The variance-Hawkes process and its application to energy markets

We define a new model using a Hawkes process as a subordinator in a standard Brownian motion. We demonstrate that this Hawkes-subordinated Brownian motion, or, more succinctly, variance-Hawkes process, can be fitted to 2018 and 2019 New York Mercantile Exchange natural gas and West Texas Intermediate crude oil front-month futures log returns. This variance-Hawkes process allows clustering effects to be easily encoded into financial models’ behavior in a simple and tractable way. We also compare the simulations of a square-of-variance Hawkes process with its Itô formula. We simulate both processes and compare their distributions, trajectories and percentage errors across multiple runs. We derive the generator relating to this Hawkes-subordinated Brownian motion, calculate several moments and estimate its distribution. We also provide explicit solutions to the second moments of the Hawkes process and its intensity, as well as the cross-moment between the Hawkes process and its intensity in the case of an exponential kernel.