Warnings About Future Jumps: Properties of the Exponential Hawkes Model

Having observed a cluster of jumps in the discrete prices of a financial asset, and modeling the jump arrival times with an exponential Hawkes process, we study and quantify the probability that the cluster is going to produce further jumps. We also provide bounds for the probability of observing a given number of consecutive jumps and formalize the stochastic increasingness property of the durations between two consecutive jumps.<br><br>As an empirical exercise, we apply our results to a record of the JPM asset prices. Firstly we show that the identified jumps display dependence and clustering behavior. Secondly, we find that, under the exponential Hawkes model delivering the best QQplot, our formulas indicate that the probability that a cluster of more than 1 jump produces a further jump is mostly close to 1.