This paper shows that Edgeworth expansions for option valuation are equivalent to approximating option payoffs using Hermite polynomials. Consequently, the value of an option is the value of an infinite series of replicating polynomials. The resulting formulas express option values in terms of skewness, kurtosis, and higher moments. Unfortunately, the Hermite series diverges for fat-tailed models, so we provide an alternative spanning series based on logistic polynomials. The new moment-based formulas are a computationally efficient alternative to Fourier transform valuation and can value options even when the characteristic function is not known. Applications include a series for Heston's (1993) stochastic volatility model, and the first convergent solution for the Hull and White (1987) model.