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The Smile of certain L\'evy-type Models


  • Antoine Jacquier
  • Matthew Lorig


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.

Suggested Citation

  • Antoine Jacquier & Matthew Lorig, 2012. "The Smile of certain L\'evy-type Models," Papers 1207.1630,, revised Apr 2013.
  • Handle: RePEc:arx:papers:1207.1630

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    References listed on IDEAS

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    3. William D. Nordhaus, 2009. "The Perils of the Learning Model For Modeling Endogenous Technological Change," Cowles Foundation Discussion Papers 1685, Cowles Foundation for Research in Economics, Yale University.
    4. McDonald, Alan & Schrattenholzer, Leo, 2001. "Learning rates for energy technologies," Energy Policy, Elsevier, vol. 29(4), pages 255-261, March.
    5. McNerney, James & Doyne Farmer, J. & Trancik, Jessika E., 2011. "Historical costs of coal-fired electricity and implications for the future," Energy Policy, Elsevier, vol. 39(6), pages 3042-3054, June.
    6. Nordhaus, William D., 2007. "Two Centuries of Productivity Growth in Computing," The Journal of Economic History, Cambridge University Press, vol. 67(01), pages 128-159, March.
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    Cited by:

    1. Anastasia Borovykh & Cornelis W. Oosterlee & Andrea Pascucci, 2016. "Pricing Bermudan options under local L\'evy models with default," Papers 1604.08735,
    2. Maria Cristina Recchioni & Yu Sun & Gabriele Tedeschi, 2016. "Can negative interest rates really affect option pricing? Empirical evidence from an explicitly solvable stochastic volatility model," Working Papers 2016/23, Economics Department, Universitat Jaume I, Castellón (Spain).
    3. Peter Carr & Roger Lee & Matthew Lorig, 2017. "Pricing Variance Swaps on Time-Changed Markov Processes," Papers 1705.01069,
    4. Tim Leung & Matthew Lorig & Andrea Pascucci, 2014. "Leveraged {ETF} implied volatilities from {ETF} dynamics," Papers 1404.6792,, revised Apr 2015.

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