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An exact and explicit implied volatility inversion formula

Author

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  • Yuxuan Xia

    (School of Business, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, NJ 07310, USA)

  • Zhenyu Cui

    (School of Business, Stevens Institute of Technology, 1 Castle Point on Hudson, Hoboken, NJ 07310, USA)

Abstract

In this paper, we develop an exact and explicit (model-independent) Taylor series representation of the implied volatility based on the novel applications of an extended Faà di Bruno formula under the operator calculus setting, and the Lagrange inversion theorem. We rigorously establish that our formula converges to the true implied volatility as the truncation order increases. Numerical examples illustrate the remarkable accuracy and efficiency of the formula. The formula distinguishes from previous literature as it converges to the true exact implied volatility, is a closed-form formula whose coefficients are explicitly determined and do not involve numerical iterations.

Suggested Citation

  • Yuxuan Xia & Zhenyu Cui, 2018. "An exact and explicit implied volatility inversion formula," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 5(03), pages 1-29, September.
    • Handle: RePEc:wsi:ijfexx:v:05:y:2018:i:03:n:s2424786318500329
    DOI: 10.1142/S2424786318500329
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    References listed on IDEAS

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    1. Don M. Chance & Thomas A. Hanson & Weiping Li & Jayaram Muthuswamy, 2017. "A bias in the volatility smile," Review of Derivatives Research, Springer, vol. 20(1), pages 47-90, April.
    2. Ivan Matić & Radoš Radoičić & Dan Stefanica, 2017. "Pólya-based approximation for the ATM-forward implied volatility," International Journal of Financial Engineering (IJFE), World Scientific Publishing Co. Pte. Ltd., vol. 4(02n03), pages 1-15, June.
    3. Jim Gatheral & Ivan Matić & Radoš Radoičić & Dan Stefanica, 2017. "Tighter Bounds For Implied Volatility," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(05), pages 1-14, August.
    4. Li, Minqiang, 2008. "Approximate inversion of the Black-Scholes formula using rational functions," European Journal of Operational Research, Elsevier, vol. 185(2), pages 743-759, March.
    5. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
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    7. Hentschel, Ludger, 2003. "Errors in Implied Volatility Estimation," Journal of Financial and Quantitative Analysis, Cambridge University Press, vol. 38(4), pages 779-810, December.
    8. Alexey Medvedev & Olivier Scaillet, 2007. "Approximation and Calibration of Short-Term Implied Volatilities Under Jump-Diffusion Stochastic Volatility," The Review of Financial Studies, Society for Financial Studies, vol. 20(2), pages 427-459.
    9. Michael R. Tehranchi, 2015. "Uniform bounds for Black--Scholes implied volatility," Papers 1512.06812, arXiv.org, revised Aug 2016.
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