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From characteristic functions to implied volatility expansions

Part of: Limit theorems Mathematical economics

Published online by Cambridge University Press:  21 March 2016

Antoine Jacquier  and
Matthew Lorig
Antoine Jacquier*
Affiliation:
Imperial College London
Matthew Lorig*
Affiliation:
University of Washington
*
∗ Postal address: Department of Mathematics, Imperial College London, London SW7 2AZ, UK. Email address: [email protected]
∗∗ Postal address: Department of Applied Mathematics, University of Washington, Seattle, WA 98195-3925, USA.
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Abstract

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For any strictly positive martingale S = eX for which X has a characteristic function, we provide an expansion for the implied volatility. This expansion is explicit in the sense that it involves no integrals, but only polynomials in the log-strike. We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Lévy model, Merton (1976), one infinite activity exponential Lévy model (variance gamma), and one stochastic volatility model, Heston (1993). Finally, we illustrate how our expansion can be used to perform a model-free calibration of the empirically observed implied volatility surface.

Keywords

Implied volatility expansionexponential Lévyaffine classHestonadditive process

MSC classification

Primary: 91B70: Stochastic models
Secondary: 60F99: None of the above, but in this section 60F10: Large deviations
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 47 , Issue 3 , September 2015 , pp. 837 - 857
Copyright
Copyright © Applied Probability Trust 2015 

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