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A General Asymptotic Implied Volatility for Stochastic Volatility Models

Abstract : In this paper, we derive a general asymptotic implied volatility at the first-order for any stochastic volatility model using the heat kernel expansion on a Riemann manifold endowed with an Abelian connection. This formula is particularly useful for the calibration procedure. As an application, we obtain an asymptotic smile for a SABR model with a mean-reversion term, called lambda-SABR, corresponding in our geometric framework to the Poincaré hyperbolic plane. When the lambda-SABR model degenerates into the SABR-model, we show that our asymptotic implied volatility is a better approximation than the classical Hagan-al expression . Furthermore, in order to show the strength of this geometric framework, we give an exact solution of the SABR model with beta=0 or 1. In a next paper, we will show how our method can be applied in other contexts such as the derivation of an asymptotic implied volatility for a Libor market model with a stochastic volatility.
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https://hal.archives-ouvertes.fr/hal-00004632
Contributor : Pierre Henry-Labordere <>
Submitted on : Thursday, May 12, 2005 - 11:54:11 PM
Last modification on : Friday, May 13, 2005 - 7:41:50 AM
Long-term archiving on: : Friday, September 17, 2010 - 6:38:19 PM

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Pierre Henry-Labordere. A General Asymptotic Implied Volatility for Stochastic Volatility Models. 2005. ⟨hal-00004632v2⟩

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