Short-Rate Interest Rate Models › The Cox-Ingersoll-Ross Model

John Cox, Jonathan Ingersoll, and Stephen Ross developed their model in 1985 as part of a general equilibrium framework for asset pricing. Unlike Vasicek, who specified the rate dynamics directly, the researchers derived theirs from technological assumptions and investor preferences embedded in market equilibrium. The CIR model modifies the Vasicek specification by making volatility proportional to the square root of the rate:

$$dr(t) = \kappa(\theta - r(t))\,dt + \sigma\sqrt{r(t)}\,dW(t)$$

The drift term is identical to Vasicek, preserving the mean-reverting structure. The key innovation lies in the diffusion term:

• $\kappa, \theta$ mean reversion speed and long-run mean
• $\sigma$: volatility coefficient
• $\sqrt{r(t)}$: state-dependent volatility scaling factor
• $dW(t)$: Brownian increment

The parameters $\kappa$, $\theta$, and $\sigma$ have the same interpretations as in Vasicek: mean-reversion speed, long-run mean, and volatility coefficient respectively. The defining difference is the $\sqrt{r(t)}$ term multiplying the Brownian shock.

This square-root diffusion makes volatility depend on the current rate. When $r(t)$ is close to zero, the volatility term $\sigma\sqrt{r(t)}$ becomes small, reducing the chance of large negative shocks that could push rates below zero. The process essentially "quiets down" as it approaches zero. When $r(t)$ is larger, volatility increases proportionally, matching the empirical observation that rate volatility tends to be higher during high-rate environments. Historical data shows that volatility depends on the rate level.

The economic intuition behind the square-root specification connects to the general equilibrium derivation. In the Cox-Ingersoll-Ross framework, the interest rate is linked to the marginal product of capital in the economy. When rates are low, the economy is in a depressed state with limited investment opportunities, and uncertainty about future rates is also low. When rates are high, the economy is more active, and there is greater uncertainty about the path of future rates.

The CIR model keeps rates non-negative under certain parameter restrictions. The Feller condition states:

$$2\kappa\theta \geq \sigma^2$$

This inequality places a constraint on the relationship between model parameters:

• $\kappa$ mean reversion speed
• $\theta$: long-run mean
• $\sigma$: volatility parameter

When this condition holds, the rate $r(t)$ will never hit zero. If it is violated, rates can touch zero but will immediately bounce back; they cannot become negative. This distinction matters for certain theoretical analyses but has limited practical impact, as rates near zero behave similarly in both cases.

The intuition behind the Feller condition is straightforward when viewed as a balance of forces. The condition compares two quantities: the deterministic pull toward $\theta$ (captured by $\kappa\theta$, which represents how fast the drift pushes rates up when they are near zero) and the random shocks (captured by $\sigma^2$, which measures the magnitude of random fluctuations). If volatility is too high relative to the mean-reversion strength, the random shocks can overwhelm the upward drift near zero, potentially pushing the process to zero despite the drift trying to keep it positive.

The factor of 2 in the condition arises from the mathematical properties of the square-root process. When $r$ is small, the drift is approximately $\kappa\theta$ (since $\kappa r$ is negligible), while the variance of shocks is approximately $\sigma^2 r \cdot dt$. For the drift to dominate near zero, we need $\kappa\theta \cdot dt$ to exceed the standard deviation of shocks, which leads to the Feller condition after careful analysis.

Unlike the Vasicek model, the CIR process does not yield normally distributed rates. The square-root diffusion produces a fundamentally different distribution. Specifically, $r(T)$ conditional on $r(t)$ follows a non-central chi-squared distribution, a probability distribution that arises naturally in statistics when summing squares of normal random variables:

$$r(T) | r(t) = \frac{\sigma^2(1 - e^{-\kappa(T-t)})}{4\kappa} \cdot \chi^2_d(\lambda)$$

The non-central chi-squared distribution is characterized by two parameters:

• $\chi^2_d(\lambda)$ non-central chi-squared random variable
• $d$: degrees of freedom
• $\lambda$: non-centrality parameter

For the CIR model, these parameters are:

$$\begin{aligned} d &= \frac{4\kappa\theta}{\sigma^2} \\ \lambda &= \frac{4\kappa e^{-\kappa(T-t)}}{\sigma^2(1 - e^{-\kappa(T-t)})} r(t) \end{aligned}$$

The degrees of freedom $d$ depends only on the model parameters and is constant over time. Notice that when the Feller condition holds exactly (with equality), $d = 2$. The non-centrality parameter $\lambda$ depends on the current rate $r(t)$ and decreases as the horizon increases, which reflects the fading influence of the initial condition. The remaining notation:

• $\kappa, \theta, \sigma$ CIR model parameters
• $r(t)$: current short rate
• $T-t$: time horizon

The conditional mean and variance can be computed from the properties of the non-central chi-squared distribution:

$$\begin{aligned} \mathbb{E}[r(T)|r(t)] &= r(t)e^{-\kappa(T-t)} + \theta(1 - e^{-\kappa(T-t)}) \\ \text{Var}[r(T)|r(t)] &= r(t)\frac{\sigma^2}{\kappa}(e^{-\kappa(T-t)} - e^{-2\kappa(T-t)}) + \theta\frac{\sigma^2}{2\kappa}(1 - e^{-\kappa(T-t)})^2 \end{aligned}$$

These formulas reveal an important relationship between the Vasicek and CIR models:

• $\mathbb{E}[r(T)|r(t)]$ conditional expected value of the rate
• $\text{Var}[r(T)|r(t)]$: conditional variance of the rate
• $r(t)$: current short rate

Notice that the conditional mean formula is identical to the Vasicek model; both processes have the same expected path. The mean reversion operates identically in both cases, pulling rates toward $\theta$ at speed $\kappa$. The difference lies in the variance structure and the shape of the distribution. The CIR variance has two terms: one proportional to $r(t)$ (reflecting the state-dependent volatility) and one proportional to $\theta$ (the contribution from shocks accumulated as the process approaches its steady state).

The non-central chi-squared distribution is positively skewed, unlike the symmetric normal distribution of Vasicek. This skewness becomes more pronounced when rates are low, as the distribution is bounded below by zero but can extend to high values. When rates are high, the distribution becomes more symmetric and approaches normality, consistent with the central limit theorem as the non-centrality parameter grows.

The CIR model is also affine, with bond prices taking the form $P(t,T) = A(t,T)e^{-B(t,T)r(t)}$. The derivation follows the same approach as Vasicek: substitute the assumed form into the bond pricing PDE and solve for the coefficients. The resulting functions are:

$$\begin{aligned} B(t,T) &= \frac{2(e^{\gamma(T-t)} - 1)}{(\gamma + \kappa)(e^{\gamma(T-t)} - 1) + 2\gamma} \\ A(t,T) &= \left[\frac{2\gamma e^{(\kappa + \gamma)(T-t)/2}}{(\gamma + \kappa)(e^{\gamma(T-t)} - 1) + 2\gamma}\right]^{2\kappa\theta/\sigma^2} \end{aligned}$$

The auxiliary parameter $\gamma$ that appears in these formulas combines the model parameters in a specific way:

• $A(t,T), B(t,T)$ deterministic functions for the CIR bond price
• $\gamma = \sqrt{\kappa^2 + 2\sigma^2}$: auxiliary parameter
• $\kappa, \theta, \sigma$: CIR model parameters

The parameter $\gamma$ can be interpreted as an "adjusted" mean-reversion speed that accounts for the risk premium embedded in bond prices. It is always larger than $\kappa$, which reflects that bonds are priced as if rates reverted faster than they do under the physical measure.

While more complex than the Vasicek formulas, these expressions remain analytically tractable and enable rapid bond pricing without numerical integration. As in the Vasicek model, $B(t,T)$ measures the bond's sensitivity to the short rate (duration), while $A(t,T)$ captures the effects of mean reversion and the convexity adjustment. The CIR formulas reduce to the Vasicek formulas in the limit as the square-root term becomes approximately constant (which occurs when rate levels are high relative to their volatility).