Warnings about Future Jumps: Properties of the Exponential Hawkes Model

When a financial trader observes a cluster of jumps in the price evolution of an asset, she could be interested in understanding whether the cause of such a cluster is going to produce further jumps or if instead its effects are already diminishing and the price will return to its “usual" fluctuations.

In this work, we aim to capture the dependence mechanism of the times occurrence of the jumps, in order to quantify the probability that an observed cluster did not exhaust yet. 

The risk underlying the prices of an asset is split into two categories: the risk of “continuous", “predictable" and usually relatively small price adjustments; and the risk of abrupt, “abnormal" and possibly large fluctuations. The two types of risk are modeled by two radically different processes: a Brownian motion with stochastic volatility and a jump process. Based on [3], with discrete observations a jump is detected when a return has size above a proper threshold. 

In order to capture possible clustering of the jump arrivals, inspired by [1], we use a univariate Hawkes model, where any occurred jump increases the intensity. Both after data fitting tests and in view of its properties and mathematical tractability, we focus on the specification with a one-term exponential kernel, and we obtain new theoretical properties which are useful for our purpose.

Namely, conditionally on having observed a cluster of jumps, we quantify probabilistically the residual length of the cluster. We then formalize the stochastic increasingness property of the durations between two consecutive jumps, which strengthens their positive correlation. Finally we provide bounds for the probability of observing a given number of consecutive jumps, which takes into account the error due to the fact that the observaions are in discrete time.

We perform an illustration on how to apply the obtained formulas to empirical data. In particular we focus on a record of JPM's asset prices, where the identified jump arrivals display dependence and clustering behavior. We find that, under the exponential Hawkes model delivering the best fitting, our formulas indicate a very high probability that an observed cluster of more than 1 jump did not exhaust yet.