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I have heard that the SABR volatility model was not good at pricing a constant maturity swap (CMS). How is that?

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Hi Al Khawarizmi, welcome to quant.SE. Do you have a source you can cite for your assertion? – Tal Fishman Sep 27 '11 at 15:13
That was a question @ a BNP Paribas interview. I have to admit that I had no clue about the answer although I see the advantages of the SABR dynamics and the way it improved the delta hedging (as explained in the "founding" article). I still remember the interviewer mentioning higher rates and maturities though – Averroes Sep 27 '11 at 15:29

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.

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The link is broken. Is it possible to update it? – Gordon Apr 13 at 16:47

The SABR model has an overly fat right tail. If you do the CMS replication using cash-settled swaptions you find that you need ridiculously high strikes.

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What you say is true (I experienced that) nevertheless using those ridiculous high strike swaptions, CMS Swaplet, Caplet, Floorlet using Hagan's type replication argument, this works quite well in practice (at least for my needs over EURIBOR products). How to interpret that and what model to use instead of SABR ? Best Regards – TheBridge Jan 17 '12 at 9:09

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