A regularity structure for rough volatility

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.Dedicated to Professor Jim Gatheral on the occasion of his 60th birthday.Contents arXiv:1710.07481v1 [q-fin.PR]

Moment explosions in the rough Heston model

“…Since the pioneering work by Gatheral et al [15], the literature on these non-Markovian stochastic volatility models, inspired by fractional Brownian motion, has grown rapidly. We refer, e.g., to Bayer et al [4] for many references. In the present paper we provide some results on the explosion time and the critical moments of the rough Heston model.…”

Large Deviation Principle for Volterra Type Fractional Stochastic Volatility Models

“…Recall that we denoted by K H the kernel associated with fractional Brownian motion (see (4) and (5)). Formula (9) provides the Volterra type representation of the process U H , while the function K H in (10) is the Volterra type kernel associated with U H .…”

Functional Central Limit Theorems for Rough Volatility

“…These were first introduced by Comte and Renault [17], and later studied theoretically by Djehiche and Eddahbi [20], Alòs, León and Vives [3] and Fukasawa [31], and given financial motivation and data consistency by Gatheral, Jaisson and Rosenbaum [35] and Bayer, Friz and Gatheral [8]. Since then, a vast literature has pushed the analysis in many directions [7,9,12,26,28,35,36,40,47,64], leading to theoretical and practical challenges to understand and implement these models. One of the main issues, at least from a practical point of view, is on the numerical side: absence of Markovianity rules out PDE-based schemes, and simulation is the only possibility.…”