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The Smile of certain L\'evy-type Models

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  • Antoine Jacquier
  • Matthew Lorig
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    Abstract

    We consider a class of assets whose risk-neutral pricing dynamics are described by an exponential L\'evy-type process subject to default. The class of processes we consider features locally-dependent drift, diffusion and default-intensity as well as a locally-dependent L\'evy measure. Using techniques from regular perturbation theory and Fourier analysis, we derive a series expansion for the price of a European-style option. We also provide precise conditions under which this series expansion converges to the exact price. Additionally, for a certain subclass of assets in our modeling framework, we derive an expansion for the implied volatility induced by our option pricing formula. The implied volatility expansion is exact within its radius of convergence. As an example of our framework, we propose a class of CEV-like L\'evy-type models. Within this class, approximate option prices can be computed by a single Fourier integral and approximate implied volatilities are explicit (i.e., no integration is required). Furthermore, the class of CEV-like L\'evy-type models is shown to provide a tight fit to the implied volatility surface of S{&}P500 index options.

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    File URL: http://arxiv.org/pdf/1207.1630
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    Bibliographic Info

    Paper provided by arXiv.org in its series Papers with number 1207.1630.

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    Date of creation: Jul 2012
    Date of revision: Apr 2013
    Handle: RePEc:arx:papers:1207.1630

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    Web page: http://arxiv.org/

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    Cited by:
    1. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792, arXiv.org.

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