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

Ofelia Bonesini, Antoine Jacquier, Alexandre Pannier

TL;DR

The paper develops a path-dependent PDE framework for stochastic Volterra (rough volatility) models by extending the functional Itô calculus to singular kernels. It then translates conditional expectations into classical solutions of path-dependent PDEs and uses this structure to derive sharp weak convergence rates for Euler discretisations of rough-volatility log-prices, with a rate of order $1$ for quadratic payoffs and a rate of order $\bigstar(\Delta)$, where $\bigstar(\Delta)$ depends on the Hurst parameter $H$ but not on its value in the leading regime. The approach combines a telescopic error decomposition, Malliavin integration by parts, and precise fractional-calculus estimates to handle the non-Markovian, non-semimartingale volatility process. The results legitimize PDE-based analysis for rough volatility and offer explicit, near-optimal weak rates that inform practical simulation and pricing methods. Overall, the work advances the theoretical toolbox for rough BV models and opens avenues for PDE-based numerical methods in stochastic Volterra settings.

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 $H \in (0,1/2)$. 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~$(3H+1/2)\wedge 1$ if the test function is five times differentiable; in particular these conditions are independent of the value of~$H$.

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

TL;DR

The paper develops a path-dependent PDE framework for stochastic Volterra (rough volatility) models by extending the functional Itô calculus to singular kernels. It then translates conditional expectations into classical solutions of path-dependent PDEs and uses this structure to derive sharp weak convergence rates for Euler discretisations of rough-volatility log-prices, with a rate of order for quadratic payoffs and a rate of order , where depends on the Hurst parameter but not on its value in the leading regime. The approach combines a telescopic error decomposition, Malliavin integration by parts, and precise fractional-calculus estimates to handle the non-Markovian, non-semimartingale volatility process. The results legitimize PDE-based analysis for rough volatility and offer explicit, near-optimal weak rates that inform practical simulation and pricing methods. Overall, the work advances the theoretical toolbox for rough BV models and opens avenues for PDE-based numerical methods in stochastic Volterra settings.

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 . These integrals approximate log-stock prices in rough volatility models. We obtain the optimal weak error rates of order~ if the test function is quadratic and of order~ if the test function is five times differentiable; in particular these conditions are independent of the value of~.
Paper Structure (33 sections, 58 theorems, 288 equations, 1 table)

This paper contains 33 sections, 58 theorems, 288 equations, 1 table.

Table of Contents

  1. Introduction
  2. Motivation
  3. Weak rates
  4. Path-dependent PDEs
  5. Method of proof
  6. Organisation of the paper
  7. The functional Itô formula and the path-dependent PDE
  8. Notations
  9. Definitions and the functional Itô formula
  10. A path-dependent PDE
  11. Application to rough volatility
  12. Weak rates of convergence for rough volatility
  13. The main result
  14. Decomposition of the error
  15. Proof of Theorem \ref{['th:main']}-Case 1
  16. ...and 18 more sections

Key Result

Theorem 2.10

Under Assumptions assu:wellposedness-assu:RegTimeDerivative, suppose $U\in \mathcal{C}^{1,2}_{+,\alpha}$, with $\alpha>\frac{1}{2}-H$, $H\in(0,\frac{1}{2})$. Then the spatial derivatives eq:SpatialDerivatives exist and the following functional Itô formula holds:

Theorems & Definitions (125)

  • Definition 2.2
  • Example 2.3
  • Definition 2.4
  • Definition 2.5
  • Definition 2.7
  • Remark 2.8
  • Remark 2.9
  • Theorem 2.10
  • Remark 2.11
  • Proposition 2.13
  • ...and 115 more