Functional Central Limit Theorems for Rough Volatility
30 Pages Posted: 30 Nov 2017 Last revised: 10 May 2019
Date Written: November 28, 2017
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
Suggested Citation: Suggested Citation