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.


  1. VIX
  2. VIX futures
  3. VIX power payoffs
  4. Bergomi models
  5. implied VIX$^2$ volatility
  6. volatility-of-volatility expansion
  7. at-the-money skew
  8. S&P 500/VIX joint calibration
  9. path-dependent volatility
  10. Hermite polynomials

MSC codes

  1. 91G20
  2. 91G80
  3. 60H30

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Information & Authors


Published In

cover image SIAM Journal on Financial Mathematics
SIAM Journal on Financial Mathematics
Pages: 1418 - 1485
ISSN (online): 1945-497X


Submitted: 30 July 2021
Accepted: 15 July 2022
Published online: 8 December 2022


  1. VIX
  2. VIX futures
  3. VIX power payoffs
  4. Bergomi models
  5. implied VIX$^2$ volatility
  6. volatility-of-volatility expansion
  7. at-the-money skew
  8. S&P 500/VIX joint calibration
  9. path-dependent volatility
  10. Hermite polynomials

MSC codes

  1. 91G20
  2. 91G80
  3. 60H30



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