A regularity structure for rough volatility

TL;DR

This paper introduces a novel application of Hairer's regularity structures to analyze rough volatility models in finance, capturing complex market behaviors beyond traditional stochastic methods.

Contribution

It extends the mathematical framework of regularity structures to rough volatility, enabling new analysis tools for non-Markovian and non-semimartingale models.

Findings

Regularity structures effectively analyze rough volatility models.

The approach captures stylized facts of implied volatility surfaces.

Provides a new mathematical framework for rough stochastic processes.

Abstract

A new paradigm recently emerged in financial modelling: rough (stochastic) volatility, first observed by Gatheral et al. in high-frequency data, subsequently derived within market microstructure models, also turned out to capture parsimoniously key stylized facts of the entire implied volatility surface, including extreme skews that were thought to be outside the scope of stochastic volatility. On the mathematical side, Markovianity and, partially, semi-martingality are lost. In this paper we show that Hairer's regularity structures, a major extension of rough path theory, which caused a revolution in the field of stochastic partial differential equations, also provides a new and powerful tool to analyze rough volatility models.