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A Note on the Equivalence between the Normal and the Lognormal Implied Volatility : A Model Free Approach

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  • Cyril Grunspan

Abstract

First, we show that implied normal volatility is intimately linked with the incomplete Gamma function. Then, we deduce an expansion on implied normal volatility in terms of the time-value of a European call option. Then, we formulate an equivalence between the implied normal volatility and the lognormal implied volatility with any strike and any model. This generalizes a known result for the SABR model. Finally, we adress the issue of the "breakeven move" of a delta-hedged portfolio.

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  • Cyril Grunspan, 2011. "A Note on the Equivalence between the Normal and the Lognormal Implied Volatility : A Model Free Approach," Papers 1112.1782, arXiv.org.
    • Handle: RePEc:arx:papers:1112.1782
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    References listed on IDEAS

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    1. Cyril Grunspan, 2011. "Asymptotic Expansions of the Lognormal Implied Volatility : A Model Free Approach," Papers 1112.1652, arXiv.org.
    2. Jaehyuk Choi & Kwangmoon Kim & Minsuk Kwak, 2009. "Numerical Approximation of the Implied Volatility Under Arithmetic Brownian Motion," Applied Mathematical Finance, Taylor & Francis Journals, vol. 16(3), pages 261-268.
    3. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12, January.
    4. Marco Avellaneda & Sasha Stoikov, 2008. "High-frequency trading in a limit order book," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 217-224.
    5. Jörg Kienitz & Manuel Wittke, 2010. "Option Valuation in Multivariate SABR Models," Research Paper Series 272, Quantitative Finance Research Centre, University of Technology, Sydney.
    6. Walter Schachermayer & Josef Teichmann, 2008. "How Close Are The Option Pricing Formulas Of Bachelier And Black–Merton–Scholes?," Mathematical Finance, Wiley Blackwell, vol. 18(1), pages 155-170, January.
    7. Viorel Costeanu & Dan Pirjol, 2011. "Asymptotic Expansion for the Normal Implied Volatility in Local Volatility Models," Papers 1105.3359, arXiv.org.
    8. 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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    Cited by:

    1. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2018. "Most-Likely-Path In Asian Option Pricing Under Local Volatility Models," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 21(05), pages 1-32, August.
    2. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559, arXiv.org.
    3. Christian Bayer & Juho Happola & Ra'ul Tempone, 2017. "Implied Stopping Rules for American Basket Options from Markovian Projection," Papers 1705.00558, arXiv.org, revised Jun 2017.
    4. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2017. "Most-likely-path in Asian option pricing under local volatility models," Papers 1706.02408, arXiv.org, revised Aug 2018.
    5. Jaehyuk Choi & Minsuk Kwak & Chyng Wen Tee & Yumeng Wang, 2021. "A Black-Scholes user's guide to the Bachelier model," Papers 2104.08686, arXiv.org, revised Feb 2022.
    6. Kevin Patrick Darby, 2021. "Time is Money: The Equilibrium Trading Horizon and Optimal Arrival Price," Papers 2104.05844, arXiv.org.
    7. Cyril Grunspan & Joris van der Hoeven, 2020. "Effective asymptotic analysis for finance," Post-Print hal-01573621, HAL.
    8. Jaehyuk Choi & Minsuk Kwak & Chyng Wen Tee & Yumeng Wang, 2022. "A Black–Scholes user's guide to the Bachelier model," Journal of Futures Markets, John Wiley & Sons, Ltd., vol. 42(5), pages 959-980, May.

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