Calibrating a Constant-Parameter Hull–White Model to Historical Rates

Summary

The document poses a calibration problem for a one-factor Hull–White short-rate model with constant volatility and mean-reversion speed, using historical zero rates. It proposes inferring mean reversion from the relationship between the historical volatilities of one-year and ten-year zero rates, then choosing volatility to match the model variance of the one-year rate. It asks whether those parameters would also reproduce the ten-year rate’s variability.

The proposed comparison is framed through the variance of accumulated short rates and the variance of maturity-scaled zero rates. However, the document contains only the question and no answer or derivation. It therefore does not establish that the suggested calibration is valid, provide the required variance expression, or discuss how the time-varying drift is handled. The approach should be read as an open modeling question rather than a demonstrated procedure.

Key ideas

• The proposed model uses constant volatility and constant mean-reversion speed in a one-factor Hull–White setup.
• The question suggests estimating mean reversion from historical volatility differences across maturities.
• It proposes choosing volatility by matching modeled and historical rate variability.
• No derivation or answer is provided to validate the calibration or calculate the accumulated-rate variance.

Tags

# Calibrating Hull-White model using historical data


# Calibrating Hull-White model using historical data












I'm in search of a way to calibrate a very simple Hull-White model with a constant volatility and a constant mean-reversion speed, purely based on historical zero rates.

$$dr(t) = (\theta(t) - \alpha r(t))dt + \sigma dW(t)$$

My intuition tells me that I can use the relationship between historical 1Y zero rate- and 10Y zero rate volatilities to find an implied mean reversion for the process. With this I can find the sigma which produces a variance on the 1Y return equal to the historical volatility of the 1Y zero rate. (If this works like I hope then this choice of sigma should satisfy the same condition for the 10Y zero rate).

I believe I'm looking for sigma and alpha satisfying the following relationship: $$Var[e^{\int_0^Tr(t)dt}] = Var[e^{z(0,T)T}]$$

For T=1 and T=10, where z(0,T) are the historical zero rates. The right hand side can easily be found in the historical data, so what I need is a way of expressing the left hand side in terms of sigma, alpha, theta, and T. Taking log of both sides I'm essentially looking for an expression for:

$$Var[\int_0^Tr(t)dt].$$

Does this approach make sense to begin with? And if so, does anyone know how to proceed with such a calibration?

Thanks in advance!

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