Warnings About Future Jumps: Properties of the Exponential Hawkes Model

Abstract

Having observed a cluster of jumps in the discrete prices of a financial asset, we study and quantify the probability that the cluster is going to produce further jumps. Modeling the jump arrival times with an exponential Hawkes process, we provide some bounds for the future stochastic jump intensity under different hypotheses about the concentration of the past jumps arrivals. Moreover, we formalize the stochastic increasing property of the durations between two consecutive jumps and study the relation between the intensity decay time and the length of the possible clusters. Consequently, we provide formulas for the probability that, conditionally on the observation of a cluster of jumps, the generating phenomenon will produce a further jump, and bounds for the probability of observing a given number of consecutive jumps. 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, the probability that a cluster of more than 1 jump produces a further jump is mostly close to 1.