Finance and Stochastics

, Volume 18, Issue 2, pp 349–392 | Cite as

Asymptotics of implied volatility to arbitrary order

  • Kun Gao
  • Roger LeeEmail author


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).


Implied volatility Asymptotics 

Mathematics Subject Classification (2010)


JEL Classification




The second author thanks Nizar Touzi, Mathieu Rosenbaum, and the participants at the Third SMAI European Summer School in Financial Mathematics. Both authors thank two anonymous referees for helpful comments.


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Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of ChicagoChicagoUSA

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