Rough volatility, path-dependent PDEs and weak rates of convergence

Paper: arXiv 2304.03042

Authors: Ofelia Bonesini, Antoine Jacquier, Alexandre Pannier

Abstract

In the setting of stochastic Volterra equations, and in particular rough volatility models, we show that conditional expectations are the unique classical solutions to path-dependent PDEs. The latter arise from the functional Itô formula developed by [Viens, F., & Zhang, J. (2019). A martingale approach for fractional Brownian motions and related path dependent PDEs. Ann. Appl. Probab.]. We then leverage these tools to study weak rates of convergence for discretised stochastic integrals of smooth functions of a Riemann-Liouville fractional Brownian motion with Hurst parameter $H \in (0,\frac{1}{2})$. These integrals approximate log-stock prices in rough volatility models. We obtain the optimal weak error rates of order $1$ if the test function is quadratic and of order $(3H+\frac{1}{2})\wedge1$ if the test function is five times differentiable; in particular these conditions are independent of the value of $H$.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 2.0/10
• Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a highly sophisticated theoretical framework using path-dependent PDEs and functional Itô calculus to establish optimal weak convergence rates in rough volatility models. However, it is purely theoretical and lacks empirical backtesting or real-world market data implementation.

Research Flowchart

  flowchart TD
    A["Research Goal:<br>Analyze Path-Dependent PDEs & Weak Convergence Rates in Rough Volatility"] --> B["Inputs:<br>Stochastic Volterra Equations & Riemann-Liouville fBM (H ∈ (0, 0.5))"]
    
    B --> C["Methodology:<br>Apply Functional Itô Formula (Viens & Zhang) to Rough Models"]
    
    C --> D["Theoretical Outcome:<br>Prove Conditional Expectations are Unique Solutions to Path-Dependent PDEs"]
    
    D --> E["Computational Process:<br>Analyze Discretised Stochastic Integrals approximating Log-Stock Prices"]
    
    E --> F["Optimal Weak Error Rates:<br>• Order 1 (Quadratic Test Functions)<br>• Order (3H + 1/2) ∧ 1 (C⁵ Test Functions)"]