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The Cox-Ingersoll-Ross (CIR) Model: Ensuring Positivity and Bond Valuation

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The Formal Theorem

Let the instantaneous interest rate rt r_t be defined by the Stochastic Differential Equation (SDE):
drt=κ(θ−rt)dt+σrtdWt dr_t = \kappa(\theta - r_t)dt + \sigma\sqrt{r_t}dW_t
where κ \kappa is the speed of mean reversion, θ \theta is the long-term mean, and σ \sigma is the volatility parameter. If the Feller condition 2κθ≥σ2 2\kappa\theta \geq \sigma^2 is satisfied, the process rt r_t is strictly positive for all t>0 t > 0 . Under this measure, the price of a zero-coupon bond P(t,T) P(t, T) is given by:
P(t,T)=A(t,T)e−B(t,T)rt P(t, T) = A(t, T)e^{-B(t, T)r_t}
where:
B(t,T)=2(eγ(T−t)−1)(γ+κ)(eγ(T−t)−1)+2γ B(t, T) = \frac{2(e^{\gamma(T-t)} - 1)}{(\gamma + \kappa)(e^{\gamma(T-t)} - 1) + 2\gamma}
A(t,T)=[2γe(κ+γ)(T−t)/2(γ+κ)(eγ(T−t)−1)+2γ]2κθσ2 A(t, T) = \left[ \frac{2\gamma e^{(\kappa + \gamma)(T-t)/2}}{(\gamma + \kappa)(e^{\gamma(T-t)} - 1) + 2\gamma} \right]^{\frac{2\kappa\theta}{\sigma^2}}
and γ=κ2+2σ2 \gamma = \sqrt{\kappa^2 + 2\sigma^2} .

Analytical Intuition.

In the grand theater of financial mathematics, the Vasicek model was a tragic hero, undone by its own symmetry—allowing interest rates to plummet into the unnatural abyss of negative values. Enter the Cox-Ingersoll-Ross (CIR) model, a masterpiece of stochastic engineering. Imagine rt r_t as a restless spirit tethered to a central gravity point θ \theta by a spring with tension κ \kappa . Unlike its predecessors, CIR introduces a state-dependent volatility term, σrt \sigma\sqrt{r_t} . As the interest rate approaches the absolute floor of zero, the very chaos that drives it—the diffusion—begins to evaporate. It is as if the air becomes thinner, slowing the descent until the mean-reverting 'drift' pulls it back toward the sun. This isn't just a mathematical convenience; it's a structural necessity. To ensure this spirit never even grazes the floor, we invoke the Feller Condition: 2κθ≥σ2 2\kappa\theta \geq \sigma^2 . This inequality acts as a centrifugal force, ensuring the origin is unreachable. In the realm of bond pricing, this leads to an elegant affine solution, where the price of a zero-coupon bond is an exponential decay function of the current rate, capturing the curvature of the term structure with cinematic precision.
CAUTION

Institutional Warning.

Students often confuse CIR volatility with the Vasicek model. In Vasicek, volatility is constant, allowing rates to become negative. In CIR, the rt \sqrt{r_t} term ensures volatility vanishes at the origin. Another pitfall is neglecting the Feller condition, which is strictly required to prevent the process from reaching zero.

Academic Inquiries.

01

Why is the square root term rt \sqrt{r_t} so important?

It scales the volatility by the level of the interest rate. As rt r_t approaches zero, the random shocks (diffusion) decrease to zero, allowing the positive drift to pull the rate back up.

02

What happens if the Feller condition is not met?

If 2κθ<σ2 2\kappa\theta < \sigma^2 , the process can touch zero (the origin is 'reflecting'), though it will still stay non-negative. The Feller condition ensures zero is never even reached.

03

What is the probability distribution of the CIR process?

Unlike the Gaussian distribution of the Vasicek model, the future values of the CIR process follow a non-central chi-squared distribution, which accounts for the boundary at zero.

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

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NICEFA Visual Mathematics. (2026). The Cox-Ingersoll-Ross (CIR) Model: Ensuring Positivity and Bond Valuation: Visual Proof & Intuition. Retrieved from https://nicefa.org/library/advanced-stochastic-processes/the-cox-ingersoll-ross--cir--model--ensuring-positivity-and-bond-valuation

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