A regularity structure for rough volatility

Christian Bayer,Jorg Martin,Paul Gassiat,Benjamin Stemper,Peter K Friz

doi:10.1111/mafi.12233

Abstract

AbstractA new paradigm has emerged recently in financial modeling: rough (stochastic) volatility. First observed by Gatheral et al. in high‐frequency data, subsequently derived within market microstructure models, rough volatility captures parsimoniously key‐stylized facts of the entire implied volatility surface, including extreme skews (as observed earlier by Alòs et al.) that were thought to be outside the scope of stochastic volatility models. On the mathematical side, Markovianity and, partially, semimartingality 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 provide a new and powerful tool to analyze rough volatility models.

Highlights

We are interested in stochastic volatility (SV) models given in Ito differential form (1.1)dSt/St = σtdBt ≡ vt(ω)dBt .Here, B is a standard Brownian motion and σt are known as stochastic volatility process
Many classical Markovian asset price models fall in this framework, including Dupire’s local volatility model, the SABR, Stein-Stein - and Heston model
In all named SV model, one has Markovian dynamics for the variance process, of the form dvt = g(vt)dWt + h(vt)dt; constant correlation ρ := d B, W t/dt is incorporated by working with a 2D standard Brownian motion W, W, B := ρW + ρW ≡ ρW + 1 − ρ2W

Summary

Introduction

We are interested in stochastic volatility (SV) models given in Ito differential form (1.1). Thanks to the contraction principle and fundamental continuity properties of Hairer’s reconstruction map, the problem is reduced to understanding a LDP for a suitable enhancement of the noise This approach requires (sufficiently) smooth coefficients, but comes with no growth restrictions which is quite suitable for financial modelling: we improve the Forde-Zhang (simple rough vol) short-time large deviations [19] such as to include f of exponential type, a defining feature in the works of Gatheral and coauthors [27, 4]. As a matter of fact, and without going in further detail, there is a well-defined (“Cole-Hopf”) Ito-solution u = u(t, x; ω), but if one considers the equation with ε-mollified noise, u = uε diverges with ε → 0 In this sense, there is a fundamental lack of approximation theory and no Stratonovich solution to KPZ exists. The reconstruction map uniquely maps modelled distributions to function / Schwartz distributions. (This can be seen as generalization of the sewing lemma, the essence of rough integration, see e.g. [23], which turns a collection of sufficiently compatible local expansions into one function / Schwartz distribution.) In the KPZ context, the (Cole-Hopf Ito) solution is obtained as reconstruction of the abstract (modelled distribution) solution U

The rough pricing regularity structure

The full rough volatility regularity structure

Rough Volterra dynamics for volatility

Numerical results