The VIX Future in Bergomi Models: Fast Approximation Formulas and Joint Calibration with S/&P 500 Skew^∗

Using the exponential generating function of Hermite polynomials, we expand the prices of VIX power payoffs (including VIX futures) in various Bergomi models at any order in the volatility-of-volatility. We introduce the notion of volatility of the VIX squared implied by a VIX power payoff, which we call ``implied VIX² volatility"" and also expand at any order. We cover the one-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, we provide the expansions up to order 8 in closed form; otherwise, they simply involve one-dimensional integrals, which are extremely fast to compute. Extensive numerical experiments show that the implied volatility expansion converges much faster than the price expansion and is extremely accurate for a wide range of model parameters, including typical market calibrating parameters with very large volatilities-of-volatility. It leads to new, simple approximation formulas for the price of a VIX power payoff that shed light on how those prices depend on model parameters. We combine the new expansion and the Bergomi–Guyon expansion of the vanilla smile [L. Bergomi and J. Guyon, Risk, May (2012), pp. 60–66] to calibrate the two-factor Bergomi model jointly to the term-structures of S\&P 500 at-the-money skew and VIX futures. Very interestingly, the joint fit selects (1) much larger values of volatility-of-volatility and mean reversion than those previously reported in [L. Bergomi, Risk, October (2005), pp. 67–73] and [L. Bergomi, Stochastic Volatility Modeling, CRC Press, 2016], and (2) fully correlated Brownian motions, thus producing a (Markovian) pure path-dependent volatility model with rough-like paths.