SABR Calibration, Negative Strikes, and ATM Parameter Fitting

Summary

The discussion considers whether the standard SABR implied volatility approximation can be calibrated to quotes with negative strikes. The accepted answer explains that the cited Hagan formula relies on a log-moneyness expression and does not apply when forward and strike have opposite signs; its Taylor expansion also fails in that case. Thus, the standard expression is intended for nonnegative strikes and is not a general solution for markets that permit negative rates or prices.

For a basic calibration exercise, the reply suggests fixing beta first to choose the volatility backbone: beta zero corresponds to normal volatility, while beta one corresponds to lognormal volatility. Then alpha, rho, and nu can be fitted to at-the-money market volatility, using an ATM approximation consistent with the chosen beta. The answer cautions that Hagan’s formula is a short-maturity asymptotic approximation, so calibration errors may grow at longer maturities and tenors. The thread does not provide a full calibration procedure or discuss alternatives for negative strikes.

Key ideas

• The standard Hagan SABR approximation is not suitable when forward and strike have opposite signs.
• Its log-moneyness formulation limits its use with negative strikes.
• Fix beta to select the volatility backbone before fitting alpha, rho, and nu.
• The at-the-money approximation used in fitting should match the selected beta.
• Because the formula is a short-term asymptotic approximation, errors may increase at longer maturities and tenors.

Tags

# simple SABR model & negative strikes


# simple SABR model & negative strikes












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

## Answer by FunnyBuzer (score 0, accepted)

https://quant.stackexchange.com/a/45934

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 $\beta$.

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