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Calibrating SABR and Using Black Implied Volatility for Swaptions

Article Quant Q&A · Author: Ken
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Summary

The document discusses how SABR calibration relates to Black pricing conventions for swaptions. Its central point is that the parameters in the two approaches serve different roles: Black pricing uses an implied volatility, while SABR describes volatility through parameters such as alpha, vol-of-vol, and beta. These parameters cannot be transferred directly from one model to the other as though they had the same meaning.

A practical bridge is to derive a Black-equivalent implied volatility from SABR and then use that volatility in the Black pricing formula. The replies also suggest calibrating to at-the-money volatility and using SABR to capture the pattern of out-of-the-money volatilities. The exchange is brief and does not specify a calibration procedure, conventions, or the conditions under which the approximation is accurate. It therefore offers a conceptual distinction and a modeling route, rather than a complete swaption calibration guide.

Key ideas

  • Black implied volatility and SABR parameters represent different model quantities.
  • SABR parameters should not be treated as direct replacements for Black volatility.
  • A SABR result can be expressed as a Black-equivalent implied volatility for pricing.
  • SABR calibration can use at-the-money information while modeling volatility variation across strikes.

Tags

OptionsFixed incomeDerivatives pricingVolatility
Full text
# SABR model inconsistent with Black Swaption Pricing


# SABR model inconsistent with Black Swaption Pricing












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

Ken

## Answer by Lucas Morin (score 2)

https://quant.stackexchange.com/a/11128

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:

http://en.wikipedia.org/wiki/SABR_volatility_model

## Answer by user3264325 (score -1)

https://quant.stackexchange.com/a/10784

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

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.