  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.
