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

TL;DR

This paper links path-dependent PDEs to stochastic Volterra equations in rough volatility models, deriving optimal weak convergence rates for discretized fractional Brownian motion integrals used in financial modeling.

Contribution

It establishes the unique classical solutions to path-dependent PDEs in rough volatility models and determines optimal weak convergence rates for discretized fractional Brownian motion integrals.

Findings

Conditional expectations solve path-dependent PDEs.

Optimal weak error rate of order 1 for quadratic test functions.

Weak error rate of (3H+1/2)∧1 for five-times differentiable functions.

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 It\^o 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 $H$.