  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 [Bergomi, Smile dynamics II, Risk,
2005], the rough Bergomi model [Bayer, Friz, and Gatheral, Pricing under rough
volatility, Quantitative Finance, 16(6):887-904, 2016], and an enhanced version
of the rough model that can generate realistic positive skew for VIX smiles --
introduced simultaneously by De Marco [Bachelier World Congress, 2018] and
Guyon [Bachelier World Congress, 2018] on the lines of [Bergomi, Smile dynamics
III, Risk, 2008], that we refer to as 'mixed rough Bergomi model'. Following
the methodology set up in [Gobet and Miri, Weak approximation of averaged
diffusion processes. Stochastic Process.\ Appl., 124(1):475-504, 2014], 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
to encompass VIX futures, call and put options. 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.
