The Cox-Ingersoll-Ross (CIR) Model¶
The Cox-Ingersoll-Ross model (1985) is a mean-reverting short-rate model with square-root diffusion. Under the risk-neutral measure \(\mathbb{Q}\):
Unlike Vasicek, CIR can ensure non-negative interest rates under the Feller condition while retaining analytical tractability. This short overview places the model in the general short-rate framework; all detailed derivations live in the dedicated CIR model folder.
Where to find each topic¶
Recall (see § General Short-Rate Framework): the bond pricing PDE and the expectation representation \(P(t,T) = \mathbb{E}^{\mathbb{Q}}_t[\exp(-\int_t^T r_s\,ds)]\) apply to any Markov short-rate model.
Recall (see § Affine Term Structure): CIR is the canonical square-root affine model, with bond price \(P(t,T) = A(\tau)\exp(-B(\tau)r_t)\) obtained from a quadratic Riccati ODE for \(B\) (since \(\sigma(r)^2 = \sigma^2 r\) is affine in \(r\)).
| Topic | Canonical location |
|---|---|
| SDE and square-root process | § CIR SDE and Square-Root Process |
| Feller condition \(2\kappa\theta \geq \sigma^2\) and boundary at zero | § Feller Condition and Boundary |
| Non-central \(\chi^2\) transition density | § Transition Density |
| Zero-coupon bond pricing (Riccati ODEs, \(A\) and \(B\), \(\gamma = \sqrt{\kappa^2 + 2\sigma^2}\)) | § Zero-Coupon Bond Pricing |
| Named functions and Riccati derivation | § Named Functions and Riccati |
| Yield curve dynamics | § Yield Curve Dynamics |
| Bond options via non-central \(\chi^2\) | § Bond Options |
| Caplet and swaption formulas | § Caplet and Swaption Formulas |
| Change of measure | § Change of Measure |
| Calibration | § Calibration |
| Exact simulation and Euler pitfalls | § Exact Simulation and Euler Pitfalls |
| Monte Carlo simulation | § Monte Carlo Simulation |
For side-by-side comparison with Vasicek and Hull-White (including conditional moments and tractability), see § Vasicek vs CIR vs Hull-White.
Key takeaways¶
- CIR: \(dr_t = \kappa(\theta - r_t)\,dt + \sigma\sqrt{r_t}\,dW_t\).
- Feller condition \(2\kappa\theta \geq \sigma^2\) ensures \(r_t > 0\).
- Conditional distribution is scaled non-central \(\chi^2\).
- Affine bond prices \(P = A(\tau)e^{-B(\tau)r}\) with \(\gamma = \sqrt{\kappa^2 + 2\sigma^2}\).
Further reading¶
- Cox, Ingersoll & Ross (1985), "A Theory of the Term Structure of Interest Rates".
- Brigo & Mercurio, Interest Rate Models, Chapter 3.