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Small-Time Asymptotics For Implied Volatility Under The Heston Model

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Author Info

  • MARTIN FORDE

    ()
    (Department of Mathematical Sciences, Dublin City University, Glasnevin, Dublin 9, Ireland)

  • ANTOINE JACQUIER

    ()
    (Department of Mathematics, Imperial College, 180 Queen's Gate, London; Zeliade Systems, 56 Rue Jean-Jacques Rousseau, Paris)

Abstract

We rigorize the work of Lewis (2007) and Durrleman (2005) on the small-time asymptotic behavior of the implied volatility under the Heston stochastic volatility model (Theorem 2.1). We apply the Gärtner-Ellis theorem from large deviations theory to the exponential affine closed-form expression for the moment generating function of the log forward price, to show that it satisfies a small-time large deviation principle. The rate function is computed as Fenchel-Legendre transform, and we show that this can actually be computed as a standard Legendre transform, which is a simple numerical root-finding exercise. We establish the corresponding result for implied volatility in Theorem 3.1, using well known bounds on the standard Normal distribution function. In Theorem 3.2 we compute the level, the slope and the curvature of the implied volatility in the small-maturity limit At-the-money, and the answer is consistent with that obtained by formal PDE methods by Lewis (2000) and probabilistic methods by Durrleman (2004).

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Bibliographic Info

Article provided by World Scientific Publishing Co. Pte. Ltd. in its journal International Journal of Theoretical and Applied Finance.

Volume (Year): 12 (2009)
Issue (Month): 06 ()
Pages: 861-876

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Handle: RePEc:wsi:ijtafx:v:12:y:2009:i:06:p:861-876

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Related research

Keywords: Implied volatility asymptotics; Heston; large deviation; small-time behavior;

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Cited by:
  1. Archil Gulisashvili & Josef Teichmann, 2014. "The G\"{a}rtner-Ellis theorem, homogenization, and affine processes," Papers 1406.3716, arXiv.org.
  2. Archil Gulisashvili & Peter Tankov, 2014. "Implied volatility of basket options at extreme strikes," Papers 1406.0394, arXiv.org.
  3. Zhi Guo & Eckhard Platen, 2011. "The Small and Large Time Implied Volatilities in the Minimal Market Model," Research Paper Series 297, Quantitative Finance Research Centre, University of Technology, Sydney.
  4. Ronnie Sircar & Stephan Sturm, 2011. "From Smile Asymptotics to Market Risk Measures," Papers 1107.4632, arXiv.org, revised Jul 2012.
  5. Jos\'e E. Figueroa-L\'opez & Ruoting Gong & Christian Houdr\'e, 2011. "High-order short-time expansions for ATM option prices under the CGMY model," Papers 1112.3111, arXiv.org, revised Aug 2012.
  6. Archil Gulisashvili & Peter Laurence, 2013. "The Heston Riemannian distance function," Papers 1302.2337, arXiv.org.
  7. Cristian Homescu, 2011. "Implied Volatility Surface: Construction Methodologies and Characteristics," Papers 1107.1834, arXiv.org.
  8. Stefan Gerhold, 2012. "Can there be an explicit formula for implied volatility?," Papers 1211.4978, arXiv.org.
  9. Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
  10. Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Explicit implied vols for multifactor local-stochastic vol models," Papers 1306.5447, arXiv.org, revised Mar 2014.
  11. Stefan Gerhold & Max Kleinert & Piet Porkert & Mykhaylo Shkolnikov, 2012. "Small time central limit theorems for semimartingales with applications," Papers 1208.4282, arXiv.org.
  12. Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
  13. Jos\'e E. Figueroa-L\'opez & Ruoting Gong & Christian Houdr\'e, 2012. "High-order short-time expansions for ATM option prices of exponential L\'evy models," Papers 1208.5520, arXiv.org, revised Apr 2014.

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