Why Hull–White Uses a Time-Dependent Mean
Summary
The document explains why the Hull–White interest-rate model allows its mean level to vary over time, unlike the constant long-run mean in the Vasicek model. Mean reversion describes rates being bounded by economic and political forces; it does not imply that rates fluctuate around one fixed level. A time-varying mean allows the model’s short rate to reflect a changing interest-rate environment.
The key modeling distinction is that interest rates form a curve across maturities, rather than a single price series. In the explanation, the relevant mean is therefore a curve associated with market forward rates. Vasicek’s constant long-run mean corresponds to a flat curve and cannot fit a generally non-flat initial market curve. Hull–White replaces the fixed-rate setup with a time-dependent function so the model can match the observed initial term structure. This fit is a modeling feature, not evidence that the model predicts future rates accurately; the document does not discuss calibration details, parameter estimation, or empirical performance.
Key ideas
- Mean reversion does not require interest rates to fluctuate around a constant level.
- The interest-rate market is represented by a curve across maturities, not just one rate.
- A constant long-run mean implies a flat curve and cannot generally match the observed initial term structure.
- Hull–White uses a time-dependent mean function to fit the initial forward curve.
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Full text
# Why is the mean time-dependent in the Hull-White interest rate model? # Why is the mean time-dependent in the Hull-White interest rate model? In the Vasicek interest-rate model, the interest rate reverts to a constant mean. This makes sense to me. In my conception, the mean ought to be time-invariant, since interest rates don't follow an increasing or decreasing trend in the long term. In the Hull-White modified model, the mean is a time-dependent function. I cannot understand why this is the case. ## Answer by FQuant (score 4) https://quant.stackexchange.com/a/8630 The claim that interest rates don't follow long term trends is not consistent with observed data. The idea of mean reversion is that interest rates do not rise or fall without bound, but are limited by economic and political factors. But there is no indication that this oscillation of short rates should happen around a constant mean. Allowing the mean reversion parameters to be time-dependent (as the Hull-White model does) allows the short rate (which is described by the model) to match the term structure of interest rates (forward rates). ## Answer by AFK (score 1) https://quant.stackexchange.com/a/24513 In Equity or FX, you are modelling the dynamic of a single number $S_t$ the stock price or fx spot rate. In interest rate, you are modelling the dynamic of a curve $(f(t,t+\theta))$ all the forward rates (equivalently all the discount factors) for all tenors $\theta$. So mean reversion, in the context of rates should be mean reversion around a mean curve. This mean curve is the forward curve built from the prices of deposits, futures and swaps observed in the market. In the Vasiceck model the short rate mean reverts to a long term mean $r_\infty$. This means that you are mean reverting to a flat curve. But curves observed in the market are never flat and the model cannot even fit the forward curve observed at time 0. The Hull-White model improves on this by allowing you to fit the current expectation of the rates curve and mean revert to it. This requires replacing two of your parameters (the initial and long term short rates) by a whole curve (the initial forward curve which appears in the model in the form of the tenor dependent mean $\varphi(t)$). Hope this helps
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