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) .
After revision:  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) .