From Hawkes-type processes to stochastic volatility

TL;DR

This paper introduces a Hawkes-like process and explores its scaling limit, revealing how high-frequency trading dynamics can lead to a Heston-type stochastic volatility model with negative correlation.

Contribution

It develops a new Hawkes-like process, derives its limit theorems, and connects high-frequency order flow to a stochastic volatility model resembling Heston's.

Findings

Derives functional limit theorems for the process.

Shows high-frequency order flow leads to Heston-type volatility.

Explains negative correlation as a consequence of order arrival variance.

Abstract

We introduce a Hawkes-like process and study its scaling limit as the system becomes increasingly endogenous. We derive functional limit theorems for intensity and fluctuations. Then, we introduce a high-frequency model for a price of a liquid traded financial instrument in which the nearly unstable regime leads to a Heston-type process where the negative correlation between the noise driving the proce of the instrument and the volatility can be viewed as a result of high variance of the sell-side order arrivals.