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We've walked a million miles for one of these smiles

Author

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  • L. De Leo
  • V. Vargas
  • S. Ciliberti
  • J. -P. Bouchaud

Abstract

We derive a new, exact and transparent expansion for option smiles, which lends itself both to analytical approximation and, perhaps more importantly, to congenial numerical treatments. We show that the skew and the curvature of the smile can be computed as exotic options, for which the Hedged Monte Carlo method is particularly well suited. When applied to options on the S&P index, we find that the skew and the curvature of the smile are very poorly reproduced by the standard Edgeworth (cumulant) expansion. Most notably, the relation between the skew and the skewness is inverted at small and large vols, a feature that none of the model studied so far is able to reproduce. Furthermore, the around-the-money curvature of the smile is found to be very small, in stark contrast with the highly kurtic nature of the returns.

Suggested Citation

  • L. De Leo & V. Vargas & S. Ciliberti & J. -P. Bouchaud, 2012. "We've walked a million miles for one of these smiles," Papers 1203.5703, arXiv.org, revised Apr 2012.
    • Handle: RePEc:arx:papers:1203.5703
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    Cited by:

    1. Stefan Gerhold & I. Cetin Gulum & Arpad Pinter, 2013. "Small-maturity asymptotics for the at-the-money implied volatility slope in L\'evy models," Papers 1310.3061, arXiv.org, revised May 2016.
    2. Laurent Devineau & Pierre-Edouard Arrouy & Paul Bonnefoy & Alexandre Boumezoued, 2017. "Fast calibration of the Libor Market Model with Stochastic Volatility and Displaced Diffusion," Working Papers hal-01521491, HAL.
    3. José E. Figueroa-López & Sveinn Ólafsson, 2016. "Short-term asymptotics for the implied volatility skew under a stochastic volatility model with Lévy jumps," Finance and Stochastics, Springer, vol. 20(4), pages 973-1020, October.

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