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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.

Suggested Citation

  • 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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    1. Dilip B. Madan & Peter P. Carr & Eric C. Chang, 1998. "The Variance Gamma Process and Option Pricing," Review of Finance, European Finance Association, vol. 2(1), pages 79-105.
    2. Corrado, Charles J. & Miller, Thomas Jr., 1996. "A note on a simple, accurate formula to compute implied standard deviations," Journal of Banking & Finance, Elsevier, vol. 20(3), pages 595-603, April.
    3. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    4. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12, January.
    5. Archil Gulisashvili, 2015. "Left-Wing Asymptotics Of The Implied Volatility In The Presence Of Atoms," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 18(02), pages 1-25.
    6. paolo pianca, 2005. "Simple Formulas to Option Pricing and Hedging in the Black- Scholes Model," Finance 0511005, University Library of Munich, Germany.
    7. Michael Roper & Marek Rutkowski, 2009. "On The Relationship Between The Call Price Surface And The Implied Volatility Surface Close To Expiry," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(04), pages 427-441.
    8. Latane, Henry A & Rendleman, Richard J, Jr, 1976. "Standard Deviations of Stock Price Ratios Implied in Option Prices," Journal of Finance, American Finance Association, vol. 31(2), pages 369-381, May.
    9. Kun Gao & Roger Lee, 2014. "Asymptotics of implied volatility to arbitrary order," Finance and Stochastics, Springer, vol. 18(2), pages 349-392, April.
    10. 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.
    11. Leif Andersen & Alexander Lipton, 2013. "Asymptotics For Exponential Lévy Processes And Their Volatility Smile: Survey And New Results," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 16(01), pages 1-98.
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    Cited by:

    1. 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.
    2. Cyril Grunspan & Joris van der Hoeven, 2017. "Effective asymptotic analysis for finance," Working Papers hal-01573621, HAL.
    3. 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.
    4. Cyril Grunspan & Joris van der Hoeven, 2020. "Effective asymptotic analysis for finance," Post-Print hal-01573621, HAL.
    5. 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.

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