I have heard that the SABR volatility model was not good at pricing a constant maturity swap (CMS). How is that?
Here's a research note devoted to pricing of CMS by means of a stochastic volatility model. The authors indicate in the Introduction that
an analysis of the coupon structure leads to the conclusion that CMS contracts are particularly sensitive to the asymptotic behavior of implied volatilities for very large strikes. Market CMS rates actually drive the option market in extreme strike regions and indicate that implied volatilities ﬂatten out and converge asymptotically to a constant. This behavior is not consistent with the rapidly diverging asymptotics which are implied by SABR.
Late to the game here and an already well-answered question, but an important one nonetheless. In a nutshell, stochastic volatility models are generally a weak solution for products that depend mainly on the terminal distribution (and CMS is a ubiquitous case of this) because of poor fitting of the vol surface (especially on wing strikes). Terminal swap rate models (linear, exponential ...etc) are usually a much better option.