From Characteristic Functions to Implied Volatility Expansions

25 Pages Posted: 1 Jul 2012 Last revised: 9 Jul 2013

See all articles by Antoine (Jack) Jacquier

Antoine (Jack) Jacquier

Imperial College London; The Alan Turing Institute

Matthew Lorig

University of Washington - Applied Mathematics

Date Written: July 9, 2013


For any strictly positive martingale S=e^X for which X has an analytically tractable 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 log(K/S 0 ). We illustrate the versatility of our expansion by computing the approximate implied volatility smile in three well-known martingale models: one finite activity exponential Levy model (Merton), one infinite activity exponential Levy model (Variance Gamma), and one stochastic volatility model (Heston). We show how this technique can be extended to compute approximate forward implied volatilities and we implement this extension in the Heston setting. 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, Exponential Lévy

Suggested Citation

Jacquier, Antoine and Lorig, Matthew, From Characteristic Functions to Implied Volatility Expansions (July 9, 2013). Available at SSRN: or

Antoine Jacquier

Imperial College London ( email )

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


The Alan Turing Institute ( email )

British Library, 96 Euston Road
London, NW12DB
United Kingdom

Matthew Lorig (Contact Author)

University of Washington - Applied Mathematics ( email )

Seattle, WA
United States

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