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

TL;DR

This paper develops a multilevel Monte Carlo method to efficiently price VIX options under the rough Bergomi model, significantly reducing computational costs compared to traditional Monte Carlo approaches.

Contribution

It introduces a multilevel Monte Carlo scheme combined with discretization techniques to lower the computational complexity for VIX option pricing in the rough Bergomi model.

Findings

Cost reduced from O(ε^{-4}) to O(ε^{-2} log^2(ε))

Further reduction to O(ε^{-2}) with trapezoidal discretization

Numerical experiments confirm efficiency improvements

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.