wilmott.com

The main limitation of SABR when it comes to swaption pricing, according to the literature, is the following:

[SABR] should therefore not be used to price exotic products depending on more than one forward rate. To overcome this limitation, extended versions of the LIBOR Market Model, which include stochastic volatility à la SABR, have been developed by various practitioners. As a consequence of this, the SABR (forward interest rate) should not even be used to price swaptions, as their pricing depend on the joint evolution of several forward rates. 

I have two questions regarding this statement:

1. I do not get why the joint evolution of forward rates is a problem. For pricing purposes, don't we just use a forward curve to project all cashflows of the underlying floating swap leg? Why is their joint evolution relevant?
2. If SABR has such a major drawback, does it mean that it is not used in the industry at all for swaption pricing? Has everyone in the industry resorted to using the SABR-Libor Market Model? 

.

Hi - SABR (forward interest rate) model, cannot model the joint evolution of forward rates, as it should take their correlation into account as well as evolving them under a single measure. To take those two requirements into account you need to resort to SABR-Libor Market Model.
The industry uses SABR to price swaptions, however not through a SABR (forward interest rate) model but through a SABR (forward swap) model, where you model directly the swap rate.
Hope this helps.