An introduction to Lévy processes with applications in finance

Abstract

These lectures notes aim at introducing Lévy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of Lévy processes. We analyze a `toy' example of a Lévy process, viz. a Lévy jump-diffusion, which yet offers significant insight into the distributional and path structure of a Lévy process. Then, we present several important results about Lévy processes, such as infinite divisibility and the Lévy-Khintchine formula, the Lévy-Itô decomposition, the Itô formula for Lévy processes and Girsanov's transformation. Some (sketches of) proofs are presented, still the majority of proofs is omitted and the reader is referred to textbooks instead. In the second part, we turn our attention to the applications of Lévy processes in financial modeling and option pricing. We discuss how the price process of an asset can be modeled using Lévy processes and give a brief account of market incompleteness. Popular models in the literature are presented and revisited from the point of view of Lévy processes, and we also discuss three methods for pricing financial derivatives. Finally, some indicative evidence from applications to market data is presented.