An introduction to L\'{e}vy processes with applications in finance

TL;DR

This paper introduces Lévy processes, explaining their theory and properties, and explores their applications in financial modeling and option pricing, including market data analysis.

Contribution

It provides an accessible overview of Lévy processes, their mathematical foundations, and their use in finance, including models and derivative pricing methods.

Findings

Lévy processes are useful for modeling asset prices.

Several methods for derivative pricing using Lévy processes are discussed.

Application to market data demonstrates practical relevance.

Abstract

These lectures notes aim at introducing L\'{e}vy 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\'{e}vy processes. We analyze a `toy' example of a L\'{e}vy process, viz. a L\'{e}vy jump-diffusion, which yet offers significant insight into the distributional and path structure of a L\'{e}vy process. Then, we present several important results about L\'{e}vy processes, such as infinite divisibility and the L\'{e}vy-Khintchine formula, the L\'{e}vy-It\^{o} decomposition, the It\^{o} formula for L\'{e}vy 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\'{e}vy processes in financial modeling and option pricing. We discuss how the price process of an asset can be modeled using L\'{e}vy 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\'{e}vy processes, and we also discuss three methods for pricing financial derivatives. Finally, some indicative evidence from applications to market data is presented.