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 fit to 2018 and 2019 natural gas and crude oil front-month futures log returns. This variance-Hawkes process allows financial models to easily have clustering effects encoded into their behaviour in a simple and tractable way. We also compare the simulations of a square of a variance Hawkes process with its Ito formula. We simulate both processes and compare their distributions, trajectories, and percent errors across multiple runs. We derive the generator relating to this Hawkes subordinated Brownian motion, calculate several moments, and conjecture 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.