I am currently wrapping up my thesis. My final chapter is on applying the SABR model model for pricing purposes. I am valuing a constant maturity swap by replicating its value using plain vanille European payer and receiver swaptions as described by P. Hagan (http://pds4.egloos.com/pds/200702/26/99/convexit.pdf) in equation (2.19a). I use the SABR model to inter- and extrapolate market volatility smiles.
To be more specific, I am pricing a 5Y CMS swap swapping the 10Y EURIBOR6M swap rate against a floating payment of EURIBOR3M with payments being made quarterly. I have (somewhat arbitrarily) chosen to price the CMS swap as if today is June 1st 2010, but $\beta$ should be fairly stable and a contemporary estimate would be equally helpful/interesting. My "problem" is, that using $\beta=0.25$ gives me a CMS spread of 162 bp while using $\beta=0.85$ gives me a CMS spread of 176 bp. The Bloomberg mid quote for this specific product on June 1st 2010 is 175.5 bp, but I feel that simply choosing the $\beta$ that fits better is not very... academic.