Model Parameters
CIR Model SDE
dr = kappa(theta - r)dt + sigmasqrt(r)dW
kappa = speed |
theta = long-run mean |
sigma*sqrt(r) = volatility
Bond Pricing Results
Zero-Coupon Bond Price
0.7792
per $1 face value
Zero-Coupon Yield
4.9907%
Long-Run Yield
4.9800%
r vs theta
At Equilibrium
Feller Condition: 2*kappa*theta >= sigma^2
0.0500 >= 0.0001
SATISFIED - Zero is unattainable
Parameter Interpretation
Half-life of deviation:
1.39 years
Yield curve shape:
Normal
Model-Implied Yield Curve
CIR
Vasicek
Formula Breakdown
P(t,T) = A(tau) * exp(-B(tau) * r)
where gamma = sqrt(kappa^2 + 2*sigma^2), tau = T - t
Educational Tool
This calculator uses risk-neutral model parameters (kappa, theta, sigma) directly. In practice, these parameters are calibrated from market bond prices using numerical optimization. The Feller condition determines whether zero is attainable: when satisfied, rates stay strictly positive; when violated, zero can be reached but rates remain non-negative. This tool helps you understand how the CIR model works, but should not be used for trading decisions.
Model Assumptions
- Single factor: Interest rates driven by one source of randomness
- Constant parameters: kappa, theta, sigma do not change over time
- Mean reversion: Rates always tend toward theta
- Non-negative rates: Square-root diffusion ensures r >= 0
- Continuous time: Model operates in continuous time (no jumps)
- Risk-neutral measure: Bond prices calculated under Q-measure
- No default risk: Bonds assumed to be default-free
- Zero-coupon bonds: Single payment at maturity
Download This Calculator as an Excel Template
Interactive model with editable formulas — customize, save, and share.
Get Excel Template
Frequently Asked Questions
The Cox-Ingersoll-Ross (CIR) model, introduced in 1985, is a one-factor mean-reverting interest rate model. It uses square-root diffusion (sigma*sqrt(r)*dW) which ensures interest rates remain non-negative under admissible parameters. The Feller condition governs whether rates stay strictly positive or can reach zero. CIR is widely used for modeling nominal interest rates where negative values are unrealistic.
The Feller condition states that 2*kappa*theta must be greater than or equal to sigma^2. When satisfied, zero is unattainable and rates stay strictly positive with probability 1. If violated, rates can touch zero but cannot go negative. The condition balances mean reversion strength (kappa*theta) against volatility (sigma^2). Either way, the CIR pricing formulas remain valid.
Both are mean-reverting one-factor models, but they differ in volatility structure:
Vasicek: sigma * dW (constant volatility) - rates can go negative
CIR: sigma * sqrt(r) * dW (volatility proportional to sqrt(rate)) - rates stay non-negative
Both produce closed-form bond prices, but CIR formulas involve a gamma parameter: gamma = sqrt(kappa^2 + 2*sigma^2). Use the toggle above to compare yield curves.
These three risk-neutral parameters fully characterize the CIR model:
Kappa (mean reversion speed): How quickly rates return to equilibrium. Higher kappa = faster reversion.
Theta (long-run mean): The risk-neutral equilibrium interest rate level. Rates tend toward this value over time.
Sigma (volatility parameter): Controls the magnitude of rate fluctuations. Effective volatility is sigma*sqrt(r), so volatility decreases as rates approach zero.
Note: These are pricing parameters under the risk-neutral measure, not necessarily historical estimates.
Gamma = sqrt(kappa^2 + 2*sigma^2) is an intermediate parameter that appears throughout the CIR bond pricing formulas for A(tau) and B(tau). It combines mean reversion speed and volatility into a single quantity that governs how bond prices evolve with maturity. Gamma is always greater than or equal to kappa (equality when sigma = 0). The formula breakdown section shows gamma and other intermediate values for your inputs.
Use CIR when non-negative rates matter, such as modeling nominal rates in most economies or when pricing products sensitive to rate sign (e.g., certain exotic interest rate derivatives). Vasicek is simpler and may be preferred when negative rates are acceptable (post-2014 European rates) or for teaching purposes. Both are pedagogical tools; practitioners often use more sophisticated models like Hull-White or multi-factor models for production pricing.
Disclaimer
This calculator is for educational purposes only. The CIR model is a simplified representation of interest rate dynamics. Actual bond prices depend on many factors not captured here, including credit risk, liquidity, and market conditions. This tool should not be used for trading decisions.
