An introduction to L\'evy processes with applications in finance
Abstract
These lectures notes aim at introducing L\'evy processes in an informal and intuitive way, accessible to non-specialists in the field. In the first part, we focus on the theory of L\'evy processes. We analyze a `toy' example of a L\'evy process, viz. a L\'evy jump-diffusion, which yet offers significant insight into the distributional and path structure of a L\'evy process. Then, we present several important results about L\'evy processes, such as infinite divisibility and the L\'evy-Khintchine formula, the L\'evy-It\o decomposition, the It\o formula for L\'evy 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 L\'evy processes in financial modeling and option pricing. We discuss how the price process of an asset can be modeled using L\'evy processes and give a brief account of market incompleteness. Popular models in the literature are presented and revisited from the point of view of L\'evy processes, and we also discuss three methods for pricing financial derivatives. Finally, some indicative evidence from applications to market data is presented.