What are the approaches available to price a swaption (either European or American style)? So far it seems Black method is the only one used. Thanks!
The question should not be about the swaption pricing formula, its well established and widely accepted and utilized every single day. The question you SHOULD be asking, however, is which underlying volatility model you are using. Its idential to you questioning the use of B-S in transforming vols -> prices, and prices -> vols in the equity world while you should be asking how to model volatility. In general, you should be thinking about the Brownian motion variables and not deterministic ones when modeling and choosing which model to select in pricing derivatives.
Current going practice to price swaptions is to use the Stochastic Apha, Beta, Rho (SABR) model and its derivatives (such as dynamic SABR,...).
Here is the wiki: http://en.wikipedia.org/wiki/SABR_volatility_model
But more importantly here is the original Hagan paper: