Fitting Hull–White Swaption Volatility with Time-Dependent Sigma

Summary

The document addresses a one-factor Hull–White calibration that fits mean reversion and a single volatility parameter to market normal-volatility quotes for at-the-money swaptions. The reported fit is poor across quotes, raising the question of how to represent the observed volatility term structure. The response recommends replacing constant volatility with a time-dependent volatility function, commonly modeled as piecewise constant across time intervals.

With that added flexibility, the model can reproduce the swaption quotes exactly under the response’s stated calibration conditions: calibration on a vertical or diagonal set. The answer does not claim exact fit for every possible collection of quotes or calibration design. It also says the respondent is unsure whether QuantLib has a built-in piecewise-constant-volatility Hull–White implementation, and suggests extending the volatility function if needed. Thus, the key modeling point is that constant volatility imposes a restrictive fit, while time-varying sigma can better match quoted term structures; implementation availability should be checked for the specific library version.

Key ideas

• A constant-volatility Hull–White model may fit a set of swaption normal-volatility quotes poorly.
• A time-dependent volatility function adds flexibility to match the observed volatility term structure.
• Piecewise-constant volatility is a common representation of time-varying sigma.
• The response says exact reproduction is possible when calibrating on a vertical or diagonal quote set.
• The answer is uncertain about built-in QuantLib support and suggests extending the volatility function if required.

Tags

# Quantlib HW 1f model calibration not fitting to market normal vol quotes


# Quantlib HW 1f model calibration not fitting to market normal vol quotes












I am using Quantlib python to calibrate HW 1f model parameters from normal swaption vols quoted in the market (following the code in the cookbook - I fit both the mean-reversion & vol to market quotes)

I see the HW 1f vol is an average of the market implied vol term structure (for given expiry and different maturities for example) and fits very poorly to all the quotes.

What's a recommended approach/model to match the vol term structure observed from ATM swaption vols? Does Quantlib support any other advanced models?

Thanks, Sumit

## Answer by BrownianBread (score 2, accepted)

https://quant.stackexchange.com/a/68563

You will need a time-dependent volatility function rather than a constant volatility, typically a piece-wise constant volatility function is used and can reproduce the swaption vols exactly (presuming you are calibrating on a vertical or diagonal).

I'm not sure there is an implementation of the piecewise constant vol HW in QuantLib, hopefully someone can correct if I'm wrong. You may need to extend the sigma function to be time-dependent.

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