Application of jump-diffusion and pure jump processes in financial modeling

This thesis provides a comprehensive examination of the integration of Lévy processes into asset pricing models, addressing the real complexities, jumps, and stochastic volatility present in financial markets. The research aims to bridge the gap between theoretical foundations and practical applications, offering researchers and practitioners a versatile toolkit for improving the modeling and understanding of financial markets. Using the Black-Scholes model as a starting point, the focus is on providing a more accurate representation of market dynamics by emphasizing jump-diffusion and pure jump Lévy processes. The thesis explores examples such as the Merton and Kou jump-diffusion processes, as well as pure jump models like the Hyperbolic, Normal Inverse Gaussian, Variance Gamma, and CGMY models, all of which enrich the understanding of asset pricing dynamics. Lévy processes are constructed through the linear transformations and exponential tilting of Lévy measures, aiming to better reflect the empirical characteristics of financial markets. Additionally, the thesis examines stochastic calculus for jump processes, particularly within the framework of Itô's formula, to better quantify the evolution of asset prices.