Rough differential equations for volatility

TL;DR

This paper develops a new framework for modeling rough volatility using rough differential equations by jointly lifting Brownian motion and fractional Brownian motion, enabling more flexible and correlated stochastic volatility models.

Contribution

It introduces a canonical joint lift of Brownian motion and rough paths, extending rough volatility modeling to correlated fractional Brownian motions within the RDE framework.

Findings

Successfully models rough volatility with correlated processes.

Provides a numerical scheme for solving the joint RDEs.

Calibrates a new rough volatility model to market data.

Abstract

We introduce a canonical way of performing the joint lift of a Brownian motion $W$ and a low-regularity adapted stochastic rough path $\mathbf{X}$, extending [Diehl, Oberhauser and Riedel (2015). A L\'evy area between Brownian motion and rough paths with applications to robust nonlinear filtering and rough partial differential equations]. Applying this construction to the case where $\mathbf{X}$ is the canonical lift of a one-dimensional fractional Brownian motion (possibly correlated with $W$) completes the partial rough path of [Fukasawa and Takano (2024). A partial rough path space for rough volatility]. We use this to model rough volatility with the versatile toolkit of rough differential equations (RDEs), namely by taking the price and volatility processes to be the solution to a single RDE. We argue that our framework is already interesting when $W$ and $X$ are independent, as correlation between the price and volatility can be introduced in the dynamics. The lead-lag scheme of [Flint, Hambly, and Lyons (2016). Discretely sampled signals and the rough Hoff process] is extended to our fractional setting as an approximation theory for the rough path in the correlated case. Continuity of the solution map transforms this into a numerical scheme for RDEs. We numerically test this framework and use it to calibrate a simple new rough volatility model to market data.