Before revision:  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 L\'evy model (Merton), one infinite activity exponential L\'evy 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 .
After revision:  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 L\'evy model (Merton), one infinite activity exponential L\'evy model (Variance Gamma), and one stochastic volatility model (Heston) .