Multilevel Monte Carlo simulation for VIX options in the rough Bergomi model

Florian Bourgey and Stefano De Marco

Abstract

In this paper we consider the pricing of Chicago Board Options Exchange Volatility Index (VIX) options in the rough Bergomi model. In this setting, the VIX random variable is defined by the one-dimensional integral of the exponential of a Gaussian process with correlated increments. Hence, approximate samples of the VIX can be constructed via discretization of the integral and simulation of a correlated Gaussian vector. A Monte Carlo estimator of VIX options based on a rectangle discretization scheme and exact Gaussian sampling via the Cholesky method has a computational complexity of order Ο(ε⁻⁴) when the mean squared error is set to ε². We demonstrate that this cost can be reduced to Ο(ε^-2ln²(ε)) by combining the scheme above with the multilevel method, and further reduced to the asymptotically optimal cost Ο(ε⁻²) when using a trapezoidal discretization. We provide numerical experiments highlighting the efficiency of the multilevel approach in the pricing of VIX options in such a rough forward variance setting.