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

Abstract

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 VIX$^2$ 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.