On the use of Lèvy processes in option pricing

Abstract

In this dissertation, we t various nancial models to observed stock prices and we calculate the option prices under each of these models. All of the models considered are based on Lévy processes, which are processes with independent and identically distributed increments. The processes are popular in nance due to their exibility and their desirable mathematical properties. The models considered include the celebrated Black-Scholes model, under which the log-retuns are assumed to be driven by a Brownian motion. Two other classes of models are included in this study, both of which are generalizations of the Black-Scholes model. The rst class is the geometric Lévy process models, of which the Black-Scholes is a special case. Two speci c examples within this class are considered, the two models use the normal inverse Gaussian and Meixner processes to model log-returns. The second class of model considered generalizes the Black-Scholes while modeling the passing of time using an increasing stochastic process. The two speci c examples considered models time using a Pareto and a lognormal process. The aim of this dissertation is to explore the question of which model to use in a given nancial market. To this end, we t each of the models considered to observed log-returns. Following this step we calculate the prices of options available in this market. This is done in order to compare the prices calculated under the models to the prices observed in the market. In each case the Esscher transform is used in order to calculate the equivalent martingale measure used for the calculation of the option prices. Note that this is not the approach typically employed by nancial practitioners. In practice these models are often calibrated to the observed option prices, meaning that the parameters of the models are chosen so as to minimise some distance measure between the observed and calculated option prices. In this dissertation we depart from this methodology in order to determine if the models tted to the stock prices are capable of producing realistic option prices. When analysing the results obtained we use a two fold approach. The rst step is to determine which of the models considered provides the best t to the observed log-returns (this is done by comparing the integrated squared errors between the resulting densities and a kernel density estimate), and the second step is to compare the calculated and observed option prices (using the root mean square error calculated between the two sets of option prices). We conclude that, surprisingly, the model that ts the stock price data best often does not provide an adequate t to the option prices, and vice versa.