A General Asymptotic Implied Volatility for Stochastic Volatility Models

35 Pages Posted: 14 Apr 2005

Date Written: April 2005

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 Poincare 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.

Keywords: Heat kernel expansion, hyperbolic geometry, asymptotic smile, SABR with a mean-reversion term

JEL Classification: G13

Suggested Citation

Henry-Labordere, Pierre, A General Asymptotic Implied Volatility for Stochastic Volatility Models (April 2005). Available at SSRN: https://ssrn.com/abstract=698601 or http://dx.doi.org/10.2139/ssrn.698601

Pierre Henry-Labordere (Contact Author)

Natixis - Paris, France ( email )

Paris, Paris 75
France

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