Convergence of Heavy-Tailed Hawkes Processes and the Microstructure of Rough Volatility

TL;DR

This paper proves that nearly-unstable Hawkes processes with heavy-tailed kernels converge to a rough volatility model, providing new insights into the microstructure of financial markets and the emergence of rough volatility.

Contribution

It establishes the weak convergence of heavy-tailed Hawkes processes to a rough Heston model, extending previous results to heavy-tailed kernels and stronger convergence types.

Findings

Hawkes process intensities converge to rough volatility models

Introduces new methods for establishing tightness of stochastic processes

Provides a scaling limit for market microstructure models

Abstract

We establish the weak convergence of the intensity of a nearly-unstable Hawkes process with heavy-tailed kernel. Our result is used to derive a scaling limit for a financial market model where orders to buy or sell an asset arrive according to a Hawkes process with power-law kernel. After suitable rescaling the price-volatility process converges weakly to a rough Heston model. Our convergence result is stronger than previously established ones that have either focused on light-tailed kernels or the convergence of integrated volatility process. The key is to establish the tightness of the family of rescaled volatility processes. This is achieved by introducing a new methods to establish the $C$-tightness of c\`adl\`ag processes based on the classical Kolmogorov-Chentsov tightness criterion for continuous processes.