The Hull–White Drift Adjustment for a One-Factor Short-Rate Model

Summary

The note asks how to determine the time-dependent drift parameter in the one-factor Hull–White short-rate model when pricing zero-coupon bonds. It states the short-rate process as a mean-reverting rate with a time-varying drift and constant volatility, then gives a formula for the drift in terms of the forward rate curve, mean-reversion parameter, and volatility.

The expression adds a volatility adjustment to the forward rate at time zero for maturity t. The document offers no worked market example, calibration procedure, derivation, or discussion of conventions, so it identifies the model input relationship without showing how to estimate the required curve or parameters in practice. Applying it requires care about the meaning and time indexing of the forward rate and consistency with the model’s rate conventions.

Key ideas

• The one-factor Hull–White model describes a mean-reverting short rate with time-dependent drift.
• The note gives a formula linking the drift to the initial forward curve and model parameters.
• The volatility adjustment depends on the mean-reversion parameter and time to maturity.
• Practical use requires calibrated inputs and consistent rate and time conventions.

Tags

# Zero Coupon Bond Price under Hull White Model (One Factor)


# Zero Coupon Bond Price under Hull White Model (One Factor)












While pricing Zero coupon bond using One Factor Hull White model: $$dr(t) = \left(\theta(t) - a r \right)dt + \sigma dW(t)$$

How to determine the value of $\theta(t)$ using real world example: $$θ(t)=F_t (0,t)+σ^2 \frac{1-e^{-2at}}{2a}$$

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