Small-Time Asymptotics For Implied Volatility Under The Heston Model
AbstractWe 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 InfoArticle 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 ()
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- 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.
- Zhi Guo & Eckhard Platen, 2011.
"The Small and Large Time Implied Volatilities in the Minimal Market Model,"
1109.6154, arXiv.org, revised Oct 2011.
- Zhi Jun Guo & Eckhard Platen, 2012. "The Small And Large Time Implied Volatilities In The Minimal Market Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 15(08), pages 1250057-1-1.
- 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.
- Ronnie Sircar & Stephan Sturm, 2011. "From Smile Asymptotics to Market Risk Measures," Papers 1107.4632, arXiv.org, revised Jul 2012.
- Antoine Jacquier & Patrick Roome, 2013. "The Small-Maturity Heston Forward Smile," Papers 1303.4268, arXiv.org, revised Aug 2013.
- Matthew Lorig & Stefano Pagliarani & Andrea Pascucci, 2013. "Implied vol for any local-stochastic vol model," Papers 1306.5447, arXiv.org, revised Sep 2013.
- Stefan Gerhold, 2012. "Can there be an explicit formula for implied volatility?," Papers 1211.4978, arXiv.org.
- Stefan Gerhold & Max Kleinert & Piet Porkert & Mykhaylo Shkolnikov, 2012. "Small time central limit theorems for semimartingales with applications," Papers 1208.4282, arXiv.org.
- Archil Gulisashvili & Peter Laurence, 2013. "The Heston Riemannian distance function," Papers 1302.2337, arXiv.org.
- 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 Aug 2013.
- Leif Andersen & Alexander Lipton, 2012. "Asymptotics for Exponential Levy Processes and their Volatility Smile: Survey and New Results," Papers 1206.6787, arXiv.org.
- Cristian Homescu, 2011. "Implied Volatility Surface: Construction Methodologies and Characteristics," Papers 1107.1834, arXiv.org.
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