A General Asymptotic Implied Volatility for Stochastic Volatility Models

By Pierre Henry-Labordère. 2005

136 citations recorded by the source corpus.

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.

Subfields: Finance, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/0a673637a8104695f3d522d7caca7aa1e7663222

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Citations in the corpus

ASYMPTOTICS OF IMPLIED VOLATILITY IN LOCAL VOLATILITY MODELS (2012, unknown): “Mathematical Finance, Vol. 22, No. 4 (October 2012), 591–620”
Asymptotic Implied Volatility at the Second Order with Application to the SABR Model (2009, unknown): “In this paper, we leverage on the heat kernel methods introduced for the study of stochastic processes by Varadhan [Var67a, Var67b] and used for the SABR model in [HLW05] and [HL05].”
SMALL-TIME ASYMPTOTICS FOR FAST MEAN-REVERTING STOCHASTIC VOLATILITY MODELS (2010, background): “On one hand, the theory of large deviations has been recently applied to local and stochastic volatility models [1, 2, 4, 5, 20] and has given very interesting results on the behavior of implied volatilities near maturity.”
Hermite Polynomial Based Expansion of European Option Prices (2014, unknown): “Parallel to these convergent series, for one-factor models Henry-Labordere (2005); Gatheral et al. (2012) obtain asymptotic expansions for short-term option prices.”