Option pricing in a lévy framework using the fourier transform

Abstract

The aim of this dissertation is to explore the use of the Fourier transform in the pricing of European call options when the underlying; asset price is modeled using a Levy process. We first examine the elementary theory of Levy processes up to and including the Levy-Khintchine representation, following which a discussion on how to construct a financial market model using Levy-driven asset price processes is given. Two specific Levy processes to be used as such in subsequent empirical investigations, namely the Variance-Gamma process and the CGMY process, are analyzed in light of earlier discussions. Subsequently, two pioneering Fourier transform-based option pricing methods emerging from the work of Heston (1993) and Carr and Madan (1999) respectively are discussed in detail and implemented using MATLAB and Mathematica. Finally, we carry out a number of empirical investigations primarily designed to illustrate the improved accuracy of the aforementioned methods in capturing the behaviour of market option prices over the popular Black-Scholes option pricing model as well as the computational efficiency of the method due to Carr and Madan in light of its use of the Fast Fourier transform algorithm.