Cox-Ingersoll-Ross (CIR) Interest Rate Model Lab

Q: What is the Feller Condition in the CIR model?

The Feller condition states that 2*a*b >= sigma^2. When this inequality holds, the upward mean-reverting drift at the zero boundary is sufficiently strong to prevent the short rate from ever touching zero, guaranteeing r_t > 0 in perpetuity. If 2*a*b < sigma^2, the rate can touch zero, but the origin acts as an instantaneously reflecting boundary.

Q: How does the CIR model price zero-coupon bonds analytically?

Like Vasicek, the CIR model belongs to the affine term structure class, providing an exact closed-form solution: P(t,T) = A(tau) * exp(-B(tau) * r_t), where gamma = sqrt(a^2 + 2*sigma^2), B(tau) is a hyperbolic function of gamma and tau, and A(tau) is an exponential function raised to the power (2ab / sigma^2).

Q: What is the asymptotic infinite-maturity yield in the CIR model?

The asymptotic long-term yield is R_infinity = 2ab / (a + gamma), where gamma = sqrt(a^2 + 2*sigma^2). Because gamma > a, the denominator (a + gamma) is greater than 2a, ensuring R_infinity is strictly less than the equilibrium mean b due to the convexity effect.

Simulate the Cox-Ingersoll-Ross (1985) square-root diffusion short-rate model. Verify the Feller condition ($2ab ge sigma^2$) for strictly positive interest rates, compute analytical zero-coupon bond prices, derive spot and forward term structures, and evaluate convexity drag.

CIR (1985) Model Square-Root Diffusion Feller Condition Check Non-Negative Short Rate Affine Term Structure

Calibrated Market Regimes:

1. Square-Root SDE Parameters

CIR SDE: dr = a(b - r)dt + σ √r dW

Current Short Rate ($r_0$, %)

Spot short rate $r(0) > 0$.

Long-Term Mean ($b$, %)

Equilibrium anchor rate.

Mean Reversion Speed ($a$)

Drift speed toward $b$. Half-life: 2.77 yrs.

Volatility Coeff ($sigma$)

Square-root scaling factor $sigma$.

Pricing Horizon

Benchmark Bond Maturity (Years)

Zero-coupon maturity for analytical pricing and term structure analysis.

2. Feller Condition & Analytical Valuation

FELLER SATISFIED

Benchmark Zero Yield

4.22%

10-Year Zero

Zero Price ($P)

$65.59

Par = $100

Asymptotic Yield

4.33%

R_∞ < b (4.50%)

Feller Ratio (2ab/σ²)

2.25

≥ 1.0 (Strictly Positive)

CIR Bond Pricing: $P(t,T) = A( au) e^{-B( au) r_t}$ with $gamma = sqrt{a^2 + 2sigma^2}$

Component	Analytical Formula	Value	Interpretation
Auxiliary Parameter $gamma$	`√(a² + 2σ²)`	0.2872	Eigenvalue speed accounting for square-root volatility diffusion
Duration Factor $B( au)$	`2(e^{γτ}-1) / [(γ+a)(e^{γτ}-1) + 2γ]`	3.475	Bounded duration sensitivity (approaches $2/(gamma+a)$)
Scale Factor $A( au)$	`[ 2γ e^{(a+γ)τ/2} / denom ]^{2ab/σ²}`	0.7431	Power transformation based on Feller exponent $2ab/sigma^2$
Zero-Coupon Price ($P$)	`100 × A( au) × exp(-B( au) × r_0)`	$65.59	Exact arbitrage-free fair price per $100 par

Feller Boundary Condition: When $2ab ge sigma^2$, the upward drift at $r=0$ is strong enough that the short rate is strictly positive ($r_t > 0$ almost surely).

3. CIR Term Structure & Forward Rates

Yield Curve (0 to 30 Years): Blue solid curve depicts the CIR zero-coupon spot rate $Y(0,T)$, red dashed curve shows the forward rate $f(0,T)$, and green horizontal line marks the asymptotic yield $R_infty = rac{2ab}{a+gamma}$.

4. Analytical CIR Term Structure Schedule

Exact Closed-Form

Maturity	Zero Price ($P)	Zero Yield ($Y$)	Forward ($f$)	Factor B(τ)

Maturity spot yields: $Y(0,T) = -rac{ln P(0,T)}{T}$. Note the absence of negative rate probabilities.

5. Sensitivity: Mean Reversion ($a$) vs. Volatility ($sigma$) on 10Y Yield (%)

Demonstrates how square-root volatility $sigma$ pulls down long yields through the CIR convexity adjustment factor $gamma$.

6. Sensitivity: Short Rate ($r_0$) vs. Equilibrium Mean ($b$) on 30Y Yield (%)

Evaluates ultra-long zero-coupon spot yields across varying initial short rates and central bank long-term equilibrium targets.

7. CIR Term Structure Mathematics & Square-Root Diffusion

The Square-Root Process & Non-Negativity

Published in Econometrica in 1985 by John C. Cox, Jonathan E. Ingersoll, and Stephen A. Ross, the CIR model addresses the fundamental flaw of Gaussian interest rate models:

$$dr_t = a(b - r_t)dt + sigma sqrt{r_t} dW_t$$

Because the diffusion term is scaled by $sqrt{r_t}$, volatility approaches zero as $r_t o 0$. As long as the deterministic upward drift $a cdot b > 0$ is sufficiently strong, the rate is pushed back into positive territory.

The Feller Condition

William Feller (1951) established that the boundary behavior at zero depends entirely on the ratio of drift to diffusion:

$$2ab ge sigma^2 iff rac{2ab}{sigma^2} ge 1$$

When $2ab ge sigma^2$, the boundary $r=0$ is inaccessible, ensuring $r_t > 0$ for all time. If $2ab < sigma^2$, the origin is accessible, acting as a reflecting boundary.

Affine Term Structure & Asymptotic Yield

The CIR model belongs to the general class of affine term structure models, meaning that the logarithm of bond prices is linear in the state variable $r_t$:

$$P(t, T) = A( au) e^{-B( au) r_t}, quad au = T - t$$

With $gamma = sqrt{a^2 + 2sigma^2}$, the asymptotic long-term yield is given by:

$$R_infty = lim_{ au o infty} Y(t, T) = rac{2ab}{a + gamma} = rac{2ab}{a + sqrt{a^2 + 2sigma^2}}$$

Comparison with Vasicek: In Vasicek, $R_infty = b - rac{sigma^2}{2a^2}$, which can become negative if volatility is high. In CIR, because $a, b > 0$ and $gamma > 0$, $R_infty$ is strictly positive ($R_infty > 0$), preserving market consistency across all horizons.

8. CIR Model Mastery Self-Assessment Quiz

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Model classical Redington duration matching, convexity surplus, and Fong-Vasicek structural risk.

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Simulate fixed-for-floating swap valuation, mark-to-market NPV, and par swap rates.

10. Frequently Asked Questions (FAQ)

Introduced in 1985 by John Cox, Jonathan Ingersoll, and Stephen Ross, the CIR model addresses the primary theoretical flaw of the Vasicek model: negative interest rates. By incorporating a square-root diffusion term (sigma * sqrt(r_t) * dW_t), interest rate volatility approaches zero as rates drop toward zero. If the Feller condition 2ab ≥ sigma^2 is satisfied, interest rates are strictly positive at all times.

The Feller condition states that 2*a*b ≥ sigma^2. When this inequality holds, the upward mean-reverting drift at the zero boundary is sufficiently strong to prevent the short rate from ever touching zero, guaranteeing r_t > 0 in perpetuity. If 2*a*b < sigma^2, the rate can touch zero, but the origin acts as an instantaneously reflecting boundary.