Partial Differential Equation Approach Under Geometric Jump-Diffusion Process

In this chapter we consider the solution of the integro-partial differential equation that determines derivative security prices when the underlying asset price is driven by a jump-diffusion process. We take the analysis as far as we can for the case of a European option with a general pay-off and the jump-size distribution is left unspecified. We obtain specific results in the case of a European call option and when the jump size distribution is log-normal. We illustrate two approaches to the problem. The first is the Fourier transform technique that we have used in the case that the underlying asset follows a diffusion process. The second is the direct approach using the expectation operator expression that follows from the martingale representation. We also show how these two approaches are connected.