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Small-Time Asymptotics For Implied Volatility Under The Heston Model

Author

Listed:
  • 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).

Suggested Citation

  • Martin Forde & Antoine Jacquier, 2009. "Small-Time Asymptotics For Implied Volatility Under The Heston Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 12(06), pages 861-876.
    • Handle: RePEc:wsi:ijtafx:v:12:y:2009:i:06:n:s021902490900549x
    DOI: 10.1142/S021902490900549X
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    References listed on IDEAS

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    1. Alan L. Lewis, 2000. "Option Valuation under Stochastic Volatility," Option Valuation under Stochastic Volatility, Finance Press, number ovsv, December.
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