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Asymptotics of implied volatility to arbitrary order

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  • Kun Gao
  • Roger Lee

Abstract

In a unified model-free framework that includes long-expiry, short-expiry, extreme-strike, and jointly-varying strike-expiry regimes, we generate implied volatility and implied variance approximations, with rigorous error estimates asymptotically smaller than any given power of L, where L denotes the exogenously given absolute log of an option price that approaches zero. Our results, therefore, sharpen to arbitrarily high order of accuracy (and, moreover, extend to general extreme regimes) the model-free asymptotics of implied volatility. We then apply these general formulas to particular examples: Heston (using a previously known L expansion) and Lévy (using saddlepoint methods to derive L expansions). Copyright Springer-Verlag Berlin Heidelberg 2014

Suggested Citation

  • Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    • Handle: RePEc:spr:finsto:v:18:y:2014:i:2:p:349-392
    DOI: 10.1007/s00780-013-0223-6
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    References listed on IDEAS

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    6. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," The Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    7. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
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    More about this item

    Keywords

    Implied volatility; Asymptotics; 91G20; G13;
    All these keywords.

    JEL classification:

    • G13 - Financial Economics - - General Financial Markets - - - Contingent Pricing; Futures Pricing

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