Testing Vasicek and CIR Models Across Yield Curve Shapes

Summary

The document discusses how to compare Vasicek and Cox–Ingersoll–Ross (CIR) models for fitting zero-rate curves. Rather than judging the models from a single observed curve, it recommends calibrating each across many curves with varied shapes, including flat, normal, inverted, and humped curves, then comparing their fitting errors. Where historical data does not cover enough shapes, the comparison can include constructed stress scenarios.

The answer notes that both models are parsimonious and may struggle to match a full observed curve; a model may also fail to calibrate for some shapes. The comparison has a limitation: classical CIR does not support negative interest rates, which can make a head-to-head assessment unfair in markets where rates are negative. As a complementary check, the models can generate future curves using their bond-pricing formulas, allowing assessment of the variety and plausibility of shapes they produce over time. The document gives no empirical results or prescribed error metric, so the testing framework still requires choices about calibration data and evaluation criteria.

Key ideas

• Compare model fitting errors across many observed curve shapes rather than relying on one curve.
• Use constructed stress scenarios when historical curves do not cover important shapes.
• Check whether each model can calibrate successfully across the chosen scenarios.
• Vasicek and CIR may struggle to fit a full curve because they use few parameters.
• CIR's inability to accommodate negative rates limits fair comparison in negative-rate settings.
• Generate future curves as a complementary check on the range of shapes a calibrated model can produce.

Tags

# Choosing which interest rate model to go with?


# Choosing which interest rate model to go with?












I've been assigned with the task of modelling zero rate curve. I did it with two models: Vasicek and CIR. Looking at the two curves produced, I can see that one is closer to the observed curve than the other, but I am asked to perform some quantitative tests to confirm this observation. My question is this: Do you know of any tests for such a thing ?

## Answer by ir7 (score 5, accepted)

https://quant.stackexchange.com/a/55674

Calibrate to many observed curves, over all kinds of shapes: flat, normal, inverted, and humped, and measure and compare the model fitting errors. If you can't find all the shapes in history, make them up as possible scenarios (stresses).

Classical Vasicek and CIR are parsimonious (have few parameters), so not very good at properly matching today's full observed curve. But what could be more dramatic is that one of them may fail to calibrate altogether (say maybe to an inverted curve, maybe stressed a bit).

(Note: the competition is a bit unfair, given that CIR cannot accommodate negative interest rates observed these days.)

One more note, given a set of model parameters, you can generate curves at future times (formulas for zero-coupon bond prices, $P(t,T)$, are valid for all $t\geq 0$, not just $t=0$, and all $T \geq t$). This exercise (somewhat complementary to the calibration test) shows the capacity of the model to generate a reasonable variety of curve shapes at future times.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.