The VIX Future in Bergomi Models: Analytic Expansions and Joint Calibration with S&P 500 Skew

Abstract

We derive the formal expansion of the price of a VIX future (and more generally a VIX power payoff) in various Bergomi models at any order in powers of the volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by the VIX future, which we call implied volatility, and also expand this quantity at any order. We cover the one-factor, two-factor, and skewed two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. When the initial term-structure of variance swaps is flat, the expansions are in closed form; otherwise, they involve one-dimensional integrals which are extremely fast to compute. Our extensive numerical experiments show that both the price and implied volatility expansions provide very accurate approximations for a wide range of model parameters which covers the values typically used in the equity and FX derivatives markets and even well beyond---in particular, the expansions are accurate even for very large values of the volatility-of-volatility. They also show that the implied volatility expansion converges much faster than the price expansion. Its truncation at order 1 is already so accurate that it leads to a simple formula for the price of the VIX future that can virtually be considered exact. The derivation of the expansions naturally involves the (classical or dual bivariate) Hermite polynomials and exploits their orthogonality properties; it is interesting in itself. The expansions allow us to precisely pinpoint the roles of all the model parameters (volatility-of-volatility, mean reversions, correlations, mixing fraction) in the formation of the prices of VIX futures and VIX power payoffs in Bergomi models. We also use them together with the Bergomi-Guyon expansion of the S&P 500 smile to (instantaneously) calibrate the two-factor Bergomi model jointly to the term-structures of S&P 500 at-the-money skew and VIX2 implied volatility. Our tests and the new expansions shed more light on the inability of traditional stochastic volatility models to jointly fit S&P 500 and VIX market data. The (imperfect but decent) joint fit requires much larger values of volatility-of-volatility and fast mean reversion than the ones previously reported in Bergomi (2005, 2016).