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No arbitrage SVI

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  • Claude Martini
  • Arianna Mingone

Abstract

We fully characterize the absence of Butterfly arbitrage in the SVI formula for implied total variance proposed by Gatheral in 2004. The main ingredient is an intermediary characterization of the necessary condition for no arbitrage obtained for any model by Fukasawa in 2012 that the inverse functions of the -d1 and -d2 of the Black-Scholes formula, viewed as functions of the log-forward moneyness, should be increasing. A natural rescaling of the SVI parameters and a meticulous analysis of the Durrleman condition allow then to obtain simple range conditions on the parameters. This leads to a straightforward implementation of a least-squares calibration algorithm on the no arbitrage domain, which yields an excellent fit on the market data we used for our tests, with the guarantee to yield smiles with no Butterfly arbitrage.

Suggested Citation

  • Claude Martini & Arianna Mingone, 2020. "No arbitrage SVI," Papers 2005.03340, arXiv.org, revised May 2021.
    • Handle: RePEc:arx:papers:2005.03340
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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. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2013. "Shapes of implied volatility with positive mass at zero," Papers 1310.1020, arXiv.org, revised May 2017.
    3. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    4. Tahar Ferhati, 2020. "Robust Calibration For SVI Model Arbitrage Free," Working Papers hal-02490029, HAL.
    5. Jacopo Corbetta & Pierre Cohort & Ismail Laachir & Claude Martini, 2019. "Robust calibration and arbitrage-free interpolation of SSVI slices," Decisions in Economics and Finance, Springer;Associazione per la Matematica, vol. 42(2), pages 665-677, December.
    6. Gaoyue Guo & Antoine Jacquier & Claude Martini & Leo Neufcourt, 2012. "Generalised arbitrage-free SVI volatility surfaces," Papers 1210.7111, arXiv.org, revised May 2016.
    7. Bo Zhao & Stewart Hodges, 2013. "Parametric modeling of implied smile functions: a generalized SVI model," Review of Derivatives Research, Springer, vol. 16(1), pages 53-77, April.
    8. Stefano De Marco & Claude Martini, 2018. "Moment generating functions and normalized implied volatilities: unification and extension via Fukasawa’s pricing formula," Quantitative Finance, Taylor & Francis Journals, vol. 18(4), pages 609-622, April.
    9. Pierre Cohort & Jacopo Corbetta & Claude Martini & Ismail Laachir, 2018. "Robust calibration and arbitrage-free interpolation of SSVI slices," Papers 1804.04924, arXiv.org, revised Mar 2019.
    10. Stefano De Marco & Caroline Hillairet & Antoine Jacquier, 2017. "Shapes of implied volatility with positive mass at zero," Working Papers 2017-77, Center for Research in Economics and Statistics.
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    Cited by:

    1. Arianna Mingone, 2022. "No arbitrage global parametrization for the eSSVI volatility surface," Papers 2204.00312, arXiv.org.

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