Option Pricing with Discrete Time Jump Processes

In this paper we propose new option pricing models based on two classes of discrete Lévy-type processes. By combining these Lévy processes with several volatility dynamics of the GARCH type, we aim to take into account the dynamics of financial returns in a realistic way. The associated risk neutral dynamics of the time series models is obtained through two different specifications for the pricing kernel: we provide a characterization of the change in the probability measure using the Esscher transform and the Minimal Entropy Martingale Measure. We finally assess empirically the performance of this modelling approach, using a dataset of European options based on the CAC 40. Our empirical findings show that discrete time Lévy based models behave well regardless of the volatility dynamics and of the pricing kernel. A different specification is however required to obtain the best results for options with a shorter or longer maturity