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. 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, 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.