Least squares Monte Carlo methods in stochastic Volterra rough volatility models

Abstract

In stochastic Volterra rough volatility models, the volatility follows a truncated Brownian semi-stationary process with stochastic vol-of-vol. Recently, efficient VIX pricing Monte Carlo methods have been proposed for the case where the vol-of-vol is Markovian and independent of the volatility. Following 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 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, whilst the nested Monte Carlo method requires a prohibitive number of simulations. By exploring the infinite dimensional Markovian representation of these models, we device a scalable least squares Monte Carlo for VIX option pricing. We apply our method firstly 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. Keywords— VIX, rough volatility, stochastic Volterra models, least squares Monte Carlo, volatility of volatility ∗Supported by FCT Grant SFRH/BD/147161/2019. †Partially supported by the project CEMAPRE/REM-UiDB/05069/2020 financed by FCT/MCTES through national funds. 1 ar X iv :2 10 5. 04 51 1v 1 [ qfi n. PR ] 1 0 M ay 2 02 1