A switching self-exciting jump diffusion process for stock prices

This study proposes a new Markov switching process with clustering effects. In this approach, a hidden Markov chain with a finite number of states modulates the parameters of a self-excited jump process combined to a geometric Brownian motion. Each regime corresponds to a particular economic cycle determining the expected return, the diffusion coefficient and the long-run frequency of clustered jumps. We study first the theoretical properties of this process and we propose a sequential Monte-Carlo method to filter the hidden state variables. We next develop a Markov Chain Monte-Carlo procedure to fit the model to the S&P 500. We find that self-exciting jumps occur mainly during economic recession and nearly disappear in periods of economic growth. Finally, we analyse the impact of such a jump clustering on implied volatilities of European options.