Rough differential equations for volatility

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 Lévy 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.

💡 Research Summary

The paper introduces a novel framework for modeling rough volatility by leveraging the theory of rough paths. The authors first identify two major obstacles in existing rough volatility models: (i) the infinite quadratic covariation between a fractional Brownian motion (or other low‑regularity driver) and the Brownian motion that drives the asset price, which prevents a Stratonovich formulation; and (ii) the lack of a canonical Lévy area when the Hurst exponent H ≤ 1/4, making classical rough‑path lifts unavailable.

To overcome these issues, Section 2 develops an “Itô‑lift” construction that jointly lifts an adapted rough path X and a multidimensional Brownian motion W into a geometric rough path. This lift incorporates all second‑order iterated integrals—including those that are not classically defined, such as ∫ W dX and ∫ X dW—by imposing integration‑by‑parts identities and preserving the geometric structure. Theorem 2.5 and 2.6 formalize the construction and show that it works when X is a stochastic rough path of low regularity, in particular the canonical lift of a one‑dimensional fractional Brownian motion possibly correlated with W.

Section 3 applies the joint lift to financial modeling. The authors propose a single system of rough differential equations (RDE) for the asset price S and its variance V: \