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A new discretization strategy is introduced for the numerical solution of partial integrodifferential equations appearing in option pricing jump diffusion models. In order to consider the unknown behaviour of the solution in the unbounded part of the spatial domain, a double discretization is proposed. Stability, consistency, and positivity of the resulting explicit scheme are analyzed. Advantages of the method are illustrated with several examples.

Since empirical studies revealed that the normality of the log returns, as assumed by Black and Scholes, could not capture features like heavy tails and asymmetries observed in market-data log-returns densities [

In order to solve the PIDE problem numerically, Andersen and Andreasen [

Tavella and Randall in [

In the outstanding paper [

An efficient solution of PIDEs for the jump-diffusion Merton model is proposed in [

Almendral and Oosterlee [

In [

In [

For the sake of clarity in the presentation we recall that in a jump-diffusion model, the modified stochastic differential equation (SDE) for the underlying asset is

In this paper we consider Merton’s jump-diffusion model for a vanilla call option with payoff function

This paper is organized as follows. Section

In Section

If

For the sake of convenience we introduce a transformation of variables to remove both the advection and the reaction terms of the PIDE problem (

Let us consider the transformation

In order to approximate the integral appearing in (

Let us denote

Taking

In this section a difference scheme for problem (

Note that the integral

Let us denote

From the previous approximations, for the internal points we have

For the sake of convenience in the study of stability we introduce a vector formulation of the scheme (

Let

Dealing with prices of contracts modeled by PIDE, the solution must be nonnegative. In this section we show that numerical solution provided by scheme (

We begin with the following result.

With previous notation, assume that stepsizes

From (

Note that as matrix

With the hypotheses and notation of Lemma

The next result will be used below to guarantee stability.

Matrices

Under conditions (C1) and (C2) of Lemma

By Lemma

Hence and from the definition of

Let

For the sake of clarity and as there are many definitions of stability in the literature we recall our concept of stability in the next definition.

Let

If the property (

With the previous notation, the numerical solution

Note that scheme (

We say that a numerical difference scheme is consistent with a PIDE, if the exact theoretical solution of the PIDE approximates well to the exact solution of the difference scheme as the stepsize discretization tends to zero, [

Let us write the scheme (

Let us denote

Let us assume that

In an analogous way, let us explain the local consistency error of the unbounded integral

Thus

In the following examples the code was run on

Consider the vanilla call option problem (

Satisfying and breaking stability conditions.

The next example shows the robustness of our numerical scheme under changes of the jump intensity

Taking the same parameters of Example

Variation of the jump intensity

In the next example, the error is the difference between the numerical solution

Consider the vanilla call option problem (

Absolute errors with several values of

The next Example

Taking the problem of Example

Absolute errors with several values of

This work introduces a new discretization strategy for solving partial integrodifferential equations which involves the discretization of the unknown in the unbounded part of the integral. This fact increases the accuracy of the numerical solution in the boundary of the numerical domain as it is shown in Example

This paper was supported by the Spanish M.E.Y.C. Grant DPI2010-20891-C02-01.