I am confused on the following:

When we price swaption, the market convention is to use Black's Model which assumes forward swap rate is following Black's model under the Q(t) measure.

When we tries to use SABR, the forward swap rate is again model depending on the value of Beta.

If we were to calibrate SABR for Swaption volatility, will there be a inconsistency with specifying the model for forward swap rate?

Thanks for clarifying



To calibrate BS you compute volatility $\sigma$, to calibrate SABR you compute implied $\alpha$, the volvol and $\beta$, the skewness. These parameters does not play the same role. So you can't really use the parameters of one models to calibrate another.

But you can build equivalent parmaters, i.e. compute an equivalent vol under SABR to use BS pricing formula.

Look at the asymptotic solution paragraph:



My understanding is you can calibrate with atm vol which should be close with either model then the otm vols are taken care of by the beta parameter from sabr


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