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Shapes of implied volatility with positive mass at zero

Author

Listed:
  • Stefano De Marco

    (CMAP; Ecole Polytechnique)

  • Caroline Hillairet

    (CREST; Ensae; Université Paris Saclay)

  • Antoine Jacquier

    (Imperial College London)

Abstract

We study the shapes of the implied volatility when the underlying distribution has an atom at zero. We show that the behaviour at small strikes is uniquely determined by the mass of the atom at least up to the third asymptotic order, regardless of the properties of the remaining (absolutely continuous, or singular) distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can—a priori—distinguish between mass at the origin and a heavy-left-tailed distribution. An atom at zero is found in stochastic models with absorption at the boundary, such as the CEV process, and can be used to model default events, as in the class of jump-to-default structural models of credit risk. We numerically test our model-free result in such examples. Note that while Lee’s moment formula [21] tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as [2, 14] do not apply in this setting essentially due to the breakdown of Put-Call symmetry—and we rely here on an alternative treatment of the problem.

Suggested Citation

  • 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.
  • Handle: RePEc:crs:wpaper:2017-77
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    References listed on IDEAS

    as
    1. Archil Gulisashvili & Elias M. Stein, 2009. "Implied Volatility In The Hull–White Model," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 303-327, April.
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    Cited by:

    1. Claude Martini & Arianna Mingone, 2020. "No arbitrage SVI," Papers 2005.03340, arXiv.org, revised May 2021.
    2. Jaehyuk Choi & Lixin Wu, 2021. "A note on the option price and ‘Mass at zero in the uncorrelated SABR model and implied volatility asymptotics’," Quantitative Finance, Taylor & Francis Journals, vol. 21(7), pages 1083-1086, July.
    3. Vimal Raval & Antoine Jacquier, 2021. "The Log Moment formula for implied volatility," Papers 2101.08145, arXiv.org.
    4. Choi, Jaehyuk & Wu, Lixin, 2021. "The equivalent constant-elasticity-of-variance (CEV) volatility of the stochastic-alpha-beta-rho (SABR) model," Journal of Economic Dynamics and Control, Elsevier, vol. 128(C).
    5. Cyril Grunspan & Joris Van Der Hoeven, 2020. "Effective Asymptotics Analysis For Finance," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 23(02), pages 1-23, March.
    6. Michael R. Tehranchi, 2020. "A Black–Scholes inequality: applications and generalisations," Finance and Stochastics, Springer, vol. 24(1), pages 1-38, January.
    7. Cyril Grunspan & Joris van der Hoeven, 2020. "Effective asymptotic analysis for finance," Post-Print hal-01573621, HAL.

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    Keywords

    Atomic distribution; heavy-tailed distribution; Implied Volatility; smile asymptotics; absorption at zero; CEV model;
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