Small-Time Asymptotics for Implied Volatility Under a General Local-Stochastic Volatility Model

Posted: 27 Nov 2009 Last revised: 4 Jun 2012

See all articles by Martin Forde

Martin Forde

Dublin City University - Department of Mathematical Sciences

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Date Written: November 26, 2009

Abstract

We build on of the work of Henry-LabordĀµere and Lewis on the small-time behaviour of the return distribution under a general local-stochastic volatility model with zero correlation. We do this using the Freidlin-Wentzell theory of large deviations for stochastic differential equations, and then converting to a differential geometry problem of computing the shortest geodesic from a point to a vertical line on a Riemmanian manifold, whose metric is induced by the inverse of the diffusion coefficient. The solution to this variable endpoint problem is obtained using a transversality condition, where the geodesic is perpendicular to the vertical line under the aforementioned metric. We then establish the corresponding small-time asymptotic behaviour for call options using Holder's inequality, and the implied volatility. This avoids the use of possibly non-differentiable viscosity solutions, and we show that the distance function is actually a classical solution to the non-linear eikonal equation, which is analysed at length in Busca et al. We also derive a series expansion for the implied volatility in the small-maturity limit, in powers of the log-moneyness, and we show how to calibrate such a model to the observed implied volatility smile in the small-maturity limit. Finally, we introduce the hyperbolic stochastic volatility model, which satisfies the boundedness and Lipschitz conditions required to be able to apply the Freidlin-Wentzell theory, and we give numerical examples.

Keywords: local volatility, stochastic volatility, asymptotics, differential geometry, Freidlin-Wentzell theory

JEL Classification: G12, G13, C6

Suggested Citation

Forde, Martin and Jacquier, Antoine, Small-Time Asymptotics for Implied Volatility Under a General Local-Stochastic Volatility Model (November 26, 2009). Applied Mathematical Finance, Vol. 18, No. 6, 2011, Available at SSRN: https://ssrn.com/abstract=1514125

Martin Forde

Dublin City University - Department of Mathematical Sciences ( email )

Dublin
Ireland

Antoine Jacquier (Contact Author)

Imperial College London ( email )

South Kensington Campus
London SW7 2AZ, SW7 2AZ
United Kingdom

HOME PAGE: http://wwwf.imperial.ac.uk/~ajacquie/

The Alan Turing Institute ( email )

British Library, 96 Euston Road
London, NW12DB
United Kingdom

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