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Shapes of implied volatility with positive mass at zero

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  • Stefano De Marco
  • Caroline Hillairet
  • Antoine Jacquier

Abstract

We study the shapes of the implied volatility when the underlying distribution has an atom at zero and analyse the impact of a mass at zero on at-the-money implied volatility and the overall level of the smile. We further show that the behaviour at small strikes is uniquely determined by the mass of the atom up to high asymptotic order, under mild assumptions on the remaining distribution on the positive real line. We investigate the structural difference with the no-mass-at-zero case, showing how one can--theoretically--distinguish between mass at the origin and a heavy-left-tailed distribution. We numerically test our model-free results in stochastic models with absorption at the boundary, such as the CEV process, and in jump-to-default models. Note that while Lee's moment formula tells that implied variance is at most asymptotically linear in log-strike, other celebrated results for exact smile asymptotics such as Benaim and Friz (09) or Gulisashvili (10) do not apply in this setting--essentially due to the breakdown of Put-Call duality.

Suggested Citation

  • 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.
  • Handle: RePEc:arx:papers:1310.1020
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    File URL: http://arxiv.org/pdf/1310.1020
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    References listed on IDEAS

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    1. Delia Coculescu & Hélyette Geman & Monique Jeanblanc, 2008. "Valuation of default-sensitive claims under imperfect information," Finance and Stochastics, Springer, vol. 12(2), pages 195-218, April.
    2. Archil Gulisashvili & Elias M. Stein, 2009. "Implied Volatility In The Hull-White Model," Mathematical Finance, Wiley Blackwell, vol. 19(2), pages 303-327.
    3. repec:dau:papers:123456789/409 is not listed on IDEAS
    4. Eric Renault & Nizar Touzi, 1996. "Option Hedging And Implied Volatilities In A Stochastic Volatility Model," Mathematical Finance, Wiley Blackwell, vol. 6(3), pages 279-302.
    5. Campi, Luciano & Polbennikov, Simon & Sbuelz, Alessandro, 2009. "Systematic equity-based credit risk: A CEV model with jump to default," Journal of Economic Dynamics and Control, Elsevier, vol. 33(1), pages 93-108, January.
    6. Jim Gatheral & Antoine Jacquier, 2014. "Arbitrage-free SVI volatility surfaces," Quantitative Finance, Taylor & Francis Journals, vol. 14(1), pages 59-71, January.
    7. S. Benaim & P. Friz, 2009. "Regular Variation And Smile Asymptotics," Mathematical Finance, Wiley Blackwell, vol. 19(1), pages 1-12.
    8. Ian Martin, 2011. "Simple Variance Swaps," NBER Working Papers 16884, National Bureau of Economic Research, Inc.
    9. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    10. 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.
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