Least squares Monte Carlo methods in stochastic Volterra rough volatility models

Henrique Guerreiro and João Guerra

Abstract

In stochastic Volterra rough volatility models, the volatility follows a truncated Brownian semistationary process with stochastic volatility of volatility (vol-of-vol). Recently, efficient Chicago Board Options Exchange Volatility Index (VIX) pricing Monte Carlo methods have been proposed for cases where the vol-of-vol is Markovian and independent of the volatility. Using recent empirical data, we discuss the VIX option pricing problem for a generalized framework of these models, where the vol-of-vol may depend on the volatility and/or may not be Markovian. In such a setting, the aforementioned Monte Carlo methods are not valid. Moreover, the classical least squares Monte Carlo faces exponentially increasing complexity with the number of grid time steps, while the nested Monte Carlo method requires a prohibitive number of simulations. By exploring the infinite-dimensional Markovian representation of these models, we devise a scalable least squares Monte Carlo for VIX option pricing. We apply our method first under the independence assumption for benchmarks and then to the generalized framework. We also discuss the rough vol-of-vol setting, where Markovianity of the vol-of-vol is not present. We present simulations and benchmarks to establish the efficiency of our method as well as a comparison with market data.