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# The small-maturity smile for exponential Levy models

## Author

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• Jose E. Figueroa-Lopez
• Martin Forde

## Abstract

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.

## Suggested Citation

• Jose E. Figueroa-Lopez & Martin Forde, 2011. "The small-maturity smile for exponential Levy models," Papers 1105.3180, arXiv.org, revised Dec 2011.
• Handle: RePEc:arx:papers:1105.3180
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File URL: http://arxiv.org/pdf/1105.3180
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## Citations

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Cited by:

1. Cristian Homescu, 2011. "Implied Volatility Surface: Construction Methodologies and Characteristics," Papers 1107.1834, arXiv.org.
2. Jos'e E. Figueroa-L'opez & Yankeng Luo & Cheng Ouyang, 2011. "Small-time expansions for local jump-diffusion models with infinite jump activity," Papers 1108.3386, arXiv.org, revised Jul 2014.

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