Paper: arXiv 2412.21192

Authors: Ofelia Bonesini, Emilio Ferrucci, Ioannis Gasteratos, Antoine Jacquier

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 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.

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 5.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework applying rough differential equations (RDEs) to rough volatility by extending lead-lag approximations and joint lifts of Brownian motion and rough paths. While highly advanced mathematically and conceptually novel, the empirical section focuses mainly on numerical validation and a single calibration example rather than exhaustive backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Model Rough Volatility using Rough Differential Equations RDEs] --> B[Theory: Joint Lift of Brownian Motion W & Fractional Rough Path X]
    C[Inputs: Equity Market Data & Fractional Brownian Motion] --> D
    B --> D[Methodology: Extend Lead-Lag Scheme for Correlated Paths]
    D --> E[Computation: RDE Numerical Scheme & Model Calibration]
    E --> F[Outcome: Robust Joint Price-Volatility Framework & Successful Calibration]

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4 of 90 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.