Long memory self-exciting jump diffusion for asset prices modeling

We propose a model for asset prices driven by a self-exciting jump diffusion process. The novel feature of our model is the shocks ruled by a Hawkes process with a Mittag-Leffler memory kernel. The Mittag-Leffler kernel is a relaxation function used in fractional calculus to describe complex memory effect. In particular, it generalizes the exponential memory that ensures Hawkes process has the Markov property. Despite its interesting characteristics, the Hawkes process with the Mittag-Leffler kernel is a non-markov process. Nevertheless, we derive a closed form expression for the moment generating function of log-returns. It is obtained by representing the Hawkes process with Mittag-Leffler kernel as an infinite dimensional Markov process. Furthermore, we provide a change of measure and price European options by exploiting the fast Fourier transform technique. Applied to the times series of S&P 500 daily values, we illustrate the efficiency of the proposed model to produce excess of kurtosis, skewness compared to model with memoryless. We show that the prices of call option on S&P 500 are sensitive to the memory parameter of our model.