Functional Central Limit Theorems for Rough Volatility

Abstract

We extend Donsker’s approximation of Brownian motion to fractional Brownian motion with Hurst exponent H∈(0,1) and to Volterra-like processes. Some of the most relevant consequences of our ‘rough Donsker (rDonsker) Theorem’ are convergence results for discrete approximations of a large class of rough 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 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.

Keywords: functional limit theorems, Gaussian processes, invariance principles, fractional Brownian motion, rough volatility, binomial trees

JEL Classification: G20, G99, G60, B25