Combining a self-exciting point process with the truncated generalized Pareto distribution: An extreme risk analysis under price limits

In this paper, we introduce a general framework of the self-exciting point process with the truncated generalized Pareto distribution to measure the extreme risks in the stock markets under price limits. We incorporate the predictable marks, defined as the variance of mark distribution depending on the previous events via the intensity, into the model setting. The proposed process can well accommodate many important empirical characteristics, such as the thick-tailness, extreme risk clustering and price limits. We derive a closed-form solution for the objective likelihood, based on which the proposed model can be estimated via the standard maximum likelihood estimation algorithm. Furthermore, the closed-form measures of the Value-at-Risk and Expected Shortfall are also derived. For empirical illustration, we use the China Securities Index 300 (with ±10% price restriction) in the analysis. In general, the results from both in-sample fitting and out-of-sample forecasting measures show that the proposed process can explain the empirical data well. We also investigate the cascade effect of the China stock market by introducing the branching process to distinguish the endogenous risks from the exogenous risks.