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Proceedings of the National Institute for Mathematical SciencesVol. 2, No. 10(2008), pp.000000

A HIGH ORDER COMPACT SCHEME FOR OPTION PRICING WITH JUMPS

HAI-WEI SUN AND SPIKE T. LEE

Abstract. In this paper we consider a partial integro-differential equation (PIDE), which arisesfrom the option pricing problem in jump-diffusion model. A fourth order compact finite differencescheme is proposed to discretize the spatial variable of this PIDE. The computation of the discretizedsystem involves matrix-vector multiplication, where the matrix generated from the jump integralterm is Toeplitz-like. Hence fast Fourier transform can be used to perform these multiplicationsefficiently. Numerical results are given to show that our approach indeed has fourth order accuracy

in space, which is much better than the classical second order central difference scheme.

1. Introduction

In this paper, we consider option pricing under the jump-diffusion model presented by Merton [10]. Inthis model, the asset return follows a standard Wiener process driven by a compound Poisson processwith normally distributed jumps. Furthermore, by choosing the parameters of the jump processproperly, volatility smiles and skews can be generated under this model [3]. The value of a contingentclaim under a jump-diffusion process satisfies a partial integro-differential equation (PIDE). This kindof equation generally contains differential operators and a non-local integral term.

Numerical solutions for the PIDE were widely studied in the past several years [1, 2, 3, 8, 16]. Butthese methods are at most second order accuracy for both space and time. For time direction, Feng

and Linetsky [6] recently suggested an implicit-explicit (IMEX) scheme, where the differential partis treated implicitly and the integral part explicitly, with a new high order extrapolation approach.Their method is independent of the choice of spatial discretization and can be added to any PIDEsolver based on the IMEX scheme.

For spatial direction, most of the schemes ever proposed exploit standard central difference dis-cretization, which only has second order convergence. In this paper, we apply a fourth order compactscheme (FOCS) for pricing European call options. The PIDE is discretized in time by an IMEXmethod. Inspired by Feng and Linetskys idea [6], we employ the Richardson extrapolation, which isless accurate than theirs but comparatively straightforward, to achieve high order accuracy in time.Moreover, we exploit numerical quadrature method for evaluating the jump integral term. It guaran-tees the Toeplitz-like structure of the integral operator such that a fast algorithm can be applied. Thisrest of the paper is arranged as follows. In Section 2 we review the jump-diffusion model for option

pricing and the IMEX scheme with extrapolation approach for the PIDE. In Section 3 we propose theFOCS for the PIDE. Issues regarding the evaluation of the non-local integral term are discussed inSection 4. In Section 5 we give some numerical results.

The research was partially supported by the research grant RG-UL/07-08S/Y1/JXQ/FST and CG016/08-09W/SHW/FST from University of Macau. This is a brief review article for 2008 NIMS International Conferencein October, 2008 joint with The 4th East Asia SIAM Conference. Some of the contents in the article may appearsomewhere else.

c2007 National Institute for Mathematical Sciences

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High Order Scheme for Option Pricing with Jumps 1

2. Option Pricing in Jump-Diffusion Model and the IMEX Scheme

In jump-diffusion model, the risk-neutral dynamics of the asset price S() with

[0, T] (T is theexpiration) can be described by the following diffusion process:

dS

S= d + dz + ( 1)dq,

where is the drift rate, is the stock return volatility, 1 is an impulse function making S jumpto S, dz is the increment of a Brownian motion and dq is a Poisson process with arrival intensity (dq = 0 with probability 1 d and dq = 1 with probability d). We denote the expectation ofthe impulse function by = E( 1). Let V(S(), ) be the value of a contingent claim on the assetS(). Then V(S, ) can be computed by solving a PIDE on [0, +) [0, T]:(1) V = 2

2S2VSS + (r )SVS (r + )V +

0

V(S,)g()d,where r is the risk-free interest rate and g() is the density function of the jump size distribution.

It is common to change the variables as follows:

S = ex, = ez and t = T .Then equation (1) is reduced to a forward PIDE on (, +) [0, T]:

(2) Vt =2

2Vxx + (r

2

2)Vx (r + )V +

V(y, t)f(y x)dy.

where y = x + z, f(y x) = g(eyx)eyx and V(y, t) = V(ey, T t). We refer the readers to see [8]for details.

For a European call option, the initial condition is given by

V(x, 0) = max(ex K, 0),where K is the strike price. Boundary conditions are V(x, t) 0, as x ,

V(x, t) ex Kert, as x +.Note that European put options can be handled in a similar way.

We consider the jump-diffusion model presented by Merton [10]. The jump size distribution isnormal with mean and standard deviation , and g() is given by

g() =e(log())

2/22

2

.

After changing the variable, we have

(3) f(y x) = g(eyx)eyx = e(yx)2/22

2

.

Note that f is the probability density function of the Gaussian distribution, hence the followingproperty holds:

f(y x)dy = 1.In the following, we introduce the IMEX scheme for a PIDE. Suppose a PIDE is given by:

Vt = DV + LV,

where D is a differential operator and L is an integral operator. As L usually makes the systemdense, we utilize the IMEX scheme which is noted for avoiding dense matrices inversion. First the


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2 H. Sun & S. Lee

time interval [0, T] is divided into J time steps, with t = T /J and tj = jt, j = 0, 1, 2,...,J. LetVj = V(tj), j = 0, 1, 2,...,J. We then apply the IMEX scheme, where the differential part is treated

implicitly and the integral part is treated explicitly for j = 0, 1, 2,...,J 1:Vj+1 Vj

t= DVj+1 + LVj ,

or

(4) Vj+1 tDVj+1 = Vj + tLVj .Our ultimate goal is to find VJ = V(T) . At each time step, the linear system (4) is solved to

determine the vector Vj+1 with V0 being the given initial guess. However, the IMEX scheme isonly first order accurate in time. If the IMEX scheme is unconditionally stable, we can employ theRichardson extrapolation method to achieve higher accuracy. Denote the solution obtained at timeT with step size t/2p1, p = 1, 2,...,s, by VJp,1, where s is the stage number. Then the Richardsonextrapolation formula is given by

(5) VJp,q =2p1VJp,q1 VJp1,q1

2p1 1 , p = 2, 3, ..., s, q = 2, 3,...,p.

Graphically, it can be illustrated as follows:

VJ1,1VJ2,1 V

J2,2

......

. . .

VJs,1 VJs,2 ... V

Js,s

By this process, we have achieved a better approximation VJs,s which is of order O(ts) after sextrapolation stages.

3. The Fourth Order Compact Scheme

We now proceed to consider the FOCS for option pricing. The FOCS was proposed by Gupta [7],which is known for restraining the numerical oscillations [11, 12, 13, 14].

Recall that the PIDE (2) on (, +) [0, T] is given byVt = aVxx + bVx + cV + V f,

where

a =1

22, b = r

2

2, c = (r + ),

and

V f =

V(y, t)f(y x)dy.

First the infinite x-domain (,) is truncated to a finite computational domain [xmin, xmax]using mesh size

x =xmax xmin

I0.

Then the computational grid with N0 = I0 + 1 points is denoted by

x = {x0, x1, x2,..., xK ,...,xI01, xI0},


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where x0 = xmin, xK = log K and xI0 = xmax. Divide the time interval [0, T] into J time steps, witht = T /J and tj = jt, j = 0, 1, 2,...,J. Define V

j = V(x, tj), j = 0, 1, 2,...,J. Then the IMEX

scheme is applied as discussed in Section 2:Vj+1 Vj

t= aVj+1xx + bV

j+1x + cV

j+1 + Vj f.We then consider the following semi-discretized equation:

(6) atVj+1xx btVj+1x + (1 ct)Vj+1 = Vj + tVj f.Since the right-hand side of the above equation is given for the current time step, it can be treated asa convection-diffusion equation. Thus FOCS can be applied for (6).

Let Vji = V(xi, tj) for i = 1, 2,...,I0 1. We derive the three point FOCS as follows,Vj+1i [1 (i + i) + t(i + i + r + )] + (i ti)Vj+1i1 + (i ti)Vj+1i+1(7)

= Vji [1

(i + i)] + iV

ji1 + iV

ji+1 + t

V(y, tj)(y

xi, x)dy,

where

i =1

12 bx

24a, i =

1

12+

bx

24a, i =

dix2

ei2x

, i =di

x2+

ei2x

,

di = a +b2x2

12a+

cx2

12, ei = b +

bcx2

12aand

(8) (y x, x) =

f(y x) + bx2

12a

f(y x)x

+x2

12

2f(y x)x2

.

Consequently, we can write the discretized equation (7) in matrix form for j = 0, 1, 2,...,J 1:(9) M1V

j+1 = M2Vj + tVj ,

where Vj = (Vj1 , Vj2 ,...,VjI01) . Note that M1 and M2 are tridiagonal matrices when the com-

putational grid is uniform. At each time step, a tridiagonal system can be solved by using the LUdecomposition. After solving the linear system (9), the resulting VJ = V(T) is regarded as theapproximation to the true solution.

In order to achieve high order accuracy for the time direction, as discussed in Section 2, in thefollowing we study the unconditional stability of the IMEX scheme with the FOCS (7).

Lemma 3.1. [9] For the kernel function (y x, x) in (8), we have

(y x, x)dy = 1.

According to lemma 3.1, we have the following theorem by using the Fourier stability analysis.

Theorem 3.2. [9] The FOCS (7) is unconditionally stable.

4. Evaluation of the Integral Term

Evaluation of the integral term is required to solve the linear system (9) at each time step. Thereforea higher order approximation to the convolution integral is also necessary. The numerical quadraturemethod is used for evaluating the integral term in our case. The truncation is achieved by extendingx to a uniform grid y with mesh size x as follows:

y = {yk R : yk = ymin + ky, k = 0, 1,...,n, y = x = xmax xminI0

},


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4 H. Sun & S. Lee

where y0 = ymin and yn = ymax. Note that y equals to the mesh size x. Hence we can write theintegral in discrete form by using fourth order composite Simpsons rule for i = 1, 2,...I0 1,

V(y, t)(y xi, x)dy = y3

[V(y0, t)(y0 xi, x) + 2n/21k=1

V(y2k, t)(y2k xi, x)(10)

+ 4

n/2k=1

V(y2k1, t)(y2k1 xi, x) + V(yn, t)(yn xi, x)],

Furthermore, (10) can be written in matrix form LVy, where

L =

(y0 x1, x) (y1 x1, x) (y2 x1, x) . . . (yn x1, x)(y0 x2, x) (y1 x2, x) (y2 x2, x) . . . (yn x2, x)

......

......

(y0 xI01, x) (y1 xI01, x) (y2 xI01, x) . . . (yn xI01, x)

and

Vy =y

3(V(y0,t), 4V(y1,t), 2V(y2,t), 4V(y3,t), 2V(y4,t),...,V(yn,t))

.

Note that we have the following relation for i = 1, 2,...I0 1, k = 0, 1,...,n:xi+1 xi = x = y = yk+1 yk.

Therefore yk xi = yk+1 xi+1 and (yk xi, x) = (yk+1 xi+1, x). Thus L is a Toeplitzmatrix, and the fast Fourier transform can be applied to compute LVy [4, 5].

Remark 4.1. There exists a non-smooth region around the strike price xK when pricing a Europeancall option. Consequently it wil l affect the high order accuracy. Thus it is necessary to concentrate

grid points near xK . The local mesh refinement strategy [9] is employed for numerical testing in thenext section to guarantee the high accuracy.

5. Numerical Results

In this section, we give the numerical results of pricing European call options under Mertons jump-diffusion model [10]. In our numerical tests, we have to make sure the error in temporal directionis small enough not to affect the convergence rate of the spatial discretization. Thus sixth orderRichardson extrapolation method is utilized to obtain sixth order accuracy in time. Assume theextrapolation stage number s = 6 and t = 102 at the first stage. The value VJ6,6 in (5), which is of

order O(t6) after six extrapolation stages is accepted as the approximation to VJ.The input parameters are T = 0.25, K = 100, = 0.25, r = 0.05, = 0.9 and = 0.45. N is the

number of points in the x-direction and x is the mesh size. Error at xK is the difference between thetrue solution (in [10]) and the approximation at strike price xK . l

error denotes the infinity normerror between the true solution and the approximation.

The results in Table 1 show that the standard central difference scheme clearly attains second orderaccuracy. As comparison, we can see from Table 2 that the FOCS with local mesh refinement restoresfourth order convergence.

References

[1] A. Almendral and C. Oosterlee, Numerical Valuation of Options with Jumps in the Underlying, Applied NumericalMathematics, Vol. 53 (2005), pp. 118.

[2] K. Amin, Jump Diffusion Option Valuation in Discrete Time, Journal of Finance, Vol. 48 (1993), pp. 18331863.[3] L. Andersen and J. Andreasen, Jump-Diffusion Processes: Volatility Smile Fitting and Numerical Methods for

Option Pricing, Review of Derivatives Research, Vol. 4 (2000), pp. 231262.


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N x Error at xK l error Order

33 1/5 1.62 1.62 -

65 1/10 4.51e-1 4.51e-1 1.847129 1/20 9.76e-2 1.06e-1 2.083257 1/40 2.38e-2 2.67e-2 1.992513 1/80 5.91e-3 6.69e-3 1.998

1025 1/160 1.47e-4 1.67e-3 1.999

Table 1. Infinity norm error and convergence rates of central difference scheme forpricing a European call option under Mertons jump-diffusion model.

N x Error at xK l error Order

33 1/5 1.69e-2 2.20e-2 -65 1/10 1.82e-3 4.67e-3 2.234

129 1/20 1.16e-4 3.87e-4 3.593257 1/40 5.49e-6 2.77e-5 3.802513 1/80 6.61e-7 1.88e-6 3.883

1025 1/160 4.66e-8 1.23e-7 3.935

Table 2. Infinity norm error and convergence rates of FOCS with local mesh refine-ment for pricing a European call option under Mertons jump-diffusion model.

[4] R. Chan and M. Ng, Conjugate Gradient Methods for Toeplitz Systems, SIAM Review, Vol. 38 (1996), pp. 427482.[5] R. Chan and X. Jin, An Introduction to Iterative Toeplitz Solvers, SIAM, Philadelphia (2007).[6] L. Feng and V. Linetsky, Pricing Options in Jump-Diffusion Models: An Extrapolation Approach, Oper. Res. 56

(2008), pp. 304325.[7] M. Gupta, A fourth-order Poisson solver, J. Comput. Phys. Vol. 55(1) (1984), pp. 166172.[8] Y. dHalluin, P. Forsyth and K. Vetzal, Robust Numerical Methods for Contingent Claims under Jump Diffusion

Processes, IMA Journal of Numerical Analysis, Vol. 25 (2005), pp. 87112.[9] S. Lee and H. Sun, Fourth Order Compact Scheme with Local Mesh Refinement for Option Pricing in Jump-

Diffusion Model, Manuscript.[10] R. Merton, Option Pricing When Underlying Stock Returns Are Discontinuous, Journal of Financial Economics,

Vol. 3 (1976), pp. 125144.[11] W. Spotz and G. Carey, High-Order Compact Finite Difference Schemes for Computational Mechanics, PhD Thesis,

University of Texas at Austin (1995).[12] W. Spotz and G. Carey, High-Order Compact Scheme for the Steady Stream-Function Vorticity Equations, Int. J.

Numer. Methods Engrg., Vol. 38 (1995), pp. 34973512.[13] J. Zhang, An Explicit Fourth-Order Compact Finite Difference Scheme for Three Dimensional Convection-Diffusion

Equation, Communications in Numerical Methods in Engineering, Vol. 14 (1998), pp. 263280.[14] J. Zhang L. Ge and M. Gupta, Fourth Order Compact Scheme for 3D Convection Diffusion Equation with Boundary

Layers on Nonuniform Grids, Neural, Parallel & Scientific Computationss, Vol. 8, Num. 3-4 (2000), pp. 373392.[15] J. Zhang, H. Sun and J. Zhao, High Order Compact Scheme with Multigrid Local Mesh Refinement Procedure for

Convection Diffusion Problems, Comput. Methods Appl. Mech. Engrg., Vol. 191 (2002), pp. 46614674.[16] X. Zhang, Numerical Analysis of American Option Pricing in A Jump-Diffusion Model, Mathematics of Operations

Research, Vol. 22(3) (1997), pp. 668690.

Department of Mathematics, University of Macau, Macao, China.

E-mail address : [email protected]

Department of Mathematics, University of Macau, Macao, China.

E-mail address : [email protected]

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