CIR Model: Square-Root Diffusion and Non-Negative Interest Rates

Published: April 13, 2026
Ryan O'Connell, CFA, FRM
Article by Ryan O'Connell, CFA, FRM

When pricing interest rate derivatives or managing bond portfolio risk, the choice of interest rate model matters. The Cox-Ingersoll-Ross (CIR) model has been a go-to framework for practitioners who need rates to stay non-negative while retaining closed-form bond pricing. This guide covers the CIR stochastic differential equation, the critical Feller condition, zero-coupon bond pricing, and how CIR compares to other short-rate models like Vasicek.

What Is the Cox-Ingersoll-Ross (CIR) Model?

The CIR model is a one-factor short-rate model that describes how the instantaneous interest rate evolves over time. It was developed by John Cox, Jonathan Ingersoll, and Stephen Ross as an extension of the earlier Vasicek model, with one crucial innovation: the square-root diffusion term that prevents interest rates from going negative.

Key Concept

The CIR model combines mean reversion (rates tend toward a long-term average) with state-dependent volatility (rate fluctuations scale with the current rate level). This makes it an affine term structure model with closed-form solutions for zero-coupon bond prices.

The model belongs to the family of equilibrium models derived from general equilibrium arguments, unlike no-arbitrage models like Hull-White that are calibrated to match the current yield curve exactly. CIR remains foundational in fixed income theory, underlying much of modern interest rate risk management and bond pricing methodology.

The CIR Stochastic Differential Equation

The CIR model describes the evolution of the short rate r(t) through the following stochastic differential equation:

CIR Stochastic Differential Equation
dr = κ(θ − r)dt + σ√r dW
The short rate follows mean-reverting dynamics with square-root diffusion

Where:

  • r — the instantaneous short rate
  • κ (kappa) — speed of mean reversion (how fast rates return to the mean)
  • θ (theta) — long-term mean level of interest rates
  • σ (sigma) — volatility coefficient
  • dW — increment of a standard Wiener process (Brownian motion)

The critical difference from the Vasicek model is the σ√r term. In Vasicek, volatility is constant (σ). In CIR, volatility scales with the square root of the current rate level. This state-dependent volatility has an important economic interpretation: the conditional variance of rate changes is σ2r dt, meaning rate fluctuations are larger when rates are high and smaller when rates are low.

Pro Tip

The square-root process (also called the Feller process) appears throughout finance. It’s the same process used for variance in the Heston stochastic volatility model. Understanding CIR helps you grasp a broader class of models.

Why the CIR Model Prevents Negative Interest Rates

The mathematical structure of the CIR model guarantees that interest rates remain non-negative for all time, provided they start non-negative. Here’s the intuition:

As the rate r approaches zero, the diffusion term σ√r vanishes (since √0 = 0). This means randomness disappears near zero. Meanwhile, the drift term κ(θ − r) remains positive (since θ > 0 and r ≈ 0), pushing rates back up. The combination ensures rates cannot cross into negative territory.

Key Concept

Under standard assumptions (κ > 0, θ ≥ 0, σ > 0, r(0) ≥ 0), the CIR process is always non-negative. The Feller condition (discussed below) determines whether zero is attainable or unattainable, but even when attainable, rates cannot go negative.

This non-negativity property was historically considered a major advantage over Vasicek. However, when the ECB introduced negative deposit rates in June 2014 and the BOJ followed with negative policy rates in January 2016, CIR’s inability to model negative rates became a limitation for EUR, JPY, and CHF markets.

The Feller Condition

The Feller condition is a parameter constraint that determines whether the short rate can reach zero:

Feller Condition
2κθ ≥ σ2
When satisfied, zero is unattainable; when violated, zero can be reached but rates remain non-negative

If the Feller condition is satisfied: The drift toward θ is strong enough relative to volatility that the rate stays strictly positive almost surely. Zero is an unattainable boundary.

If the Feller condition is violated: The rate can touch zero, but it immediately reflects back into positive territory. Zero is an attainable but reflecting boundary. Rates still cannot go negative.

Important Distinction

Violating the Feller condition does not mean rates can go negative. It only changes boundary behavior at zero. For calibration and simulation, the Feller condition matters for realistic rate paths, but even violation produces a valid non-negative process.

Zero-Coupon Bond Pricing Under the CIR Model

One of CIR’s major advantages is that it admits closed-form solutions for zero-coupon bond prices. The price P(t,T) of a zero-coupon bond maturing at time T, given current time t and short rate r, is:

CIR Zero-Coupon Bond Price
P(t,T) = A(τ) × e−B(τ)r
Where τ = T − t is time to maturity

The functions A(τ) and B(τ) are given by:

B(τ) Function
B(τ) = 2(eγτ − 1) / [(κ + γ)(eγτ − 1) + 2γ]
A(τ) Function
A(τ) = [2γ e(κ+γ)τ/2 / ((κ+γ)(eγτ−1) + 2γ)]2κθ/σ2

Where γ is defined as:

Gamma Parameter
γ = √(κ2 + 2σ2)
Pro Tip

These formulas use risk-neutral parameters, not physical/historical parameters. When calibrating CIR to market bond prices, you’re finding the risk-neutral κ, θ, and σ that match observed prices, which generally differ from parameters estimated from historical rate data.

How to Calculate CIR Bond Prices

Let’s work through a concrete example of pricing a zero-coupon bond using the CIR model. Consider a scenario where the Fed has raised rates to 5%, but market expectations suggest a long-term equilibrium around 4%.

CIR Bond Pricing Example

Scenario: A fixed income desk prices a 5-year Treasury STRIP using CIR-calibrated parameters.

Given parameters:

  • Current short rate: r = 5% (0.05)
  • Long-term mean: θ = 4% (0.04)
  • Mean reversion speed: κ = 0.3
  • Volatility: σ = 0.05
  • Time to maturity: τ = 5 years

Step 1: Calculate γ

γ = √(0.32 + 2 × 0.052) = √(0.09 + 0.005) = √0.095 ≈ 0.3082

Step 2: Calculate B(τ)

eγτ = e0.3082×5 = e1.541 ≈ 4.669

B(5) = 2(4.669 − 1) / [(0.3 + 0.3082)(4.669 − 1) + 2 × 0.3082]

B(5) = 7.338 / [(0.6082)(3.669) + 0.6164] = 7.338 / 2.848 ≈ 2.577

Step 3: Calculate A(τ)

The exponent is 2κθ/σ2 = 2(0.3)(0.04)/(0.0025) = 9.6

The base is 2γe(κ+γ)τ/2 / [denominator] = (0.6164)(4.574) / 2.848 ≈ 0.990

A(5) = 0.9909.6 ≈ 0.908

Step 4: Calculate bond price

P = 0.908 × e−2.577 × 0.05 = 0.908 × e−0.1289 = 0.908 × 0.879 ≈ 0.799

The 5-year zero-coupon bond is priced at approximately 79.9% of face value, implying a continuously compounded yield of about 4.5%.

Try the CIR Bond Price Calculator

CIR Model vs Vasicek Model

The CIR and Vasicek models are the two foundational one-factor short-rate models. Understanding their differences is essential for choosing the right model.

CIR Model

  • Square-root diffusion: σ√r dW
  • Rates are always non-negative
  • Volatility is state-dependent (heteroskedastic)
  • Closed-form zero-coupon bond pricing
  • Cannot model negative rate environments

Vasicek Model

  • Constant diffusion: σ dW
  • Rates can go negative
  • Volatility is constant (homoskedastic)
  • Closed-form zero-coupon bond pricing
  • Suitable for negative rate environments

Before 2014, CIR’s non-negativity was viewed as more realistic. After the ECB and BOJ implemented negative policy rates, Vasicek’s ability to produce negative rates became a feature rather than a bug for modeling EUR, JPY, and CHF rates.

Limitations of the CIR Model

While CIR is mathematically elegant, it has several practical limitations:

Key Limitations

Cannot model negative rates: In EUR, JPY, and CHF markets post-2014, this is a fundamental constraint. Extensions like shifted CIR exist but add complexity.

Single factor: CIR uses one factor (the short rate) to drive the entire yield curve. Real yield curves exhibit more complex dynamics—level, slope, and curvature shifts—that affect duration hedging and require multi-factor models to capture accurately.

Time-homogeneous parameters: The parameters κ, θ, and σ are constant. This means CIR cannot exactly fit an arbitrary initial yield curve. The Hull-White model extends Vasicek with time-varying parameters to enable exact yield curve calibration.

Calibration challenges: Estimating three risk-neutral parameters (κ, θ, σ) from market prices can yield unstable estimates, especially in short samples or when the yield curve has unusual shapes.

Common Mistakes

When working with the CIR model, watch out for these common errors:

1. Confusing Feller violation with negative rates: Violating the Feller condition means rates can touch zero, not that they can go negative. CIR rates are always non-negative regardless of parameters.

2. Using physical-measure parameters for bond pricing: Bond prices require risk-neutral parameters, not the parameters estimated from historical rate data. Using historical estimates directly produces incorrect prices.

3. Applying CIR to negative-rate markets: If you’re modeling EUR or JPY rates, standard CIR fundamentally cannot capture the negative rate environment. Use Vasicek, Hull-White, or shifted CIR instead.

4. Naive Euler simulation: Simple Euler discretization can produce negative rates in simulation even though the continuous-time process is non-negative. Use exact transition sampling (which exploits the known chi-squared distribution of CIR) or positivity-preserving schemes for accurate paths.

Frequently Asked Questions

Both CIR and Vasicek are one-factor mean-reverting short-rate models with closed-form bond pricing. The key difference is the diffusion term: Vasicek uses constant volatility (σ dW), while CIR uses square-root volatility (σ√r dW). This makes CIR non-negative but state-dependent in its volatility. Vasicek can produce negative rates, which was historically seen as a flaw but became useful for modeling EUR/JPY markets after 2014.

The Feller condition is the parameter constraint 2κθ ≥ σ2. When satisfied, the short rate stays strictly positive (zero is unattainable). When violated, rates can touch zero but immediately reflect back up (zero is attainable but reflecting). Either way, rates remain non-negative. The condition matters for calibration because violating it can produce rate paths that spend time at zero, which may not match market behavior.

No. The square-root diffusion term mathematically prevents negative rates. As rates approach zero, the volatility term vanishes while the positive drift remains, pushing rates back up. This was historically viewed as an advantage but became a limitation when central banks implemented negative policy rates. For negative-rate environments, consider the Vasicek model or shifted extensions of CIR.

CIR calibration fits risk-neutral parameters (κ, θ, σ) directly to observed zero-coupon bond prices or interest rate derivative prices (caps, floors, swaptions). The standard approach minimizes the squared pricing errors between model-implied and market-observed bond prices across maturities. The challenge is obtaining stable parameter estimates, especially when the yield curve has unusual shapes or limited tenor coverage.

CIR stands for Cox-Ingersoll-Ross, named after John C. Cox, Jonathan E. Ingersoll Jr., and Stephen A. Ross, who developed the model in their 1985 paper “A Theory of the Term Structure of Interest Rates.” The model was derived from general equilibrium arguments and remains one of the canonical short-rate models in fixed income theory.
Disclaimer

This article is for educational and informational purposes only and does not constitute investment advice. The CIR model is a theoretical framework with known limitations. Parameter estimates and bond prices shown are illustrative examples. Always conduct your own research and consult a qualified financial advisor before making investment decisions.