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

We provide explicit approximation formulas for VIX futures and options in forward variance models, with particular emphasis on the family of so-called Bergomi models: the one-factor Bergomi model, the rough Bergomi model, and an enhanced version of the rough model that can generate realistic positive skew for VIX smiles-introduced simultaneously by De Marco and Guyon at the Bachelier World Congress 2018, that we refer to as 'mixed rough Bergomi model'. Following the methodology set up in previous works of Gobet and Miri on asymptotic approximations for integrated diffusion processes, we derive weak approximations for the law of the VIX, leading to option price approximations under the form of explicit combinations of Black-Scholes prices and greeks. As new contributions, we cope with the fractional integration kernel appearing in rough models, and treat the case of non-smooth payoffs, so as to encompass VIX futures, call, and put options. We stress that our approach does not rely on small maturity asymptotics, and can therefore be applied to any VIX option maturity. Our results are illustrated by several numerical experiments over a wide range of model parameter configurations, and by calibration tests to VIX market data.