (2.9) is the basis of different time stepping schemes depending upon how we approximate the matrixexponential functions and integral term. For a nonlinear stiff system, Cox and Matthews [2] developed time steppingschemes by first presenting a sequence of formulae that gives a multi-step type higher order approximations. This approachhas a benefit of generating a family of high order numerical schemes with potentially good performance, except a few com-putational difficulties, also pointed out by Kassam and Trefethen [9] and Khaliq et al. [10].

For the case when A is a scalar or a diagonal matrix, several schemes based on Runge–Kutta time stepping were developedby Cox and Matthews [2] but we shall consider only the following fourth order scheme (ETDRK):

unþ1 ¼ e�kAun þ1

k2 ð�AÞ�3 Fðun; tnÞ �4þ kAþ e�kAð4þ 3kAþ k2A2Þh i

þ 2 Fðan; tn þ k=2Þ þ Fðbn; tn þ k=2Þð Þ�

2� kAþ e�kAð�2� kAÞ� �

þ Fðcn; tn þ kÞ �4þ 3kA� k2A2 þ e�kAð4þ kAÞh i

; ð3:1Þ

where

an ¼ e�kA=2un � A�1ðe�kA=2 � IÞFðun; tnÞ; ð3:2Þbn ¼ e�kA=2un � A�1ðe�kA=2 � IÞFðan; tn þ k=2Þ; ð3:3Þcn ¼ e�kA=2an � A�1ðe�kA=2 � IÞ 2Fðbn; tn þ k=2Þ � Fðun; tnÞð Þ: ð3:4Þ

This scheme can easily be implemented when A is a scalar or a diagonal matrix with no zero entry on the diagonal. But for anondiagonal matrix A, there can be computational difficulties when A has eigenvalues close to zero since it is required tocompute ð�AÞ�1 and ð�AÞ�3. Also these schemes require calculation of matrix exponential functions e�kA and e�kA=2 whichis nontrivial when A is nondiagonal.

4. Padé approximations and modified time stepping schemes

In this section we shall present a modification of the fourth order Cox–Matthews scheme using Padé approximations ofthe matrix exponential functions. An advantage, we find, of using Padé approximations is that the factors A�1 and A�3 cancelout.

124 M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136

4.1. Padé approximations

Use of Padé approximations of the matrix exponential functions is an important and widely used tool (see [14,20,21]).Padé approximations are derived by expanding a function as a ratio of two power series and determining coefficients of boththe numerator and the denominator. These are generalizations of the power series approximations and are usually superiorto Taylor expansions when functions contain poles, because the use of rational functions allows them to be well represented.Let Pn;mðzÞ and Qn;mðzÞ be two polynomials of degree n and m, respectively. Then the rational function Rn;mðzÞ ¼ Pn;mðzÞ

Qn;mðzÞis an

approximation of order nþm to a function f ðzÞ if,

f ðzÞ ¼ Pn;mðzÞQn;mðzÞ

þ Oðznþmþ1Þ:

The ðnþmÞth order rational Padé approximation, denoted by ðn;mÞ-Padé, of the exponential function e�z can be written asRn;mðzÞ ¼ Pn;mðzÞ

Q n;mðzÞ, where

Pn;mðzÞ ¼Xn

j¼0

ðmþ n� jÞ!n!

ðmþ nÞ!j!ðn� jÞ! ð�zÞj; and Qn;mðzÞ ¼Xm

j¼0

ðmþ n� jÞ!m!

ðmþ nÞ!j!ðm� jÞ! ðzÞj:

Definition. A rational approximation RðzÞ of e�z is said to be A-acceptable if jRðzÞj < 1 whenever RðzÞ < 0 and L-acceptable if,in addition, jRðzÞj ! 0 as RðzÞ ! �1.

It is a well known property that Rn;mðzÞ ¼ e�z þ Oðjzjmþnþ1Þ as z! 0 [20]. The rational Padé approximation Rn;mðzÞ to e�z is:(see [15])

(i) A-acceptable if n ¼ m,(ii) L-acceptable if n ¼ m� 1 or n ¼ m� 2.

Because of the practical purpose, the following examples of Padé approximations of e�z are of particular interest:

R2;2ðzÞ ¼12� 6zþ z2

12þ 6zþ z2 ¼ 1þ �12z12þ 6zþ z2 ;

R0;4ðzÞ ¼24

24þ 24zþ 12z2 þ 4z3 þ z4 :

Most of the spatial discretizations of two asset problems do not lead to M-matrices unless the cross derivative term is zeroand, therefore, can cause oscillations in the numerical solution. These oscillations can be extinguished if the schemes arenonoscillatory, that is, RðRðzÞÞ > 0 for RðzÞ > 0 where RðzÞ ¼ e�z is the amplification symbol of the one step method. Withz ¼ xþ iy and w ¼ RðzÞ ¼ uþ iv , Fig. 1 displays u ¼ RðRðzÞÞ for the (0,4)-Padé and (2,2)-Padé approximations. While notentirely nonoscillatory, the (0,4)-Padé approximation, possessing L-acceptable amplification symbol, rapidly damps oscilla-tions, however, they persist for the A-stable methods corresponding to the A-acceptable (2,2)-Padé approximation, see [13].It should also be noted that the graph of (0,4)-Padé approximation converges to zero rapidly whereas it converges to one forthe (2,2)-Padé approximation.

4.2. Modified time stepping schemes

4.2.1. L-stable methodWe use fourth order (0,4)-Padé approximation of e�z to construct an L-stable method based on the fourth order ETDRK

scheme.

unþ1 ¼ R0;4ðkAÞun þ P1ðkAÞFðun; tnÞ þ P2ðkAÞ Fðan; tn þ k=2Þ þ Fðbn; tn þ k=2Þð Þ þ P3ðkAÞFðcn; tn þ kÞ;

where

R0;4ðkAÞ ¼ 24ð24I þ 24kAþ 12k2A2 þ 4k3A3 þ k4A4Þ�1;

P1ðkAÞ ¼ kð4I � k2A2Þð24I þ 24kAþ 12k2A2 þ 4k3A3 þ k4A4Þ�1;

P2ðkAÞ ¼ kð4I þ 2kAþ k2A2Þð24I þ 24kAþ 12k2A2 þ 4k3A3 þ k4A4Þ�1;

P3ðkAÞ ¼ kð4I þ 4kAþ k2A2 þ k3A3Þð24I þ 24kAþ 12k2A2 þ 4k3A3 þ k4A4Þ�1

and

an ¼ eR0;4ðkAÞun þ ePðkAÞFðun; tnÞ;bn ¼ eR0;4ðkAÞun þ ePðkAÞFðan; tn þ k=2Þ;cn ¼ eR0;4ðkAÞan þ ePðkAÞ 2Fðbn; tn þ k=2Þ � Fðun; tnÞð Þ

0

10

20

30

40 −10

−5

0

5

10

−1

0

1

y

(0,4) Pade

x

u

0

10

20

30

40 −10

−5

0

5

10

−1

0

1

y

(2,2) Pade

x

u

Fig. 1. Amplification symbols of (0,4)-Padé (top) and (2,2)-Padé (bottom).

M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136 125

with

eR0;4ðkAÞ ¼ 384ð384I þ 192kAþ 48k2A2 þ 8k3A3 þ k4A4Þ�1;ePðkAÞ ¼ kð192I þ 48kAþ 8k2A2 þ k3A3Þð384I þ 192kAþ 48k2A2 þ 8k3A3 þ k4A4Þ�1

:

The rational functions R0;4ðkAÞ and eR0;4ðkAÞ are (0,4)-Padé approximations of e�kA and e�kA=2 respectively. The rationalfunctions P1ðkAÞ; P2ðkAÞ, and P3ðkAÞ are obtained from the second, third and fourth term of (3.1) respectively and ePðkAÞ isobtained from the second term of (3.2). An important observation we find that R0;4ðkAÞ and corresponding PiðkAÞ sharethe same denominator and eR0;4ðkAÞ and corresponding ePðkAÞ do the same. Similar properties are also possessed by thefollowing A-stable method.

4.2.2. A-stable methodUsing (2,2)-Padé approximation of e�z, we have constructed an A-stable method.

unþ1 ¼ R2;2ðkAÞun þ P1ðkAÞFðun; tnÞ þ P2ðkAÞ Fðan; tn þ k=2Þ þ Fðbn; tn þ k=2Þð Þ þ P3ðkAÞFðcn; tn þ kÞ;

126 M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136

where

R2;2ðkAÞ ¼ ð12I � 6kAþ k2A2Þð12I þ 6kAþ k2A2Þ�1;

P1ðkAÞ ¼ kð2I þ kAÞð12I þ 6kAþ k2A2Þ�1;

P2ðkAÞ ¼ 2kð12I þ 6kAþ k2A2Þ�1;

P3ðkAÞ ¼ kð2I � kAÞð12I þ 6kAþ k2A2Þ�1

and

an ¼ eR2;2ðkAÞun þ ePðkAÞFðun; tnÞ;bn ¼ eR2;2ðkAÞun þ ePðkAÞFðan; tn þ k=2Þ;cn ¼ eR2;2ðkAÞan þ ePðkAÞ 2Fðbn; tn þ k=2Þ � Fðun; tnÞð Þ

with

eR2;2ðkAÞ ¼ ð48I � 12kAþ k2A2Þð48I þ 12kAþ k2A2Þ�1;ePðkAÞ ¼ 24kð48I � 12kAþ k2A2Þ�1

:

5. Parallel/serial version of the new schemes

The schemes mentioned above involve inverses of higher order matrix polynomials which can cause computational dif-ficulties. All the poles of the above mentioned schemes are distinct and we take advantage of this characteristic to resolvethese difficulties. We also construct parallel versions of the schemes using the algorithms developed by Gallopoulos and Saad[4], and Khaliq et al. [11]. Poles as well as weights of R0;4ðkAÞ and R2;2ðkAÞ are distinct and occur in complex conjugate pairs.Using the property, zþ �z ¼ 2RðzÞ, we require only to consider one pole and corresponding weight from each pole–weightpair to construct parallel versions of the schemes. A parallel version of the scheme constructed using the rational approxi-mations R0;4ðkAÞ and R2;2ðkAÞ of e�kA saves computational cost by almost 50% as compared to the parallel versions constructedby using rational approximations with all real distinct poles, see [22].

To compute unþ1 for the L-stable method, we utilize:

R0;4 zð Þ ¼ 2X2

j¼1

Rwj

z� cj

� �

and corresponding fPiðzÞg3

i¼1 takes the form,

Pi zð Þ ¼ 2X2

j¼1

Rwij

z� cj

� �; i ¼ 1;2;3;

where fcjg are the poles of R0;4 as well as of Pi with corresponding weights wj and wij respectively.To compute an; bn, and cn, we use:

eR0;4 zð Þ ¼ 2X2

j¼1

R~wj

z� ~cj

� �

and the corresponding ePðzÞ as

eP zð Þ ¼ 2X2

j¼1

ReXj

z� ~cj

!;

where f~cjg are the poles of eR0;4 as well as of eP with corresponding weights f ~wjg and feXjg respectively. Similarly, the fourthorder A-stable method can also be written in parallel using the same approach.

5.1. Algorithm

For i ¼ 1;2

Step 1. Solve

ðkA� ~ciIÞNai ¼ ~wiun þ keXiFðun; tnÞ;

for Nai and then compute an as: an ¼ 2P2

i¼1RðNaiÞ.

M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136 127

Step 2. Solve

ðkA� ~ciIÞNbi ¼ ~wiun þ keXiFðan; tn þ k=2Þ;

for Nbi and then compute bn as: bn ¼ 2P2

i¼1RðNbiÞ.Step 3. Similarly, to compute cn, we first solve

ðkA� ~ciIÞNci ¼ ~wian þ keXi 2Fðbn; tn þ k=2Þ � Fðun; tnÞð Þ;

for Nci and then compute cn as: cn ¼ 2P2

i¼1RðNciÞ.Step 4. Finally, to compute unþ1, we first solve

ðkA� ciIÞNui ¼ wiun þ kw1iFðun; tnÞ þ 2kw2i Fðan; tvn þ k=2Þ þ Fðbn; tn þ k=2Þð Þ þ kw3iFðcn; tn þ kÞ;

for Nui and then compute unþ1 as: unþ1 ¼ 2P2

i¼1RðNuiÞ.

5.2. Poles and weights

To construct parallel versions of the schemes, poles and weights are computed using Maple10.Poles and corresponding weights for R0;4ðzÞ and fPiðzÞg3

i¼1 are:

c1 ¼ �1:72944423106769þ i0:888974376121862;c2 ¼ �0:270555768932292� i2:50477590436244;w1 ¼ 0:541413348429182� i1:58885918222330;w2 ¼ �0:541413348429154� i0:248562520866115;w11 ¼ 0:244153693956274� i0:0497524711964030;w12 ¼ �0:244153693956268� i0:0750708534900480;w21 ¼ �0:0240066687966667� i0:210771761184790;w22 ¼ 0:0240066687966698þ i0:110830774318527;w31 ¼ 0:473042583717175þ i0:293424221840328;w32 ¼ 0:0269574162828241� i0:165188084403066

and for eR0;4ðzÞ, and ePðzÞ are:

~c1 ¼ �3:45888846213543� i1:77794875224371;~c2 ¼ �0:541111537864595� i5:00955180872487;~w1 ¼ 1:08282669685827þ i3:17771836444659;~w2 ¼ �1:08282669685831� i0:497125041732246;eX1 ¼ �0:621169602486758� i0:599415294095229;eX2 ¼ 0:121169602486770� i0:203064159380992:

Poles and corresponding weights for R2;2ðzÞ and fPiðzÞg3i¼1 are:

c1 ¼ 3:0þ i1:73205080756888;w1 ¼ 6:0� i10:3923048454132;w11 ¼ 0:5� i1:44337567297406;w21 ¼ �i0:577350269189624;w31 ¼ �0:5þ i0:288675134594812

and eR2;2ðzÞ, and ePðzÞ have:

~c1 ¼ 6þ i3:46410161513776;~w1 ¼ 12:0� i20:7846096908265;eX1 ¼ �i3:46410161513774:

6. Models for American options

Most common approach to solve American options problems is to formulate a linear complementary problem and thensolve it numerically (see [3,17–19,24]). Using Heston’s stochastic volatility model with suitable assumptions on the markets,a linear complementarity problem with two-dimensional parabolic partial differential operator is derived for pricing

128 M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136

American options, for example, see [8,23,24]. For the Heston model, the governing PDE arises for pricing options under sto-chastic volatility is:

ouotþ 1

2yx2 o2u

ox2 þ12r2y

o2uoy2 þ qryx

o2uoxoy

þ rxouoxþ jðh� yÞ � kr

ffiffiffiyp½ � ou

oy� ru ¼ 0; ð6:1Þ

where kðx; y; tÞ ¼ ky is the market price of volatility risk. The other parameters are, risk free interest rate r, mean level of thevariance h, rate of reversion on the mean level j and volatility of the variance r. Correlation between price of the underlyingasset and its variance is denoted by q. Detailed derivation of the formula and its closed form solution for the Europeanoptions can be found in [6].

Eq. (6.1) is solved backward in time from the expiry date of the option t ¼ T to the current time t ¼ 0. SubstitutingT � t ¼ t�, we transform Eq. (6.1) to a more familiar form (forward in time) as:

ouot�� 1

2yx2 o2u

ox2 �12r2y

o2uoy2 � qryx

o2uoxoy

� rxouox� ½jðh� yÞ � kr

ffiffiffiyp � ou

oyþ ru ¼ 0: ð6:2Þ

For simplicity of notation, we shall use t instead of t�. Let gðx; yÞ be the price of the option at maturity, called payoff. Trans-forming the problem into forward in time, gðx; yÞ defines the initial condition:

uðx; y; 0Þ ¼ gðx; yÞ: ð6:3Þ

Payoff for an American put option is:

gðx; yÞ ¼maxfE� x;0g; ð6:4Þ

where E is the strike price. The boundary conditions at x ¼ 0 and y ¼ 0 are given by,

uð0; y; tÞ ¼ gð0; yÞ and uðx; 0; tÞ ¼ gðx; 0Þ: ð6:5Þ

As x!1;u! 0 and as y!1;uy ! 0 and therefore the solution of

ouot� 1

2yx2 o2u

ox2 � rxouoxþ ru ¼ 0 ð6:6Þ

is considered as the boundary as y!1, see [24].Because of the early exercise constraint, price of an American option has to be at least the same as the payoff function,

that is, uðx; y; tÞP gðx; yÞ for x P 0 and 0 6 t 6 T , T being the maturity time. Penalty methods force the solution towards afeasible one by penalizing the violations of the early exercise constraint. Most stochastic models of the underlying assets foroption pricing will result in a convection–diffusion problems. Therefore any type of constraint can be forced using a suitabledefinition of the source or sink term. We shall use a penalty term suggested by Zvan et al. [24]:

1�

maxfg � u;0g; 0 6 � 66 1 ð6:7Þ

with Eq. (6.2) for pricing American options under stochastic volatility. We shall also demonstrate the performance of ourmethod for pricing two assets American options under constant volatility. The following penalty term is used for thisproblem:

�Cuþ �� q

; ð6:8Þ

where C P rE > 0 is a penalty term constant and qðx1; x2; . . . ; xnÞ ¼ E�Pn

i¼0aixi, where ai are payoff parameters, for moredetails, see [17]. In order to compare our results with those available in the literature, we are using the same penalty termfor the corresponding problem.

7. Spatial discretization

In this section we shall present the spatial discretization of the Heston stochastic volatility model with nonlinear penaltyterm. The spatial discretization of the two asset Black–Scholes equation for pricing American options can be done in a similarfashion.

Although the option pricing problems are defined in an unbounded domain fðx; y; tÞ; x P 0; y P 0;0 6 t 6 Tg, we shallconsider the following finite computational domain fðx; y; tÞ;0 6 x 6 X;0 6 y 6 Y ;0 6 t 6 Tg, where X and Y are chosen largeenough so that the computational error due to this truncation is negligible.

Let m;n, and l be the number of grid points in the x-, y-, and t-direction, respectively. Let hx ¼ X=m;hy ¼ Y=n and k ¼ T=land let xi ¼ ihx; i ¼ 0; . . . ;m; yj ¼ jhy; j ¼ 0; . . . ;n, and tk ¼ ks; k ¼ 0; . . . ; l. Let the values of the finite difference approximationsof uðx; y; tÞ at these grid be denoted by:

ui;j uðxi; yj; tÞ: ð7:1Þ

M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136 129

Assuming that uðx; y; tÞ is twice differentiable w.r.t. x and y, we replace the partial derivatives with respect to x and y by thesecond order central differences. A standard approach to approximate the cross derivative term is:

o2uoxoy

¼ 14hxhy

ðuðxþ hx; yþ hy; tÞ � uðx� hx; yþ hy; tÞ � uðxþ hx; y� hy; tÞ þ uðx� hx; y� hy; tÞÞ

þ OðhxhyÞ; as hx ! 0; hy ! 0: ð7:2Þ

Instead we shall use the Taylor expansions of uðxþ hx; yþ hyÞ and uðx� hx; y� hyÞ and write the cross derivative term interms of derivatives w.r.t. x and w.r.t. y only. The same approach has also been used by Ikonen [8]. Numerical experimentsshow that this approach is computationally more efficient.

Using the Taylor expansion of uðxþ hx; yþ hyÞ

uðxþ hx; yþ hyÞ uðx; yÞ þ hxouoxþ hy

ouoyþ 1

2h2

xo2uox2 þ

12

h2yo2uoy2 þ hxhy

o2uoxoy

;

we can write the cross derivative term as:

o2uoxoy

1hxhy

uðxþ hx; yþ hyÞ � uðx; yÞ � hxouox� hy

ouoy� 1

2h2

xo2uox2 �

12

h2yo2uoy2

!:

Similarly using the Taylor expansion of uðx� hx; y� hyÞ, we can write:

o2uoxoy

1hxhy

uðx� hx; y� hyÞ � uðx; yÞ þ hxouoxþ hy

ouoy� 1

2h2

xo2uox2 �

12

h2yo2uoy2

!:

Convex combination of these two approximations yields,

o2uoxoy

whxhy

uðxþ hx; yþ hyÞ � uðx; yÞ � hxouox� hy

ouoy� 1

2h2

xo2uox2 �

12

h2yo2uoy2

!

þ 1�whxhy

uðx� hx; y� hyÞ � uðx; yÞ þ hxouoxþ hy

ouoy� 1

2h2

xo2uox2 �

12

h2yo2uoy2

!; ð7:3Þ

where the weighing parameter w lies in [0,1]. Replacing the cross derivative term approximation (7.3) in Eq. (6.2) and addingthe penalty term (6.7) (denoted by Fðu; tÞ), we obtain

ouotþ �1

2yx2 þ qryx

hx

2hy

� �o2uox2 þ �1

2r2yþ qryx

hy

2hx

� �o2uoy2 þ �rx� qryx

1hyþ 2wqryx

1hy

� �ouox

þ �jðh� yÞ þ krffiffiffiyp� qryx

1hxþ 2wqryx

1hx

� �ouoyþ r þ qryx

1hxhy

� �u� 1�w

hxhyqryxuðx� hx; y� hy

� whxhy

qryxuðxþ hx; yþ hyÞ þ Fðu; tÞ ¼ 0: ð7:4Þ

For simplicity of notation, we let

Aðx; yÞ ¼ �12

yx2 þ qryxhx

2hy;

Bðx; yÞ ¼ �12r2yþ qryx

hy

2hx;

Cðx; yÞ ¼ �rx� qryx1hyþ 2wqryx

1hy;

Dðx; yÞ ¼ �jðh� yÞ þ krffiffiffiyp� qryx

1hxþ 2wqryx

1hx;

Gðx; yÞ ¼ r þ qryx1

hxhy;

Hðx; yÞ ¼ �1�whxhy

qryx;

Kðx; yÞ ¼ � whxhy

qryx

and rewrite Eq. (7.4) as:

ouotþ Aðx; yÞ o

2uox2 þ Bðx; yÞ o

2uoy2 þ Cðx; yÞ ou

oxþ Dðx; yÞ ou

oyþ Gðx; yÞuþ Hðx; yÞuðx� hx; y� hyÞ

þ Kðx; yÞuðxþ hx; yþ hyÞ þ Fðu; tÞ ¼ 0:

130 M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136

Using the following spatial derivative approximations:

ouox

� �i;j

12hx

uiþ1;j � ui�1;j� �

;ouoy

� �i;j

12hy

ui;jþ1 � ui;j�1� �

;

o2uox2

!i;j

1

h2x

ui�1;j � 2ui;j þ uiþ1;j� �

;o2uoy2

!i;j

1

h2y

ui;j�1 � 2ui;j þ ui;jþ1� �

and evaluating Ai;j ¼ Aðxi; yjÞ; Bi;j ¼ Bðxi; yjÞ;Ci;j ¼ Cðxi; yjÞ;Di;j ¼ Dðxi; yjÞ;Gi;j ¼ Gðxi; yjÞ;Hi;j ¼ Hðxi; yjÞ, and Ki;j ¼ Kðxi; yjÞ, at theabove mentioned grid points, we obtain the following initial-value problem:

dudtþAu ¼ Fðu; tÞ; uð0Þ ¼ g; ð7:5Þ

where A is n:m n:m block tridiagonal matrix,

A ¼

M1 U1

L2 M2 U2

. .. . .

. . ..

Lm�1 Mm�1 Um�1

Lm Mm

266666664

377777775

and tridiagonal matrices Ui;Mi, and Li are of the form:

Ui ¼

Ai1

h2xþ Ci1

2hxKi1

0 Ai2

h2xþ Ci2

2hxKi2

. .. . .

. . ..

0 Ain�1

h2xþ Cin�1

2hxKin�1

KinAin

h2xþ Cin

2hx

266666666664

377777777775; for i ¼ 1; . . . ;m� 1;

Mi ¼

�2Ai1

h2x� 2Bi1

h2yþ Gi1

Bi1

h2yþ Di1

2hy

Bi2

h2y� Di2

2hy

�2Ai2

h2x� 2Bi2

h2yþ Gi2

Bi2

h2yþ Di2

2hy

. .. . .

. . ..

Bin�1

h2y� Din�1

2hy

�2Ain�1

h2x� 2Bin�1

h2yþ Gin�1

Bin�1

h2yþ Din�1

2hy

2Bin

h2y

�2Ain

h2x� 2Bin

h2yþ Gin

2666666666664

3777777777775;

for i ¼ 1; . . . ;m,

Li ¼

Ai1

h2x� Ci1

2hx0

Hi2Ai2

h2x� Ci2

2hx0

. .. . .

. . ..

Hin�1Ain�1

h2x� Cin�1

2hx0

HinAin

h2x� Cin

2hx

266666666664

377777777775; for i ¼ 2; . . . ;m� 1

and

Lm ¼

2 Am1

h2x

Km1

Hm2 2 Am2

h2x

Km2

. .. . .

. . ..

Hmn�1 2 Amn�1

h2x

Kmn�1

Hmn þ Kmn 2 Amn

h2x

266666666664

377777777775

and u ¼ ½u1 u2 u3 � � � um�T with ui ¼ ½ui;1 ui;2 ui;3 � � � ui;n� for i ¼ 1;2; . . . ;m.

M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136 131

8. Numerical experiments

Results obtained in the previous sections are illustrated on two important problems from financial mathematics, pricingAmerican options under stochastic volatility and two asset American options pricing problem. Convergence results forthese problems are obtained using the fourth L-stable method. ‘‘Difference” in the convergence tables is the absolutevalue of the maximum change in the solution as the temporal grid is refined and ‘‘Ratio” is the ratio of successive differences.In the numerical experiments we also indicate a crucial quantity in financial mathematics, the Delta of an option, i.e., rate ofchange of the option value with respect to the asset price (see [7,23]). For Example 8.1, we approximate the Delta valuesusing:

Table 1Examplscientifi

Referen

Yousuf

Clarke

Ikonen

Oosterl

Zvan et

Table 2ExamplDx ¼ 0:

s ¼ Dt

0.050.0250.01250.006250.003120.00156

ouoxðxi; yj; tkÞ

ukiþ1;j � uk

i�1;j

xiþ1 � xi�1:

8.1. American put option under stochastic volatility

We consider the Heston stochastic volatility model (6.2) and add the penalty term (6.7). This problem is solved usingA-stable as well as L-stable methods. Reference values given in Table 1 are computed using the L-stable method with theparameter values: E ¼ 10; T ¼ 0:25; r ¼ 0:1;j ¼ 5; h ¼ 0:16;r ¼ 0:9; k ¼ 0:0;q ¼ 0:1; � ¼ 0:005 and weighing parameterw ¼ 0:75 (see Eq. (7.3)). These values are in good agrement with the values given in the literature, see [1,8,18,24]. Table 2shows convergence of the L-stable method. Fig. 2 shows that the A-stable method exhibit unwanted oscillations at the strikeprice whereas Fig. 3 shows the smooth results computed by using the L-stable method. Figs. 4 and 5 are the graphs of Delta ofthe option. Unwanted oscillations get worse when we compute Delta values of the option, as is shown in Fig. 4.

8.2. Two asset American put option

We consider a two asset put option with nonlinear penalty term �Cuþ��q (see [12,17])

ouot� 1

2r2

1x2 o2uox2 �

12r2

2y2 o2uoy2 � qr1r2xy

o2uoxoy

� ðr � D1Þxouox� ðr � D2Þy

ouoyþ ru� �C

uþ �� q¼ 0 ð8:1Þ

with the following initial and boundary conditions,

e 8.1: Reference prices computed using the L-stable method on a (400, 80, 10) grid at the variance y ¼ 0:0625 and y ¼ 0:25 and the prices published inc literature.

ce y x ¼ 8 x ¼ 9 x ¼ 10 x ¼ 11 x ¼ 12

0.0625 1.9994 1.1073 0.5181 0.2126 0.08180.25 2.0788 1.3336 0.7960 0.4484 0.2429

and Parrott [1] 0.0625 2.000 1.1080 0.5316 0.2261 0.09070.25 2.0733 1.3290 0.7992 0.4536 0.2502

[8] 0.0625 2.0000 1.1076 0.5200 0.2137 0.08200.25 2.0784 1.3336 0.7960 0.4483 0.2428

ee [18] 0.0625 2.0000 1.1070 0.5170 0.2120 0.08150.25 2.0790 1.3340 0.7960 0.4490 0.2430

al. [24] 0.0625 2.000 1.1076 0.5202 0.2138 0.08210.25 2.0784 1.3337 0.7961 0.4483 0.2428

e 8.1: Convergence table for American option under stochastic volatility using L-stable method with8;Dy ¼ 0:2; r ¼ 0:05;j ¼ 5; h ¼ 0:16;r ¼ 0:9; k ¼ 0:0;q ¼ 0:1; � ¼ 0:005 and weighing parameter w ¼ 0:75.

Difference Ratio Order

0.000e+00 – –2.1518e�002 – –1.8528e�003 11.61 3.541.0903e�004 16.99 4.09

5 5.9149e�006 18.43 4.2025 4.2613e�007 13.88 3.79

0 5 10 15 200

0.5

1

−10

−5

0

5

10

15

Volatility

Asset

Opt

ion

Valu

e

Fig. 2. American put option under stochastic volatility using A-stable method.

0 5 10 15 200

0.5

1

0

2

4

6

8

10

12

Volatility

Asset

Opt

ion

Valu

e

Fig. 3. American put option under stochastic volatility using L-stable method.

132 M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136

uðx; y; 0Þ ¼ gðx; yÞ; x; y P 0; ð8:2Þuðx; 0; tÞ ¼ g1ðx; tÞ; x P 0; ð8:3Þuð0; y; tÞ ¼ g2ðy; tÞ; y P 0; ð8:4Þlimx!1

uðx; y; tÞ ¼ G1ðy; tÞ; y P 0; ð8:5Þ

limy!1

uðx; y; tÞ ¼ G2ðx; tÞ; x P 0; ð8:6Þ

where

qðx; yÞ ¼ E� ða1xþ a2yÞ; and gðx; yÞ ¼maxfqðx; yÞ; 0g: ð8:7Þ

We solve Eq. (8.1) along with (8.2)–(8.6) using the algorithm (5.1) with the parameters: spatial domain 0 6 x 6 4;0 6 y 6 4,time domain 0 6 t 6 1, constant volatilities r1 ¼ 0:2;r2 ¼ 0:3, dividend values D1 ¼ 0:05;D2 ¼ 0:01, payoff parameters

0 5 10 15 20

0

0.5

1−20

−15

−10

−5

0

5

10

15

20

AssetVolatility

Del

ta V

alue

Fig. 4. Delta of American put option under stochastic volatility using A-stable method.

Table 3Example 8.2: Computed prices of two asset American put option using the L-stable method on a (30, 30, 40) grid with � ¼ 0:01. (xi ¼ ihx; yj ¼ jhy , withhx ¼ hy ¼ 4=30 ¼ 0:1333.)

x=y x5 x10 x15 x20 x25

y5 0.46830 0.20847 0.04512 0.00766 0.00199y10 0.11637 0.02342 0.00445 0.00157 0.00090y15 0.01594 0.00330 0.00130 0.00086 0.00062y20 0.00283 0.00112 0.00076 0.00061 0.00046y25 0.00089 0.00057 0.00046 0.00039 0.00032

0 5 10 15 20

0

0.5

1−1

−0.8

−0.6

−0.4

−0.2

0

AssetVolatility

Del

ta V

alue

Fig. 5. Delta of American put option under stochastic volatility using L-stable method.

M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136 133

a1 ¼ 0:4;a2 ¼ 0:6, interest rate r ¼ 0:1, and the strike price E ¼ 1. The small regularization parameter 0 < � 66 1 is taken as� ¼ 0:01 and the penalty term constant C is used as C ¼ rE ¼ 0:1 in our numerical experiments. The boundary conditionsg1ðx; yÞ; g2ðx; yÞ are solutions of the corresponding single asset American put option problems, and G1ðy; tÞ;G2ðx; tÞ are taken

01

23

4 01

23

4

0

0.2

0.4

0.6

0.8

1

Asset 2Asset 1

Opt

ion

Valu

e

Fig. 6. Payoff for two assets American put option.

Table 4Example 8.2: Convergence table for two asset American put option using the L-stable method with � ¼ 0:01.

s ¼ Dt Difference Ratio Order

0.05 0.0000e+000 – –0.025 3.8428e�006 – –0.0125 3.8412e�007 10.00 3.320.00625 2.9126e�008 13.19 3.720.003125 1.9066e�009 15.28 3.930.0015625 1.1692e�010 16.31 4.03

01

23

4 01

23

4

0

0.2

0.4

0.6

0.8

1

Asset 2Asset 1

Opt

ion

Valu

e

Fig. 7. Two assets American put option using A-stable method.

134 M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136

identically equal to zero. Graph of payoff function with a1 ¼ 0:6;a2 ¼ 0:4 is given in Fig. 6. Although the strike price is E ¼ 1,the corner or edge in Fig. 6 is not at x ¼ 1 and y ¼ 1 because of a1 and a2. Graph shown in Fig. 7 using A-stable method con-tains unwanted oscillations at the corner where L-stable method produces a smooth graph shown in Fig. 8. In Table 3, optionvalues are given at different mesh points. Convergence results using L-stable method are given in Table 4.

01

23

4 01

23

4

0

0.2

0.4

0.6

0.8

1

Asset 2Asset 1

Opt

ion

Valu

e

Fig. 8. Two assets American put option using L-stable method.

M. Yousuf / Applied Mathematics and Computation 213 (2009) 121–136 135

9. Conclusion

Presented L-stable method is based on Padé approximations of the matrix exponential functions. For the spatial discret-ization we used a special seven point finite difference stencil on uniform grids. Use of the splitting technique makes themethod more efficient, stable, and accurate. High order method can be implemented with essentially the same computa-tional complexity as the first order implicit method. The proposed method of this article shows sufficiently accurate conver-gence rate. Also, our numerical experiments show that the option prices computed are in good agreement with the pricespresented in the literature. Moreover, the PDE approach allows reliable option valuation at all points of the time-asset grid.The L-stable method shall perform well and produce a smooth solution even if discontinuities are introduced in the timedomain, for example barrier options.

Acknowledgement

This work is supported by the Junior Faculty Project # JF 070005, King Fahd University of Petroleum and Minerals, Dhah-ran, Saudi Arabia.

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