Statistical and probabilistic properties of Hawkes processes and their application to the large variations of financial assets prices

In this work we study variables and events that can be exploited to forecast catastrophic episodes. Our objective is forecasting the occurrence of relevant depreciations of financial assets, given the observation of clusters of previous jumps in the series of the prices. In fact we believe that in periods preceding catastrophic episodes the probability of a jump increases after the occurrence of some jumps. Assuming a jump diffusion model for the log returns of a given asset we filter out and study the point process counting the number of the occurred jumps. We find that while the jumps of the S&P500 index are compatible with a Poisson process, the ones of the JPM asset are well described by a Hawkes model with an exponential kernel, which is characterized by the ability of capturing self-excitation mechanisms. We investigate on the dependence properties of the durations between consecutive jump times, as well as on the clusters length, and we furnish narrow bounds for the probability of the occurrence of a cluster as well as for the occurrence of a further jump after a cluster. The application of such formulas to the JPM jumps dataset gives quantication of the risk of jump which can be useful for porfolio management and forecasting aims.