Functional central limit theorems for rough volatility

TL;DR

This paper introduces a novel approximation method for simulating rough volatility models, enabling efficient Monte Carlo simulations and the development of binomial trees for early exercise options, validated by numerical experiments.

Contribution

It extends Donsker's approximation to fractional Brownian motion, providing a theoretical foundation and practical algorithms for rough volatility simulation.

Findings

Efficient simulation algorithm for rough volatility models.

Validation against benchmark Hybrid scheme with remarkable agreement.

First binomial tree scheme for rough volatility models with early exercise options.

Abstract

The non-Markovian nature of rough volatility processes makes Monte Carlo methods challenging and it is in fact a major challenge to develop fast and accurate simulation algorithms. We provide an efficient one for stochastic Volterra processes, based on an extension of Donsker's approximation of Brownian motion to the fractional Brownian case with arbitrary Hurst exponent $H \in (0,1)$. Some of the most relevant consequences of this `rough Donsker (rDonsker) Theorem' are functional weak convergence results in Skorokhod space for discrete approximations of a large class of rough stochastic volatility models. This justifies the validity of simple and easy-to-implement Monte-Carlo methods, for which we provide detailed numerical recipes. We test these against the current benchmark Hybrid scheme~\cite{BLP17} and find remarkable agreement (for a large range of values of~$H$). This rDonsker Theorem further provides a weak convergence proof for the Hybrid scheme itself, and allows to construct binomial trees for rough volatility models, the first available scheme (in the rough volatility context) for early exercise options such as American or Bermudan options.