We propose a very fast and accurate algorithm for pricing swaptions when the underlying term structure dynamics are affine. The efficiency of the algorithm stems from the fact that the moments of the underlying asset (i.e., a coupon bond) possess simple closed-form solutions. These moments uniquely identify the cumulants of the distribution. The probability distribution of the asset's future price is then estimated using an Edgeworth-expansion technique. The speed of the approach follows from the fact that no numerical integrations are ever performed, while the accuracy stems from the fact that the cumulants decay very quickly. Using as an example a three-factor Gaussian model, we obtain prices of a 2-10 swaption in less than .05 seconds, with an absolute error of less than a few parts in $10^{-6}$. An added benefit of our approach is that prices of swaptions across multiple strikes can be estimated at (virtually) no additional computational cost. Finally, the method intrinsically provides an estimate of the pricing error.