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The Feller Condition and Positivity in the CIR Model

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The Formal Theorem

Consider the Cox-Ingersoll-Ross (CIR) stochastic differential equation given by drt=κ(θ−rt)dt+σrtdWt dr_t = \kappa(\theta - r_t)dt + \sigma \sqrt{r_t} dW_t , where κ,θ,σ>0 \kappa, \theta, \sigma > 0 are constants. The process rt r_t remains strictly positive for all t≥0 t \geq 0 if and only if the Feller condition holds:
2κθ≥σ2 2\kappa\theta \geq \sigma^2

Analytical Intuition.

Imagine the interest rate rt r_t as a particle diffusing in a one-dimensional landscape, tethered to a mean θ \theta by a restoring force κ \kappa . The 'diffusion' term σrtdWt \sigma \sqrt{r_t} dW_t represents random fluctuations; crucially, as rt r_t approaches zero, the volatility vanishes. However, the deterministic drift κθdt \kappa \theta dt pushes the process back toward positive territory. The Feller condition is a delicate, high-stakes equilibrium. If the volatility σ2 \sigma^2 is too aggressive, the random 'kicks' can overcome the drift, forcing the process to hit the absorbing boundary at zero. When 2κθ≥σ2 2\kappa\theta \geq \sigma^2 , the pull toward the mean is powerful enough to outpace the diffusion near the origin. It creates an 'entropic barrier'—the drift essentially pushes the particle away from zero faster than the noise can push it toward it, ensuring that rt r_t never reaches the abyss of non-positivity. It is the mathematical boundary between a process that can touch zero and one that is forever bounded away from it.
CAUTION

Institutional Warning.

Students often mistake the Feller condition for a requirement for stationarity. While it dictates boundary behavior at zero, the process remains mean-reverting regardless of whether 2κθ≥σ2 2\kappa\theta \geq \sigma^2 . The condition specifically guarantees the avoidance of the zero-state, preventing the square root term from collapsing the model's dynamics.

Academic Inquiries.

01

What happens if 2\kappa\theta < \sigma^2?

The process rt r_t will reach zero with probability 1. While the square root diffusion is well-defined at zero, the process can hover there, losing its ability to recover effectively.

02

Is the CIR model valid for negative interest rates?

No, the CIR model is specifically designed to enforce non-negativity. If market data suggests negative rates, the CIR model must be extended, such as in the Shifted-CIR or Hull-White models.

Standardized References.

  • Definitive Institutional SourceCox, J. C., Ingersoll, J. E., & Ross, S. A., A Theory of the Term Structure of Interest Rates.

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Institutional Citation

Reference this proof in your academic research or publications.

NICEFA Visual Mathematics. (2026). The Feller Condition and Positivity in the CIR Model: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-feller-condition-and-positivity-in-the-cir-model

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