Numerical Analysis of Jump Diffusion Models: A Partial Differential Equation Approach

We discuss a number of numerical methods that approximate the solution of the Partial Integro Differential Equation (PIDE) that models contingent claims with jumps. Starting with the Merton jump model for the underlying asset we motivate how to find the corresponding PIDE that models a derivative quantity on that asset. We pose the PIDE in a form that becomes amenable to a solution using the finite difference method (FDM). We discuss a number of schemes and we focus on one-factor models and we concentrate on improving the Black Scholes for European options. The models can be extended to multi-factor option pricing problems but a discussion is outside the scope of this article. The main goal of this article is to show the relationship between jump models, PIDEs and numerical analysis. We see this synergy as a new development in computational finance and new techniques need to be developed and elaborated upon in the coming years.