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Fixed Income and Interest-Rate Models / Short-Rate Models

The CIR Model: Square-Root Diffusion, Feller Condition, and Non-Negative Rates

The Vasicek model has one structural flaw for a floor-trading desk: it assigns a positive probability to negative rates. For most interest-rate derivatives, this barely matters. But for options struck near zero — floors on overnight rates, corridor trades, barrier swaptions — the ability to rule out negative rates analytically changes the pricing by a meaningful amount. The Cox-Ingersoll-Ross (CIR) model (1985) addresses this by replacing Vasicek's constant diffusion coefficient σ\sigma with σr\sigma\sqrt{r}. The square-root diffusion naturally collapses toward zero as r→0r \to 0, preventing rr from going negative under the Feller condition. The cost is a Riccati ODE for the bond price coefficient B(τ)B(\tau) rather than the linear ODE in Vasicek, and bond option pricing requires non-central chi-squared quantiles rather than Gaussian ones — but both are still computable in closed form.
Learning objectives. By the end of this note you should be able to:
  • write down the CIR SDE and explain why σr\sigma\sqrt{r} makes the model non-negative;
  • state the Feller condition 2kθ≥σ22k\theta \ge \sigma^2 and explain what happens when it fails;
  • give the conditional distribution of r(t)r(t) given r(s)r(s) (non-central chi-squared, rescaled);
  • state the CIR bond price formula P(t,T)=A(t,T)e−B(t,T)r(t)P(t,T) = A(t,T)e^{-B(t,T)r(t)} with explicit AA and BB;
  • compare Vasicek and CIR along the key dimensions: rate non-negativity, volatility structure, analytical complexity;
  • identify the CIR model as a special case of the affine term-structure framework.

In this lesson

    1. The square-root diffusion SDE
    1. The Feller condition: when does rr stay strictly positive?
    1. Distribution of the short rate
    1. Affine bond price formula
    1. Vasicek versus CIR: comparison
    1. Bond options under CIR

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