# What SABR $\beta$ to use for EURIBOR swaption smiles

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.

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Hi, just to be sure have you incorporated the Basis Swap that exists between E6M and E3M ? Regards –  TheBridge Feb 21 '11 at 14:50
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## 1 Answer

There's a paper by Fabio Mercurio called "Smiling at Convexity" which discusses this and proposes doing basically what you've done, namely setting beta to match the market prices of CMS swaps.

In the Hagan et. al. SABR paper they discuss ways of setting beta based on plotting ATM vols versus the forward rates. The idea here is to plot log vol vs log fwd rate, the slope of which is 1 - $\beta$. This gives you a rough idea, but it's very noisy, so you wouldn't really be able to distinguish (say) $\beta$ = 0.5 and beta = 0.6.

The convexity adjustment will be very sensitive to how you extrapolate the smile, which in the SABR model is strongly affected by the value of beta. You also have to be a bit careful with how far you take the extrapolation, the SABR model will just keep the vol going up forever as you integrate the strike up to infinity.

EDIT: Just to be clear, there's no unique answer, different desks will do different things. Fitting the CMS swaps is probably a good idea, but given only one parameter $\beta$ you would only get a best fit. I've not tried to do this, but I suspect that your $\beta$ values would be very stable against moves in the the CMS market, which seems unsatisfiying, as you'd expect $\beta$ to change very slowly with time.

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Thanks a lot for the tip on the Mercurio paper! I thought I had read pretty much all the articles on the subject, but this was missing. –  Søren Skov Feb 23 '11 at 9:20
I agree with the rest of your answer. The plotting method discussed by Hagan et al. (for potential outside readers: javaquant.net/papers/hagan_2002_managing.pdf) is indeed not very "fulfilling" if you ask me. The SABR beta seems to require a bit more attention, if you NEED to get the "correct" value - if you merely want to fit an observed smile (say, for interpolation) pretty much any beta will do. Thanks a lot for your very relevant reply! –  Søren Skov Feb 23 '11 at 9:28
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