The theory of option pricing in the GARCH framework has been developed in recent years. However, an efficient method for computing option prices in this framework remains lacking. In this article, a fast analytical approximation is developed for computing European option prices in the GARCH framework. The approach, following that of Jarrow and Rudd (1982), uses the Edgeworth expansion of the risk-neutral density function. Analytical expressions for the first four moments of the cumulative asset return over any horizon under the GARCH model are derived in this paper. A numerical analysis shows that these moment formulas are accurate under fairly general conditions. The analytical GARCH option pricing formula based on the Edgeworth expansion is found to work well for short-maturity options. For long-maturity options, the approximate formula is generally satisfactory, except when the volatility dynamic of the GARCH model exhibits an extremely high level of persistence.