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Interpreting Alpha and Nu in QuantLib’s SABR Model

Article Quant Q&A · Author: Alexander CHP
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Summary

This note clarifies how QuantLib names two parameters in the SABR volatility model. The QuantLib parameter alpha represents the volatility process’s initial value, while nu represents the volatility of that process, often called vol of vol. The answer distinguishes these meanings from parameter labels that may be encountered in other descriptions of SABR, where naming can differ and cause confusion.

It also explains what the ql.sabrVolatility function returns: implied Black volatility for a specified strike, forward, expiry, and SABR parameter set. It does not return realized volatility of volatility. The note gives a direct conceptual answer but does not discuss calibration, parameter estimation, implementation details, or how results vary with conventions and inputs.

Key ideas

  • QuantLib’s SABR alpha is the initial value of the volatility process.
  • Nu represents the volatility of the volatility process.
  • The SABR volatility function returns implied Black volatility for the supplied strike, forward, expiry, and parameters.
  • The function does not report realized vol of vol.

Tags

OptionsVolatilityDerivatives pricing
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# QuantLib interpreting parameters nu and alpha in SABR model


# QuantLib interpreting parameters nu and alpha in SABR model












I would like to find out what represent "nu" and "alpha" parameters in QuantLib according to Wikipedia here: https://en.wikipedia.org/wiki/SABR_volatility_model

Is nu from QuantLib the diffusion parameter "alpha" of the volatility dynamics from Wiki? Or is it the initial vol, sigma0 of the dynamics? Same for alpha in QuantLib: is the initial vol (sigma0) or the alpha from wikipedia?

In other words, which parameter from QuantLib SABR is the initial sigma0 , alpha or nu?

Number 2: Also ql.sabrVolatility measures the implied Black Volatility of the forward rate given a certain expiry and parameters, or measures the realized vol of vol?

## Answer by user35980 (score 2, accepted)

https://quant.stackexchange.com/a/82220

In quantlib, $\alpha$ is the initial vol (the $t=0$ value of the volatility process) and $\nu$ is the volatility of said process (the "vol of vol").

`ql.sabrVolatility` returns the implied volatility for a given strike, forward, time to expiry and set of SABR parameters ($\alpha,\beta,\nu,\rho$).

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.