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SABR Question

Why does the market take the $\beta$ parameter as a "constant"?

  • I see most brokers quoting SABR parameters nowadays.
  • I've seen many banks use $\beta$=0.5 as a rule.
  • I've seen quants select a $\beta$ based on best fit to calibration instruments.

What is most correct to the spirit of the paper, and explain any issues to anticipate with IMR/IPV processes.

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Managing Smile risk from Hagan et. al.

Generally if you pre-select $\beta$, it is from a priori considerations.

  • $\beta = 1$ corresponds to stochastic lognormal
  • $\beta = 0$ is stochastic normal
  • $\beta=1/2$ CIR

In the SABR model, beta is usually calibrated first, followed by the other 3 parameters. Frequently, instead of calibrating beta, it is simply assumed to have $\beta=1/2$ (since CIR is widely used). That said, JPY it is also natural to select 0 for JPY due to negative rates.
The paper explains both. "Aesthetic" consideration (a priori - fixed) or determined form historical observations.

Click here for an intuitive explanation of the SABR model. How to estimate $\beta$ can be seen in this answer.

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