The Smile in Stochastic Volatility Models

15 Pages Posted: 3 Dec 2011 Last revised: 6 May 2012

See all articles by Lorenzo Bergomi

Lorenzo Bergomi

Societe Generale

Julien Guyon

Bloomberg L.P.; Columbia University - Department of Mathematics; New York University - Courant Institute of Mathematical Sciences

Date Written: December 2, 2011

Abstract

We consider general stochastic volatility models with no local volatility component and derive the general expression of the volatility smile at order two in volatility-of-volatility. We show how, at this order, the smile only depends on three dimensionless numbers whose precise expressions as functionals of the model's spot/variance and variance/variance covariance functions we provide.

Finally we assess the accuracy of our order two expansion using realistic levels of volatility-of-volatility.

JEL Classification: G13

Suggested Citation

Bergomi, Lorenzo and Guyon, Julien, The Smile in Stochastic Volatility Models (December 2, 2011). Available at SSRN: https://ssrn.com/abstract=1967470 or http://dx.doi.org/10.2139/ssrn.1967470

Lorenzo Bergomi (Contact Author)

Societe Generale ( email )

Paris-La Défense, Paris 92987
France

Julien Guyon

Bloomberg L.P. ( email )

731 Lexington Avenue
New York, NY 10022
United States

Columbia University - Department of Mathematics ( email )

3022 Broadway
New York, NY 10027
United States

New York University - Courant Institute of Mathematical Sciences ( email )

New York University
New York, NY 10012
United States

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