Lévy Processes in Finance

Abstract

This thesis deals with applications of Lévy processes in the field of mathematical finance. The class of Lévy processes generalizes the widely used Brownian motion in the sense that the distribution of their increments can model the statistically observed heavy-tailed and skewed returns of a broad range of exchange-traded assets.<br /> The problem of the incompleteness of a Lévy driven market is dealt with in the first part. Incompleteness leads to non-uniqueness of the martingale measure and hence to the problem of selecting a specific one. Two different solutions are proposed: The first is an ad hoc change of measure based on qualitative economic considerations whereas the second gives a simple method of how to fit a carefully chosen family of martingale measures to observed option prices.<br /> While the first part has focused on scalar modelling issues, the second one is concerned about multidimensional Lévy processes. A two-dimensional asset price model is proposed where the marginal distributions incorporate heavy tails and skewness and, in addition, the dependence between the two assets is modeled by a mixture of the well-known linear dependence and a computationally tractable form of tail dependence. For this model a method of approximating prices of basket options is proposed and compared to a fast Monte Carlo method as well as to prices obtained in the standard two-dimensional diffusion model. Finally, it is shown how hedging of multidimensional options can be done in a Lévy model.