Warnings about future jumps: properties of the exponential Hawkes model

We analyze jump risks in financial asset prices modeled by Ito semimartingales with an exponential Hawkes process as the jump counter. First, using little information, we estimate the probability that an observed jump cluster is not yet exhausted. Second, we make explicit the conditional density of consecutive jump durations and prove that durations stochastically increase. Third, we provide bounds for jump probabilities in consecutive time intervals. Application to 5-minute U.S. returns shows that cluster depletion probabilities strongly correlate with the expected yearly jump count, and improve jump forecasts.