Short Rate Models: Vasicek and CIR

Interest rates don't wander off forever in one direction — a central bank sets policy precisely to keep rates anchored around some level it considers appropriate. A short-rate model tries to capture that behavior mathematically: rates jiggle around randomly day to day, but there's a persistent pull back toward a long-run average, the way a stretched rubber band pulls back toward its resting length. Two of the earliest and still most widely taught versions of this idea are the Vasicek and CIR models.

The same idea, two versions

The Vasicek model writes the rate's instantaneous change as:

$$dr = a(b - r)\,dt + \sigma\,dW$$

In words: the rate $r$ drifts toward the long-run level $b$ at speed $a$ (how far away it currently is, times a pull strength), plus a random shock of constant size $\sigma$ driven by a random walk $dW$. Because the random shock's size doesn't shrink near zero, Vasicek can technically push the rate below zero — historically seen as a flaw, though less so once real-world rates actually went negative.

CIR modifies only the random term:

$$dr = a(b - r)\,dt + \sigma\sqrt{r}\,dW$$

In words: the same mean-reversion, but now the randomness scales with $\sqrt{r}$ — as the rate gets close to zero, the random shocks shrink toward zero too, so the rate can approach zero but (under the right parameter conditions) never cross it.

Watch how the simulated paths above wander but keep snapping back toward a central level — that pull is exactly the $a(b-r)$ term in both formulas, and it's the defining feature that separates these models from a plain random walk.

Worked example

Suppose $a = 0.5$, $b = 4\%$, and the rate today is $r = 6\%$. The expected drift over the next instant is:

$$a(b - r) = 0.5 \times (4\% - 6\%) = -1\%$$

The rate is expected to drift downward, toward the 4% long-run level, at a pace proportional to how far above it currently sits (2 percentage points, scaled by the speed parameter 0.5). If the rate were instead already at 4%, the drift term would be zero — no pull in either direction, only the random shock remains.

What this means in practice

These one-factor models are simple enough to derive closed-form bond prices from, which made them foundational teaching and pricing tools, though real trading desks today typically use richer multi-factor models to capture the yield curve's full shape. Still, the core intuition — rates mean-revert, and how fast they do determines how far they can realistically move before snapping back — underlies almost every more advanced interest-rate model built since.