A short note on option pricing with Lévy Processes.

In this paper, we provide exact formulas for the pricing of European options under the risk neutral measure, whereas under the historic measure the data follow two types of models : a GARCH process with Lévy innovations, or a GARCH process with Poisson jumps. This approach aims to take realistic account of the jumps that are observed in the markets and to introduce them into the theory of pricing in incomplete markets. We assume that the "pricing kenel" that can move from measurement historical risk-neutral measure can be obtained from the Esscher transform (Siu et al., 1994), or using the MEMM transformation introduced by Elliott and Madam (1998). We show how these two types of "pricing kernels" impact on the options prices and through an example we quantify the difference.