Short Rate Models: Vasicek And CIR

A thermostat doesn't hold a room at exactly 70 degrees — it constantly nudges the temperature back toward that setpoint, correcting harder the further the room has drifted, while the room also picks up random jostling from doors opening and people walking through. A short rate model treats the interest rate the same way: not a fixed number, and not a random walk that wanders forever, but a value that gets pulled back toward some long-run average, disturbed by ongoing random noise. The Vasicek and CIR models are the two simplest, most widely taught versions of that idea, and the entire difference between them comes down to one question: is the noise allowed to push the rate below zero?

The Vasicek model

$$dr_t = a(b - r_t)\, dt + \sigma\, dW_t .$$

In plain English: $r_t$ is the instantaneous short rate right now. $b$ is the long-run average rate the model pulls toward — the thermostat's setpoint. $a$ is the speed of pull: how hard the model corrects when $r_t$ strays from $b$: a bigger $a$ means faster reversion. $\sigma$ is the size of the random noise, and $dW_t$ is that noise itself, a random shove of typical size $\sqrt{dt}$. The term $a(b - r_t)$ is positive (pulling the rate up) whenever $r_t$ is below $b$, and negative (pulling it down) whenever $r_t$ is above $b$ — exactly a thermostat's correction. This is the same Ornstein-Uhlenbeck process used elsewhere in finance, applied specifically to interest rates.

Worked example 1 — mean reversion in action

Take $a = 0.5$, $b = 4\%$, $\sigma = 1\%$, starting rate $r_0 = 2\%$, and one-year time steps ($dt = 1$). Simulate two steps using a specific, deterministic pair of shocks so the arithmetic is checkable by hand.

Step 1, shock $Z_1 = +1$ (one standard deviation up): $r_1 = r_0 + a(b - r_0)\,dt + \sigma Z_1 = 0.02 + 0.5(0.04 - 0.02)(1) + 0.01(1) = 0.02 + 0.01 + 0.01 = 0.04$, or 4%. Notice the pull term alone, $0.5 \times 0.02 = 0.01$, moved the rate a full percentage point closer to the 4% target — mean reversion is already doing real work in a single step.

Step 2, shock $Z_2 = -1$: $r_2 = r_1 + a(b - r_1)\,dt + \sigma Z_2 = 0.04 + 0.5(0.04 - 0.04)(1) + 0.01(-1) = 0.04 + 0 - 0.01 = 0.03$, or 3%. Once the rate sits exactly at the long-run average, the pull term vanishes entirely — there's nothing left to correct — and the rate only moves because of the pure noise term.

Worked example 2 — the negative-rate problem, and CIR's fix

Vasicek's noise term, $\sigma\, dW_t$, has a fixed size no matter how low the rate has already fallen — which means a large enough downward shock can push it below zero. Take $a = 0.5$, $b = 1\%$, $\sigma = 3\%$, $r_0 = 0.5\%$, and a two-standard-deviation downward shock, $Z = -2$: $r_1 = 0.005 + 0.5(0.01 - 0.005)(1) + 0.03(-2) = 0.005 + 0.0025 - 0.06 = -0.0525$, or $-5.25\%$. A clearly unrealistic result for most economic environments before the era of deliberately negative central bank rates, and exactly the criticism leveled at Vasicek from day one.

CIR (Cox-Ingersoll-Ross) fixes this with one change: scale the noise by the square root of the current rate.

$$dr_t = a(b - r_t)\, dt + \sigma \sqrt{r_t}\, dW_t .$$

Rerun the identical scenario under CIR, same $a$, $b$, $\sigma$, $r_0 = 0.5\%$, same shock $Z = -2$: the noise term is now $\sigma \sqrt{r_0}\, Z = 0.03 \times \sqrt{0.005} \times (-2) = 0.03 \times 0.0707 \times (-2) = -0.00424$. So $r_1 = 0.005 + 0.0025 - 0.00424 = 0.00326$, or 0.326% — still positive. As the rate approaches zero, $\sqrt{r_t}$ shrinks toward zero too, so the random shove shrinks right along with it, and (given the parameters satisfy a technical condition called the Feller condition) the rate can get arbitrarily close to zero but never actually cross it. Same shock, same starting point, same pull toward the mean — the only difference is that CIR's noise term knows to quiet down near zero, and that single square root is the entire model's improvement over Vasicek.

Sample paths above. Both Vasicek and CIR look like this — wandering, but always pulled back toward a central level — the difference worked example 2 highlighted only shows up down near zero, where Vasicek's paths can cross into negative territory and CIR's cannot.

What this means in practice

Both models give closed-form (exact, no simulation needed) formulas for zero-coupon bond prices and are still taught as the entry point to interest rate modeling because the mean-reversion mechanic — rates don't drift forever, they get pulled back toward some economically sensible level — is genuinely how central banks behave, targeting a policy rate and correcting when the economy strays from it. In practice, trading desks mostly use richer extensions (like the Hull-White model, Vasicek with a time-varying mean that can be calibrated to match today's whole yield curve exactly) rather than the plain versions here, but every one of those extensions inherits the same mean-reverting skeleton these two models introduce.

Practice

1. Using Vasicek with $a = 0.4$, $b = 3\%$, $\sigma = 1.5\%$, $r_0 = 5\%$, and a shock of $Z = 0$ (no noise, pure reversion), compute $r_1$ after one year.
2. Using CIR with the same $a$, $b$, $\sigma$, but $r_0 = 3\%$ (already at the mean) and a shock of $Z = -1$, compute the size of the noise term $\sigma\sqrt{r_0}\,Z$ — how does it compare to what plain Vasicek would have given for the same shock?