Option Pricing Driven by Lévy Processes

Abstract

The methodology of pricing financial derivatives, particularly stock options, was first introduced by Bachelier and developed by Black, Scholes and Merton, who gave the explicit formula for option pricing. Recent developed models such as jump-diffusion, Heston and Variance Gamma are also widely studied within the quantitative finance field and are proven to be applicable to a certain degree in real markets. A brief understanding of option pricing with stochastic processes is given in this thesis. Risk neutral valuation and notion of finding an equivalent martingale measure provide a framework under which derivatives are priced. Basic procedures of constructing a Brownian motion and stochastic integral from fundamental blocks are introduced. Infinitely divisible distributions and Lévy processes are detailedly discussed, including Lévy-Itô decomposition and the notion of subordination. Exponential-Lévy model and Fourier transform methods are presented to illustrate different approaches to option pricing. Simulation of AAPL stock prices based on estimated parameters from historical data under jump diffusion model is compared with empirical data to test the fitness of the model. Stock prices by minimal measure and Esscher transform measure are computed under geometric Lévy processes. Finally, univariate Variance Gamma process model is extended to Sato's two factor model for multivariate option pricing. The focus of this thesis is to give a detailed analysis of different option pricing models using mathematical and statistical concepts and theories, accompanied with simulations and empirical data to test the fitness of models. Extensions to numerous popular models are also discussed.