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Uniform bounds for Black--Scholes implied volatility

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  • Michael R. Tehranchi

Abstract

In this note, Black--Scholes implied volatility is expressed in terms of various optimisation problems. From these representations, upper and lower bounds are derived which hold uniformly across moneyness and call price. Various symmetries of the Black--Scholes formula are exploited to derive new bounds from old. These bounds are used to reprove asymptotic formulae for implied volatility at extreme strikes and/or maturities.

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  • Michael R. Tehranchi, 2015. "Uniform bounds for Black--Scholes implied volatility," Papers 1512.06812, arXiv.org, revised Aug 2016.
    • Handle: RePEc:arx:papers:1512.06812
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    Cited by:

    1. 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.
    2. 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.
    3. Cyril Grunspan & Joris van der Hoeven, 2017. "Effective asymptotic analysis for finance," Working Papers hal-01573621, HAL.
    4. Dan Stefanica & Radoš Radoičić, 2017. "An Explicit Implied Volatility Formula," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 20(07), pages 1-32, November.
    5. Cyril Grunspan & Joris van der Hoeven, 2020. "Effective asymptotic analysis for finance," Post-Print hal-01573621, HAL.

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