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Weak approximations and VIX option price expansions in forward variance curve models

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  • Florian Bourgey
  • Stefano De Marco
  • Emmanuel Gobet

Abstract

We provide explicit approximation formulas for VIX futures and options in forward variance models, with particular emphasis on the family of so-called Bergomi models: the one-factor Bergomi model [Bergomi, Smile dynamics II, Risk, 2005], the rough Bergomi model [Bayer, Friz, and Gatheral, Pricing under rough volatility, Quantitative Finance, 16(6):887-904, 2016], and an enhanced version of the rough model that can generate realistic positive skew for VIX smiles -- introduced simultaneously by De Marco [Bachelier World Congress, 2018] and Guyon [Bachelier World Congress, 2018] on the lines of [Bergomi, Smile dynamics III, Risk, 2008], that we refer to as 'mixed rough Bergomi model'. Following the methodology set up in [Gobet and Miri, Weak approximation of averaged diffusion processes. Stochastic Process.\ Appl., 124(1):475-504, 2014], we derive weak approximations for the law of the VIX, leading to option price approximations under the form of explicit combinations of Black-Scholes prices and greeks. As new contributions, we cope with the fractional integration kernel appearing in rough models and treat the case of non-smooth payoffs, so to encompass VIX futures, call and put options. We stress that our approach does not rely on small-time asymptotics nor small-parameter (such as small volatility-of-volatility) asymptotics, and can therefore be applied to any option maturity and a wide range of parameter configurations. Our results are illustrated by several numerical experiments and calibration tests to VIX market data.

Suggested Citation

  • Florian Bourgey & Stefano De Marco & Emmanuel Gobet, 2022. "Weak approximations and VIX option price expansions in forward variance curve models," Papers 2202.10413, arXiv.org, revised May 2022.
    • Handle: RePEc:arx:papers:2202.10413
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    References listed on IDEAS

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