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Estimating Hull–White Parameters from Historical Rate Data

Article Quant Q&A · Author: QuackQuack
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Summary

The document asks how to calibrate the one-factor Hull–White interest-rate model under a historical measure for counterparty credit risk calculations. It distinguishes this from risk-neutral calibration, which commonly fits market prices of swaptions or caps, and asks what historical data to use and how to estimate mean reversion and volatility.

The replies offer only a brief outline: derive the time-dependent drift from the forward curve, then estimate volatility and mean-reversion speed from historical forward-rate observations. They also point to external references and older Matlab notes, but do not explain the estimation procedure, specify data construction choices, or present calibration results. The discussion therefore serves as a starting point rather than a complete recipe. Any historical estimates would depend on rate-series selection, sampling frequency, and model assumptions; these details are not addressed here.

Key ideas

  • Historical calibration estimates Hull–White parameters from observed rate histories rather than option prices.
  • The time-dependent drift can be derived from the forward curve.
  • Volatility and mean-reversion speed are to be estimated from historical forward-rate data.
  • The discussion does not specify a detailed estimation method or resolve data-selection choices.

Tags

Fixed incomeStatisticsDerivatives pricing
Full text
# Historical calibration of Hull-White model


# Historical calibration of Hull-White model












I have a question concerning 1-factor Hull-White model. For my master project I need to calibrate it to compute Counterparty credit risk metrics. I know that the model might be calibrated either for risk-neutral measure (in CVA applications) using market-traded swaptions or caps or for historical measure. This is where I am stuck at the moment. Can anyone offer a good paper / or some idea how to calibrate mean reversion parameter a and variance sigma to the historical data? What historical data do I have to use? And how to perform the calibration step-by-step?

If anyone also has also a solution in Matlab and could share it - that would be also highly appreciated =))

Thanks in advance

## Answer by Kiwiakos (score 1)

https://quant.stackexchange.com/a/33339

You can find Matlab code in these notes: http://cosweb1.fau.edu/~jmirelesjames/MatLabCode/Lecture_notes_2008d.pdf I wrote them 10 years ago and have not revisited since, but it should work.

## Answer by NSZ (score 0)

https://quant.stackexchange.com/a/32710

I hope you can find the answer to your question here: How to calibrate Hull-White from zero curve? However if you want a step-by-step procedure I would resume it like that:

- Calibrate your parameter $\theta(t)$ starting from the forward curve and using the formula in the link above.

- Calibrate volatility from the historical series of the forward curve

- Similarly estimate the mean reversion speed

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.