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From characteristic functions to implied volatility expansions


  • Antoine Jacquier
  • Matthew Lorig


For any strictly positive martingale $S = \exp(X)$ for which $X$ has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential L\'evy model (Merton), one infinite activity exponential L\'evy model (Variance Gamma), and one stochastic volatility model (Heston). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

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  • Antoine Jacquier & Matthew Lorig, 2012. "From characteristic functions to implied volatility expansions," Papers 1207.0233,, revised Jun 2014.
  • Handle: RePEc:arx:papers:1207.0233

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    1. Antoine Jacquier & Aleksandar Mijatović, 2014. "Large Deviations for the Extended Heston Model: The Large-Time Case," Asia-Pacific Financial Markets, Springer;Japanese Association of Financial Economics and Engineering, vol. 21(3), pages 263-280, September.
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    6. Benoit Mandelbrot, 2015. "The Variation of Certain Speculative Prices," World Scientific Book Chapters,in: THE WORLD SCIENTIFIC HANDBOOK OF FUTURES MARKETS, chapter 3, pages 39-78 World Scientific Publishing Co. Pte. Ltd..
    7. Martin Forde & Antoine Jacquier & Aleksandar Mijatovic, 2009. "Asymptotic formulae for implied volatility in the Heston model," Papers 0911.2992,, revised May 2010.
    8. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, June.
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    11. Pavel Cizek & Wolfgang Karl Härdle & Rafal Weron, 2005. "Statistical Tools for Finance and Insurance," HSC Books, Hugo Steinhaus Center, Wroclaw University of Technology, number hsbook0501, June.
    12. Alexey Medvedev & Olivier Scaillet, "undated". "Approximation and Calibration of Short-Term Implied Volatilities under Jump-Diffusion Stochastic Volatility," Swiss Finance Institute Research Paper Series 06-08, Swiss Finance Institute, revised Jan 2006.
    13. Eric Benhamou, 2002. "Option pricing with Levy Process," Finance 0212006, EconWPA.
    14. L. Rogers & M. Tehranchi, 2010. "Can the implied volatility surface move by parallel shifts?," Finance and Stochastics, Springer, vol. 14(2), pages 235-248, April.
    15. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, vol. 81(3), pages 637-654, May-June.
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