Change of measure in a Heston-Hawkes stochastic volatility model

TL;DR

This paper extends the Heston stochastic volatility model by incorporating a compound Hawkes process, establishing conditions for equivalent martingale measures and analyzing a specific case with exponential jump sizes.

Contribution

It introduces a novel combination of the Heston model with a compound Hawkes process and proves a general result on the existence of equivalent martingale measures.

Findings

Established existence of equivalent martingale measures for the combined model

Analyzed a specific case with exponentially distributed jump sizes

Provided theoretical foundation for future applications in finance and related fields

Abstract

We consider the stochastic volatility model obtained by adding a compound Hawkes process to the volatility of the well-known Heston model. A Hawkes process is a self-exciting counting process with many applications in mathematical finance, insurance, epidemiology, seismology and other fields. We prove a general result on the existence of a family of equivalent (local) martingale measures. We apply this result to a particular example where the sizes of the jumps are exponentially distributed.