Parameter estimation of Lévy processes

Abstract

Levy processes have become increasingly popular in mathematical finance because of their ability to capture the leptokurtic shape of stock returns and also the jumps observed in stock prices. In this dissertation we will present some of the theory and major results of Levy processes. In particular we shall focus on the Normal Inverse Gaussian and the Meixner process. Then we shall be looking at different parameter estimation methods for Levy processes, which can be split into two major categories: the parametric approach and nonparametric approach. For the nonparametric approach we shall consider a projection estimator proposed by Comte and Genon-Catalot [14] and also an estimator introduced by Rubin and Tucker [ 44]. In the parametric approach we consider the Integrated Sum of Squared Estimation proposed by Heathcote [28] and a Stochastic Programming method presented by Sant and Caruana [ 45]. Finally these methods of estimation are implemented on the Malta Stock Exchange Index and some results are compared were possible.