Hull-White Model

Vasicek's model says the short rate is pulled, on average, toward some long-run level $\theta$ that you fix once and for all. That's a reasonable story for a rate that wanders around a stable equilibrium, but it has a practical flaw: with one constant $\theta$, the model produces one particular shape of yield curve, and that shape almost never matches the actual curve trading in the market today. A trading desk pricing a bond option needs a model that reproduces today's curve exactly — otherwise the model is mispricing the simplest instruments before it even gets to the option. Hull-White's fix is to stop treating the mean-reversion target as one fixed number and instead let it be a function of time, $\theta(t)$, chosen precisely so the model's own curve matches the market's.

The model

The short rate evolves as

$$dr_t = \big(\theta(t) - a\, r_t\big)\, dt + \sigma\, dW_t,$$

where $r_t$ is the instantaneous short rate, $a$ is the speed of mean reversion (how fast the rate gets pulled back after a shock), $\sigma$ is the constant volatility of the rate, and $dW_t$ is the random shock driving it (Brownian motion). In plain English: the rate drifts toward a moving target $\theta(t)$ at speed $a$, with random noise of size $\sigma$ layered on top — and because $\theta(t)$ is allowed to change with time instead of being one fixed number, the model has just enough freedom to force its average path to line up with whatever yield curve the market is actually quoting today, a property called being "arbitrage-free to the initial curve."

Worked example 1 — solving for the drift target

Suppose the market curve implies an instantaneous forward rate of 5.00% at $t = 2$ years, with its slope (derivative with respect to maturity) equal to 0.30% per year at that point. With mean-reversion speed $a = 0.10$ and volatility $\sigma = 1.0\%$, the standard Hull-White fitting formula gives $\theta(t) = \frac{\partial f(0,t)}{\partial t} + a f(0,t) + \frac{\sigma^2}{2a}\big(1 - e^{-2at}\big)$. Plugging in: $0.30\% + 0.10 \times 5.00\% + \frac{0.0001}{0.20}(1 - e^{-0.40}) = 0.30\% + 0.50\% + 0.0005 \times (1 - 0.6703) = 0.80\% + 0.0005 \times 0.3297 = 0.80\% + 0.0165\% \approx 0.8165\%$. This $\theta(2)$ is the specific pull-target the model needs at $t=2$ to stay consistent with the observed curve there.

Worked example 2 — mean reversion changes the path's shape

With $a = 0.10$ (slow reversion) versus $a = 0.50$ (fast reversion), the same starting shock decays very differently: after a 1% rate shock, the expected excess rate after 5 years is $1\% \times e^{-0.10 \times 5} = 1\% \times 0.6065 = 0.6065\%$ still lingering under slow reversion, but only $1\% \times e^{-0.50 \times 5} = 1\% \times 0.0821 = 0.0821\%$ remaining under fast reversion — the fast-reverting model has essentially forgotten the shock, while the slow-reverting one still carries most of it five years later. This single parameter $a$ is what controls how "sticky" rate shocks are, and it's typically calibrated to match the shape of the volatility term structure, not just the yield curve.

Drag the mean-reversion speed and watch simulated paths snap back to their target faster or slower — that's exactly the $a$ parameter above, and Hull-White uses the same mechanics with a target that moves over time to match the curve.

What this means in practice

Hull-White remains one of the most widely used short-rate models for pricing Bermudan swaptions and callable bonds because it fits today's curve exactly and stays analytically tractable (bond prices and even European swaption prices have closed forms), while still needing only a lattice or Monte Carlo simulation for the early-exercise features that make those products hard.