A switching self-exciting jump diffusion process for stock prices

Notice that stochastic volatility models also explain assymmetry and high kurtosis. However Cont and Tankov (2004) mention on page 6 that Brownian models with stochastic volatility cannot explain the presence of jumps in price due to the continuity of sample paths.

Notice that if \(C\in [u^{*},\omega _{2})\), the function \(F_{\omega _{1}}(C)\) is equal to

We can think to relate thresholds to regimes. For example, if we denote by \(\tilde{\sigma }_{i}^{ML}\) the volatility of the SGBM in the most likely state at time \(t_{i}\), we can assume that \(g(\alpha _{k},i)=\sqrt{\Delta }\tilde{\sigma }_{i}^{ML}\Phi ^{-1}(\alpha _{k})\quad k=1,2\). However, when this method is applied to the S&P 500 data set, fewer jumps are detected during recessions than in periods of growth. This counter intuitive result is explained by the fact that thresholds are proportional to the volatility in each regime. As the volatility is much important during recessions than in other periods, thresholds are also much higher. Consequence: less log-returns exceed thresholds during bad economic times. This observation motivates us to not relate thresholds to regimes and to use instead the standard deviation of the whole sample. This reduces the accuracy of the POT method. However, given that we use it to find an acceptable starting point for the MCMC algorithm, this loss of accuracy has a limited impact on final conclusions.

In theory the acceptance rate can be improved to any desired level. Increasing the number of particles raises considerably the computational time of the estimation procedure.