Proposed here is a very fast and accurate algorithm for pricing swaptions when the underlying term structure dynamics are affine. The algorithm is efficient because the moments of the underlying asset (e.g., a coupon bond) have 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 approach is fast because no numerical integrations are ever performed; it is accurate because the cumulants decay very quickly. Using as an example a three-factor Gaussian model, the authors obtain prices of a 2-10 swaption in under 0.05 seconds, with an absolute error of only a few parts in 10-6. An added benefit of the approach is that prices of swaptions across multiple strikes can be estimated at virtually no additional computational cost. Finally, the method provides an intrinstic estimate of the pricing error, and remains feasible even when the number of factors is infinite.