CoxIngersollRoss

Cox–Ingersoll–Ross model, commonly abbreviated CIR, is a mathematical model used in finance to describe the evolution of interest rates and other positive-valued quantities. It was introduced by John Cox, Jonathan Ingersoll, and Stephen Ross in 1985 to capture mean reversion and nonnegativity of interest rates.

The model treats the short rate r_t as a diffusion process solving the stochastic differential equation dr_t = kappa (theta - r_t) dt + sigma sqrt(r_t) dW_t, where kappa > 0 is the speed of mean reversion, theta >= 0 is the long-run mean level, sigma >= 0 is the volatility, and W_t is a standard Brownian motion. The square-root diffusion term ensures the process remains nonnegative under the Feller condition 2 kappa theta >= sigma^2.

Under a risk-neutral measure for pricing bonds and derivatives, zero-coupon bond prices have an affine form P(t,T) = A(t,T) exp(-B(t,T) r_t), where B(t,T) = (1 - e^{-kappa (T-t)})/kappa and A(t,T) depends on kappa, theta, sigma, and the time to maturity. The transition density of r_t is known to be noncentral chi-squared.

CIR is used for short-rate modeling, for pricing interest-rate derivatives, and as a building block in multifactor models and term-structure frameworks. It contrasts with Gaussian models like Vasicek, which can produce negative rates. Limitations include parameter calibration and the need to ensure the Feller condition; extensions include multifactor CIR and CIR with jumps.