Warnings About Future Jumps: Properties of the Exponential Hawkes Model

We quantify specific jump risks for financial asset prices modeled by Ito semimartingales<br>where the jump counter is an exponential Hawkes process. First, when<br>a cluster of jumps is observed we quantify the probability that the cluster is not<br>yet exhausted, using very little information on event times. Second, we study the<br>positive correlation of the durations between consecutive jumps by proving that two<br>successive durations are stochastically increasing. Third, given an evenly spaced time<br>grid, we provide bounds for the probability of occurrence of jumps in consecutive intervals,<br>using the information on which unit intervals contain jumps. An application<br>to 5-minute US returns shows that the probability that an observed cluster did not<br>exhaust yet by a given date varies among assets and sectors. Further, the magnitude<br>of the probabilities is marginally linked to the trading volume, whereas they<br>arestrictly related to the expected number of jumps.