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Michael Brenndoerfer

Short-Rate Interest Rate Models › Comparing Vasicek and CIR

Michael BrenndoerferDecember 6, 20252 min read

Quantitative FinanceAll chapters

Part 7 of 15Interest Rate Models and Derivatives

Chapter 38Short-Rate Interest Rate Models › Comparing Vasicek and CIR

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Out[7]:

Visualization

Line plot comparing constant Vasicek volatility to the square-root CIR volatility function. — Instantaneous volatility comparison between Vasicek and CIR models. Vasicek has constant volatility regardless of rate level (horizontal line). CIR volatility scales with the square root of the rate, producing lower volatility when rates are low and higher volatility when rates are high.

The table below summarizes the key differences between the two models:

Comparison of key features between Vasicek and CIR short-rate models.
Feature	Vasicek	CIR
Diffusion	Constant ( $\sigma$ )	State-dependent ( $\sigma\sqrt{r}$ )
Rate distribution	Normal	Non-central chi-squared
Negative rates	Possible	Not possible (with Feller condition)
Volatility behavior	Constant	Higher when rates are high
Analytical tractability	Simpler formulas	More complex but still closed-form

Both models share the same mean-reversion structure, producing similar expected rate paths. The primary differences concern:

Distributional assumptions Vasicek assumes normal shocks regardless of rate level; CIR assumes volatility scales with rate level.
Positivity: CIR guarantees non-negative rates when the Feller condition holds; Vasicek does not.
Empirical fit: CIR's state-dependent volatility often better matches observed rate dynamics, where volatility tends to be higher during high-rate periods.