The Cox-Ingersoll-Ross Process

Short-term interest rates and variance both share an awkward property: they mean-revert like the Ornstein-Uhlenbeck process, drifting back toward some long-run average, but they also can't sensibly go negative — a rate or a variance below zero doesn't mean anything. The plain Ornstein-Uhlenbeck process, with its constant volatility, doesn't respect that boundary and can wander negative in simulation. The Cox-Ingersoll-Ross (CIR) process fixes this with one small but consequential change: it makes the noise shrink toward zero as the level itself approaches zero.

The model

The CIR process is written

$$dr_t = \kappa(\theta - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t,$$

which looks almost identical to the Ornstein-Uhlenbeck (Vasicek) model — the same pull back toward a long-run mean $\theta$ at speed $\kappa$ — except the volatility term is scaled by $\sqrt{r_t}$ instead of being constant. As $r_t$ gets close to zero, that $\sqrt{r_t}$ factor shrinks the randomness toward zero too, so the deterministic upward pull from the mean-reversion term takes over and pushes the process back up before noise can drag it below zero. Under a mild condition on the parameters (known as the Feller condition, $2\kappa\theta \ge \sigma^2$), the process is guaranteed to stay strictly positive for all time; violate that condition and the process can still touch zero, though it's reflected back up rather than going negative.

A short numeric feel for the mechanism: at $r_t = 0.04$ (a 4% rate) the volatility term scales by $\sqrt{0.04} = 0.2$; if the rate drifted down to $r_t = 0.01$, the volatility scaling shrinks to $\sqrt{0.01} = 0.1$ — half the randomness at a quarter of the level, which is exactly the self-damping effect that keeps the process away from the boundary.

What this means in practice

The CIR process is the standard building block for short-rate models (as one leg of models like CIR itself and its many extensions) and, in a different guise, for stochastic variance in models like Heston, where variance also must stay non-negative. Its main practical cost versus the simpler Vasicek model is that the square-root volatility term makes bond and option pricing formulas somewhat more involved (though still closed-form for many cases) and makes naive discretized simulation prone to producing small negative values near zero unless a specialized simulation scheme is used.