Calibrating SABR Parameters Across Option Maturities

Summary

The document asks how to calibrate a SABR volatility surface for S&P 500 index calls when the beta parameter is fixed in advance. The choice is between fitting alpha, rho, and nu separately for each expiration or fitting one parameter set across the full matrix of market implied volatilities and strikes.

The answer recommends a shared parameter set across maturities for the same underlying, while noting that a term structure for the parameters could be introduced. Its stated qualification is that when SABR implied volatilities are generated with the Hagan approximation, the parameters are assumed constant. This is a concise calibration recommendation rather than a worked procedure: it gives no objective function, data checks, fit diagnostics, or evidence comparing the two approaches. The advice is therefore tied to the stated approximation and modeling setup, and should not be read as a universal rule for every SABR implementation or volatility surface.

Key ideas

• With beta fixed, the question is whether to fit SABR parameters by maturity or jointly across maturities.
• The answer recommends one parameter set for the same underlying when using the stated Hagan approximation.
• A term structure for SABR parameters is mentioned as a possible extension.
• The document gives no calibration algorithm or comparison of fit quality across the approaches.

Tags

# SABR Calibration


# SABR Calibration












I need to generate the Volatility Surface of call options on S&P500 index, my dataset contains implied volatilities regarding various expiration dates for various strike prices.

My doubt is, given beta in advance, in order to get a surface should I recalibrate the model parameters (alfa,rho,nu) for every different exipiration date or I have to run the LSQ non linear between the matrix containing the market volatilities and the volatilities by sabr all togheter? (So my surface will be based on a single set of parameters and not one for each maturity)

Thanks for the help

## Answer by Canardini (score 1)

https://quant.stackexchange.com/a/49963

You are observing the same underlying $S_t$, therefore it has to be one set of parameters for all maturities. You could add a term structure to the parameters , however , since you are using SABR, I assume you use Hagan expansion to generate the implied vols, and for this approximation, the parameters must be constant.

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