Weak approximations and VIX option price expansions in forward variance curve models

Abstract

We provide explicit approximation formulas for VIX futures and options in stochastic forward variance models, with particular emphasis on the family of so-called Bergomi models: the one-factor Bergomi model of Bergomi [6], the rough Bergomi model of Bayer et al. [3], and an enhanced version of the rough model that can generate realistic positive skew for VIX smiles -introduced simultaneously by De Marco [13] and Guyon [20] on the lines of Bergomi [7], that we refer to as "mixed rough Bergomi model". Following the methodology set up by Gobet and Miri [17], we derive weak approximations for the law of the VIX random variable, leading to option price approximations under the form of explicit combinations of Black-Scholes prices and greeks. The new challenge we tackle is to handle the fractional integration kernel appearing in rough models and to deal with non-smooth payoffs. We stress that our approach does not rely on small-time asymptotics nor small-parameter (such as small volatility-of-volatility) asymptotics and can therefore be applied to any option maturity and a wide range of parameter configurations. Our results are illustrated by several numerical experiments and calibration tests to VIX market data.