Why does the Cox-Ingersoll-Ross model use a square-root diffusion?

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AcadiFi · Accepted Answer

The square-root diffusion term sigma*sqrt(r) solves Vasicek's biggest flaw: negative rates.

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Second advantage: volatility is state-dependent. Empirically, interest rate volatility tends to be higher when rates are higher — exactly what CIR captures. This is more realistic than Vasicek's constant sigma.

CIR is also affine, so zero-coupon bond prices retain the form P(t,T) = A(t,T) * exp(-B(t,T)*r_t), with different A and B functions involving hyperbolic functions. Closed-form pricing is preserved.

The distribution of r is non-central chi-squared rather than Gaussian. This makes some calculations messier but ensures rates stay positive.

Trade-off: CIR struggles to fit observed yield curves exactly (it's still equilibrium), so traders at shops like Palatine Fixed Income use Hull-White (an extended, time-varying version of Vasicek) for pricing callable structures, reserving CIR for academic or insurance-reserve modeling where positivity matters more than exact curve fit.

For CFA Level II, remember: CIR = mean-reverting, square-root vol, positive rates, affine, good for long-horizon simulations.

Why does the Cox-Ingersoll-Ross model use a square-root diffusion?

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