My goal is to calibrate a simple SABR model.

I do have $tenor$, $expiry$, $forward$ and "market volatilities for strike spread" ranging from -150 to 150 bps.

I think the model can only be calibrated for strike spreads greater than 0.

Is this correct?

I believe this to be true because: $$ \log (f/K) $$ is only defined if $K \gt 0$ assuming $f \gt 0$

excerpt from Hagan et al (2002) paper link

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1 Answer 1


Yes, the paper you are referring only works for non-negative strikes. In fact, the Taylor expansion does not converge when $f$ and $K$ have different signs. Hagan's formula is a short-term asymptotic approximation, meaning that the calibration error will increase with maturities and tenors. As an exercise, this is the easiest way to calibrate the standard SABR model. First fix the $\beta$ parameter to control the backbone of the volatility surface ($\beta=0$ for normal vol and $\beta=1$ for lognormal vol) and then fit the $\alpha$, $\rho$ and $\nu$ parameters to match the ATM market volatilities. Note that the asymptotic approximation for the ATM implied volatility must be consistent with the choice of $\beta$.

  • $\begingroup$ why only match ATM vols? How about the non-atm vol quotes? Isn't the idea behind SABR is to fit the smile ? $\endgroup$
    – Peaceful
    Oct 24, 2023 at 20:25
  • $\begingroup$ You calibrate the ATM swaptions volatilities because the SABR formula above is a short-term expansion ATM across tenors and maturities. Then you can use the calibrated SABR to interpolate/extrapolate across all strikes. Note this formula is just one of the many results we have on SABR and it suffers from poor performance on the wings and no no-arbitrage guarantee for very low strikes. $\endgroup$
    – FunnyBuzer
    Oct 25, 2023 at 11:41
  • $\begingroup$ thanks. I see your point. However, from data providers, apart from ATM vols, you often also get skews data (non-ATM vols). Are you saying, in the calibration process, the skew data does not need to be used at all? and what exactly would be the calibration process? If you are not using any skew data, how can you calibrate sabr to fit the the smile? $\endgroup$
    – Peaceful
    Oct 25, 2023 at 19:07

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