CIR short rate: no negative rates, by construction
Replace Vasicek's constant diffusion with $\sigma\sqrt{r_t}$ and the diffusion vanishes at zero. Negative rates disappear, the bond price is still affine, and the math costs one extra auxiliary parameter.
Vasicek's headline flaw is its Gaussian dynamics: a constant $\sigma$ means the SDE can wander to arbitrarily negative rates, which most economies treat as impossible. Cox, Ingersoll & Ross (1985) made the diffusion proportional to $\sqrt{r_t}$ — so the noise shrinks as the rate approaches zero. With a mild condition on the parameters the rate stays strictly positive forever, and the bond price keeps the same affine form $P(t,T) = A(t,T) e^{-B(t,T) r_t}$. The price for the fix is that $A$ and $B$ now involve an auxiliary $\gamma = \sqrt{\kappa^2 + 2\sigma^2}$, and the rate process is no longer Gaussian.
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Worked example
Today's short rate is $r_0 = 5\%$. Under CIR with $\kappa = 0.3$, $\theta = 0.06$, $\sigma = 0.05$ (annualised), price a $5$-year zero-coupon bond. Verify the Feller condition first.
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Reflection
CIR fixes the negative-rate issue by making the diffusion $r$-dependent. What did you lose in the process: which Vasicek tractability properties survive, and which break? Why is calibration harder under CIR even though the bond price is still closed-form?