How Hull–White Calibrates the Risk-Neutral Short-Rate Drift

Summary

The document explains the drift in the extended Vasicek, or Hull–White, short-rate model. Under the risk-neutral measure, the short rate follows a mean-reverting process with a time-dependent drift term and volatility. The drift is not itself the riskless return; it describes how the short rate evolves under the chosen pricing measure.

The model calibrates its time-dependent parameter to reproduce the market discount curve. The stated condition matches the model’s expected discount factor to the observed discount price for each maturity, and the response says this calibration can be done explicitly. The note offers a concise explanation rather than a derivation: it does not show the calibration formula or discuss assumptions, parameter estimation, or how model fit is assessed beyond matching discount factors.

Key ideas

• The Hull–White short-rate drift is defined under a risk-neutral measure.
• The drift describes the evolution of the short rate, rather than representing the riskless return itself.
• A time-varying parameter is calibrated so model discount factors match market discount prices.
• The document asserts that this calibration is explicit but does not provide the derivation.

Tags

# Hull-White Extension of Vasicek Model


# Hull-White Extension of Vasicek Model












I am reading the book Interest Rate Models by Brigo and Mercurio and try to understand the Hull White Model Extended Vasicek Model. They start off by defining the instantaneous short-rate process under the risk-neutral Measure by

\begin{align} dr(t)=[\theta (t) -a(t)r(t)]dt + \sigma dW(t) \end{align}

with $\theta $, $a, $ $\sigma$ being deterministic functions of time. I dont fully understand why those dynamics describe the risk neutral one. Doesnt it mean that the drift $\theta (t) -a(t)r(t)$ is the riskless return and if yes why?

Thanks for any help.

## Answer by Antoine Conze (score 2, accepted)

https://quant.stackexchange.com/a/39003

$\theta(t) - a(t) r(t)$ is the risk neutral drift. The Hull & White models posits the dynamics $dr(t) = (\theta(t) - a(t) r(t)) dt + \sigma dW(t)$ under the risk neutral measure $P$ and then calibrates $\theta(t)$ so that the risk neutral condition $$ E^P\left[e^{-\int_0^T r(u) du} \right]=\text{discount}(T) $$ is satisfied and $P$ is indeed the risk neutral measure. Calibration of $\theta(t)$ is exact and explicit.

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