  For any strictly positive martingale $S = \exp(X)$ 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 L\'evy model (Merton), one
infinite activity exponential L\'evy model (Variance Gamma), and one stochastic
volatility model (Heston). Finally, we illustrate how our expansion can be used
to perform a model-free calibration of the empirically observed implied
volatility surface.
