Estimation of Lévy Processes in Mathematical Finance: A Comparative Study

In the field of mathematical finance, the concern with the applications of Lévy processes has been growing for the last several years. The models are aimed at incorporating stylized empirical facts. In the classical Black-Scholes option pricing model, the log return of asset is assumed to follow the normal distribution. However, compared to the normal distribution, the empirical density of log returns typically has more mass near the origin, less in the flanks, and more in the tail. Empirical works also suggest the discontinuity of the sample path of price processes. To account for these features, several models based on Lévy processes have been propounded in the literature. Prominent examples include the Poisson jump model (Merton (1976)), the variance gamma process (Madan and Seneta (1990), Madan et al. (1998)), the normal inverse Gaussian process (Barndorff-Nielsen (1998)), and the CGMY process (Carr et al. (2002)).