Why does the CIR model prevent negative rates and how does it differ from Vasicek?

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AcadiFi · Accepted Answer

The CIR model extends Vasicek by replacing the constant volatility with a square-root term: dr = a(b - r)dt + sigma * sqrt(r) * dW. When r approaches zero, the diffusion coefficient sigma * sqrt(r) also approaches zero, so the random shock cannot drag r below zero. Meanwhile, the drift a(b - r) becomes positive (since b > 0), pushing the rate back up. This combination is formalized by the Feller condition: 2ab >= sigma^2 guarantees r stays strictly positive.

Numerical illustration: with a = 0.8, b = 3.5%, sigma = 6%, we have 2*0.8*0.035 = 0.056 and sigma^2 = 0.0036, so Feller holds comfortably. If sigma were 10%, sigma^2 = 0.01 > 0.056 and the boundary becomes attainable.

Bond prices retain analytic form P(t,T) = A(tau) * exp(-B(tau) * r), though A and B involve hyperbolic functions. The distribution of future rates is non-central chi-squared rather than Gaussian, producing skewness consistent with observed rate behavior. Volatility rising with rate level matches empirical evidence better than Vasicek's homoscedastic assumption. The main drawback is calibration complexity and the tendency to underfit rates when they are near zero for long periods, which European markets experienced 2014-2022.

Why does the CIR model prevent negative rates and how does it differ from Vasicek?

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