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


  • Cyril Grunspan


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,
  • Handle: RePEc:arx:papers:1112.1782

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    References listed on IDEAS

    1. Marco Avellaneda & Sasha Stoikov, 2008. "High-frequency trading in a limit order book," Quantitative Finance, Taylor & Francis Journals, vol. 8(3), pages 217-224.
    2. Jörg Kienitz & Manuel Wittke, 2010. "Option Valuation in Multivariate SABR Models," Research Paper Series 272, Quantitative Finance Research Centre, University of Technology, Sydney.
    3. Cyril Grunspan, 2011. "Asymptotic Expansions of the Lognormal Implied Volatility : A Model Free Approach," Papers 1112.1652,
    4. 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.
    5. 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.
    6. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12.
    7. Viorel Costeanu & Dan Pirjol, 2011. "Asymptotic Expansion for the Normal Implied Volatility in Local Volatility Models," Papers 1105.3359,
    8. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480.
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    Cited by:

    1. Louis-Pierre Arguin & Nien-Lin Liu & Tai-Ho Wang, 2017. "Most-likely-path in Asian option pricing under local volatility models," Papers 1706.02408,
    2. Dan Pirjol & Lingjiong Zhu, 2016. "Short Maturity Asian Options in Local Volatility Models," Papers 1609.07559,
    3. Christian Bayer & Juho Happola & Ra'ul Tempone, 2017. "Implied Stopping Rules for American Basket Options from Markovian Projection," Papers 1705.00558,, revised Jun 2017.

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