Does the Term-Structure of Equity At-the-Money Skew Really Follow a Power Law?

18 Pages Posted: 3 Aug 2022 Last revised: 28 Mar 2023

See all articles by Julien Guyon

Julien Guyon

Ecole Nationale des Ponts et Chaussées (ENPC)

Mehdi El Amrani

Bloomberg L.P.

Date Written: July 27, 2022


Using two years of S&P 500, Eurostoxx 50, and DAX data, we empirically investigate the term-structure of the at-the-money-forward (ATM) skew of equity indexes. While a power law (2 parameters) captures the term-structure well away from short maturities, the power law fit deteriorates considerably when short maturities are included. By contrast, 3-parameter shapes that look like power laws but do not blow up at vanishing maturity, such as time-shifted or capped power laws, are shown to fit well regardless of whether short maturities are included or not. Our study suggests that the term-structure of equity ATM skew has a power-law shape for maturities above 1 month but has a different behavior, and in particular may not blow up, for shorter maturities. The 3-parameter shapes are derived from non-Markovian variance curve models using the Bergomi-Guyon expansion. A simple 4-parameter term-structure similarly derived from the (Markovian) two-factor Bergomi model is also considered and provides even better fits. The extrapolated zero-maturity skew, far from being infinite, is distributed around a typical value of 1.5 (in absolute value).

Keywords: At-the-money skew, equity skew, variance curve models, calibration, power law, rough volatility, Bergomi model

JEL Classification: G13

Suggested Citation

Guyon, Julien and El Amrani, Mehdi, Does the Term-Structure of Equity At-the-Money Skew Really Follow a Power Law? (July 27, 2022). Available at SSRN: or

Julien Guyon (Contact Author)

Ecole Nationale des Ponts et Chaussées (ENPC) ( email )

28, rue des Saints-Peres
75343 Paris Cedex 07

Mehdi El Amrani

Bloomberg L.P.

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