Magic strikes for variance and gamma contracts, and other attainable claims

Florian Bourgey and Jim Gatheral

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Abstract: Building on the forest expansion of Al\`os, Gatheral and Radoi\v{c}i\'c and on the explicit Bergomi-Guyon smile expansion derived by Bourgey and Gatheral (2026), we derive fixed-point approximations, in terms of the implied total variance at a small number of magic strikes, for the fair values of power payoff contracts; the variance and gamma contracts arise as the endpoints of this one-parameter family, priced by a single set of formulae. We prove that such approximations exist at every order: at order $n$, $2\lceil n/2\rceil+1$ magic strikes suffice. Numerical tests under the Heston and rough Bergomi models demonstrate good accuracy even for strongly skewed smiles, and the method applies directly to interpolated market smiles without strike inversion. For the volatility contract, we show that the maturity-T smile does not determine the leading correction to the Rolloos-Arslan approximation.

Date: 2026-09
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