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

Abstract

We derive the expansion of the price of a VIX future in various Bergomi models at order 6 in small volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by the VIX future, which we call "VIX2 implied volatility", expand this quantity at order 5, and show that the implied volatility expansion converges much faster than the price expansion. We cover the one-factor, two-factor, and skewed two-factor Bergomi models and allow for maturity-dependent and/or time-dependent parameters. 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 in Bergomi models. The derivation of the expansion naturally involves the (classical or dual bivariate) Hermite polynomials and exploits their orthogonality properties. When the initial term-structure of variance swaps is flat, the expansion is a closed-form expression; otherwise, it involves one-dimensional integrals which are extremely fast to compute. The VIX2 implied volatility expansion is extremely precise for both the one-factor model and the two-factor model with independent factors, even for the very large values of volatility-of-volatility that are usual in equity derivatives markets, and can virtually be considered an exact formula in those cases. We use the new expansion 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 expansion 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).

Keywords: Vix, Vix Futures, Bergomi Models, VIX2 Implied Volatility, Analytic Expansion, Small Volatlity-of-Volatility, At-the-Money Skew, S&P 500/Vix Joint Calibration, Hermite Polynomials

JEL Classification: G13