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The small-maturity smile for exponential Levy models

  • Jose E. Figueroa-Lopez
  • Martin Forde
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    We derive a small-time expansion for out-of-the-money call options under an exponential Levy model, using the small-time expansion for the distribution function given in Figueroa-Lopez & Houdre (2009), combined with a change of num\'eraire via the Esscher transform. In particular, we quantify find that the effect of a non-zero volatility $\sigma$ of the Gaussian component of the driving L\'{e}vy process is to increase the call price by $1/2\sigma^2 t^2 e^{k}\nu(k)(1+o(1))$ as $t \to 0$, where $\nu$ is the L\'evy density. Using the small-time expansion for call options, we then derive a small-time expansion for the implied volatility, which sharpens the first order estimate given in Tankov (2010). Our numerical results show that the second order approximation can significantly outperform the first order approximation. Our results are also extended to a class of time-changed L\'evy models. We also consider a small-time, small log-moneyness regime for the CGMY model, and apply this approach to the small-time pricing of at-the-money call options.

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    File URL: http://arxiv.org/pdf/1105.3180
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    Paper provided by arXiv.org in its series Papers with number 1105.3180.

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    Date of creation: May 2011
    Date of revision: Dec 2011
    Handle: RePEc:arx:papers:1105.3180
    Contact details of provider: Web page: http://arxiv.org/

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