Jump Diffusion and {α}-Stable Techniques for the Markov Switching Approach to Financial Time Series

We perform a detailed comparison between a Markov Switching Jump Diffusion Model and a Markov Switching {\alpha}-Stable Distribution Model with respect to the analysis of non-stationary data. We show that the jump diffusion model is extremely robust, flexible and accurate in fitting of financial time series. A thorough computational study involving the two models being applied to real data, namely, the S&P500 index, is provided. The study shows that the jump-diffusion model solves the over-smoothing issue stated in (Di Persio and Frigo, 2016), while the {\alpha}-stable distribution approach is a good compromise between computational effort and performance in the estimate of implied volatility, which is a major problem widely underlined in the dedicated literature.