Weak error rates of numerical schemes for rough volatility

TL;DR

This paper investigates the weak error convergence rates of numerical schemes used to simulate rough volatility models involving fractional Brownian motion with Hurst index H, providing explicit rates for specific cases.

Contribution

It derives the convergence rates of weak errors for discretization schemes in rough volatility models, highlighting differences between exact and hybrid schemes.

Findings

Convergence rate of order (3H+0.5)∧1 for exact left-point discretization.

Convergence rate of order H+0.5 for hybrid schemes with optimal weights.

Results applicable to simulation accuracy in rough volatility modeling.

Abstract

Simulation of rough volatility models involves discretization of stochastic integrals where the integrand is a function of a (correlated) fractional Brownian motion of Hurst index $H \in (0,1/2)$. We obtain results on the rate of convergence for the weak error of such approximations, in the special cases when either the integrand is the fBm itself, or the test function is cubic. Our result states that the convergence is of order $(3H+ \frac{1}{2}) \wedge 1$ for exact left-point discretization, and of order $H+\frac{1}{2}$ for the hybrid scheme with well-chosen weights.