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

Abstract

We consider the pricing of VIX options in the rough Bergomi model [Bayer, Friz, and Gatheral, Pricing under rough volatility, Quantitative Finance 16(6), 887-904, 2016]. 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 $\mathcal O(\varepsilon^{-4})$ when the mean-squared error is set to $\varepsilon^2$. We demonstrate that this cost can be reduced to $\mathcal O(\varepsilon^{-2} \log^2(\varepsilon))$ combining the scheme above with the multilevel method [Giles, Multilevel Monte Carlo path simulation, Oper. Res. 56(3), 607-617, 2008], and further reduced to the asymptotically optimal cost $\mathcal O(\varepsilon^{-2})$ 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.