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Smiles in delta

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  • Arianna Mingone

Abstract

Fukasawa introduced in [Fukasawa, Math Financ, 2012] two necessary conditions for no butterfly arbitrage which require that the $d_1$ and $d_2$ functions of the Black-Scholes formula have to be decreasing. In this article we characterize the set of smiles satisfying these conditions, using the parametrization of the smile in delta. We obtain a parametrization of the set via one real number and three positive functions. We also show that such smiles and their symmetric smiles can be transformed into smiles in the strike space by a bijection. Our result motivates the study of the challenging question of characterizing the subset of butterfly arbitrage-free smiles.

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  • Arianna Mingone, 2022. "Smiles in delta," Papers 2209.00406, arXiv.org.
    • Handle: RePEc:arx:papers:2209.00406
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    References listed on IDEAS

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    1. Michael R. Tehranchi, 2020. "A Black–Scholes inequality: applications and generalisations," Finance and Stochastics, Springer, vol. 24(1), pages 1-38, January.
    2. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    3. L. Rogers & M. Tehranchi, 2010. "Can the implied volatility surface move by parallel shifts?," Finance and Stochastics, Springer, vol. 14(2), pages 235-248, April.
    4. Schlüter, Stephan & Fischer, Matthias J., 2009. "A tail quantile approximation formula for the student t and the symmetric generalized hyperbolic distribution," FAU Discussion Papers in Economics 05/2009, Friedrich-Alexander University Erlangen-Nuremberg, Institute for Economics.
    5. Roger W. Lee, 2004. "The Moment Formula For Implied Volatility At Extreme Strikes," Mathematical Finance, Wiley Blackwell, vol. 14(3), pages 469-480, July.
    6. Masaaki Fukasawa, 2010. "Normalization for Implied Volatility," Papers 1008.5055, arXiv.org, revised Sep 2010.
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    Cited by:

    1. Claude Martini & Arianna Mingone, 2023. "Options are also options on options: how to smile with Black-Scholes," Papers 2308.04130, arXiv.org.

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