Jump-diffusion models driven by Lévy processes

Abstract: During the past and this decade, a new generation of continuous-time financial models has been intensively investigated in a quest to incorporate the so-called stylized empirical features of asset prices like fat-tails, high kurtosis, volatility clustering, and leverage. Modeling driven by “memoryless homogeneous” jump processes (Lévy processes) constitutes one of the most plausible directions in this enterprise. The basic principle is to replace the underlying Brownian motion of the Black-Scholes model with a type of jump-diffusion process. In this chapter, the basic results and tools behind jump-diffusion models driven by Lévy processes are covered, providing an accessible overview, coupled with their financial applications and relevance. The material is drawn upon recent monographs (c.f. [18], [47], [46]) and papers in the field.