A Fourier Transform Analysis of the American Call Option on Assets Driven by Jump-Diffusion Processes

Abstract

This paper considers the Fourier transform approach to derive the implicit integral equation for the price of an American call option in the case where the underlying asset follows a jump-diffusion process. Using the method of Jamshidian (1992), we demonstrate that the call option price is given by the solution to an inhomogeneous integro-partial differential equation in an unbounded domain, and subsequently derive the solution using Fourier transforms. We also extend McKean's incomplete Fourier transform approach to solve the free boundary problem under Merton's framework, for a general jump size distribution. We show how the two methods are related to each other, and also to the Geske-Johnson compound option approach used by Gukhal (2001). The paper also derives results concerning the limit for the free boundary at expiry, and presents a numerical algorithm for solving the linked integral equation system for the American call price, delta and early exercise boundary. This scheme is applied to Merton's jump-diffusion model, where the jumps are log-normally distributed.