CAIA® Level II: when do Vasicek and CIR rate paths differ?

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Vasicek and CIR both make the short rate tend towards a long-run level. The decisive difference is the shock: Vasicek applies constant volatility, whereas CIR makes volatility shrink as the rate approaches zero. For a CAIA® Level II candidate, that difference determines the shape of future-rate outcomes, the treatment of zero and the interpretation of a model price.

Start with the same pull towards a long-run rate

The short rate is the interest rate for an instant in the model, not the yield of a particular long bond. In each model, the direction of the expected small change depends on the gap between the current rate and the long-run level. If the current rate is above that level, the drift is downward. If it is below, the drift is upward. A larger reversion-speed parameter makes the pull stronger for the same gap. The CAIA curriculum overview locates fixed-income modelling within Level II Methods and Models; the assigned curriculum reading applies that framework to these two one-factor specifications.

dr_t=\kappa(\theta-r_t)dt+\sigma dW_t

dr_t=\kappa(\theta-r_t)dt+\sigma\sqrt{r_t}\,dW_t

Read these equations in two pieces. The first term is conditional direction, not a promise that the next observed rate will move that way. The second term is a random disturbance that can overwhelm the drift over a short interval. A candidate who sees a rate above the long-run level and concludes that its next realised observation must fall has replaced an expected change with a certain one. The distinction matters when interpreting a simulated path, a distribution of possible rates or a bond value based on many possible paths.

How the two distributions part company

Vasicek uses a constant shock coefficient. Its conditional future rate has a normal distribution, so there is always some model probability below zero when the variance is positive, even if the starting rate and long-run level are positive. That mathematical property is neither an assertion that market rates will be negative nor a reason to ignore the model. It tells you to check whether the Gaussian left tail is acceptable for the question being answered. The Federal Reserve discussion of Gaussian term-structure models makes the negative-rate implication explicit.

CIR multiplies the random shock by the square root of the current short rate. When the rate is high, the same volatility coefficient creates a larger absolute shock; near zero, the absolute shock contracts. The upward drift near zero then matters more relative to that shock. The process is non-negative under its usual positive parameter assumptions. Its future-rate distribution is asymmetric rather than normal. Do not describe its volatility as constant merely because the symbol for its volatility coefficient is fixed; the realised coefficient changes with the rate itself.

The distinction at zero needs careful wording. CIR can stay strictly above zero when the parameter condition below holds. If it fails, zero can be reachable, but the square-root process is still not the same as a Gaussian process that freely wanders below zero. An exam response that calls the condition a test of non-negativity confuses a boundary that may be touched with a rate that turns negative.

2\kappa\theta\geq\sigma^2

An illustrative rate-path check

Use invented round parameters to see the mechanics. Let the current short rate be 6%, the long-run level 4% and the reversion speed 0.5 per year. The initial expected drift is 0.5 times the gap of negative 2 percentage points, or negative 1 percentage point per year. That is a local rate of change, not the exact one-year forecast. Because the gap shrinks as the expected path approaches the long-run level, extending that initial drift unchanged for the whole year would overstate the decline.

E[r_T\mid r_0]=\theta+(r_0-\theta)e^{-\kappa T}

For that illustrative one-year horizon, the exponential factor is about 0.6065. The expected rate is therefore 4% plus 2% times 0.6065, approximately 5.21%. The calculation is identical for the two models because their drift is identical. It does not imply identical uncertainty around the mean. In a Vasicek illustration with a constant volatility coefficient of 1 percentage point per square-root year, the shock scale does not change when the current rate changes. In a separate CIR illustration with coefficient 0.10 in consistent rate units, the instantaneous shock coefficient is 0.10 times the square root of the rate: about 0.0245 at a 6% rate and 0.0100 at a 1% rate.

For the CIR parameters just used, twice the reversion speed times the long-run level equals 0.04; the squared volatility coefficient equals 0.01. The Feller inequality holds. Notice that comparing the two volatility coefficients numerically would be meaningless without their different units and their different effect on the rate. A useful comparison fixes the drift and asks how each diffusion term responds when the state changes. It should not claim that one model necessarily gives a higher bond price based only on these illustrative parameters.

The horizon adds another check. Under Vasicek, uncertainty about the future rate starts at zero when the future date is the present moment, widens as the horizon extends, and approaches a finite long-run variance when reversion is positive. That is different from saying volatility is zero or that each future date has the same variance. A one-day forecast and a distant forecast can have the same long-run target but very different distributions. Under CIR the evolving rate level also affects the scale of each new shock, so describing uncertainty requires attention to the path and its state. When an exam prompt asks about a terminal rate, focus on its conditional distribution. When it asks about a bond value, remember that discounting depends on the path of rates before maturity. This separation prevents a correct statement about the terminal mean from being used as an unjustified statement about price.

What a bond-pricing question changes

Both models can lead to a closed-form zero-coupon bond price that depends on the current short rate and maturity through model-specific functions. That shared form is a convenience, not proof that the models price a given bond identically. Changing the distribution of future rates changes the expected discounting of a future cash flow. A candidate should identify the relevant model, use parameters appropriate to the pricing measure and apply that model’s own functions. A physical forecast of future rates and a risk-neutral bond price answer different questions, so a fitted historical reversion level should not be inserted into a pricing equation without considering the assumed risk premium.

Think of a unit payment at maturity. Its value depends on the accumulation of short rates along each possible future path, not just the expected terminal short rate. Two models can share the same conditional mean at the final date yet distribute earlier rates differently. Averaging a nonlinear discount factor over those paths then need not give the same value. This explains why the distributional distinction is relevant to valuation even when a question gives identical drift parameters. It also explains why substituting the expected short rate directly into a simple discount formula generally does not reproduce a model bond price.

A compact exam decision sequence

First identify the state variable: the instantaneous short rate, not a bond yield of arbitrary maturity.
Then separate drift from shock. The sign of the gap to the long-run level determines expected local direction, while the shock determines uncertainty.
For Vasicek, look for constant shock size and a normal future-rate distribution with a possible negative tail.
For CIR, look for a square-root shock, a non-negative process and the Feller condition only when strict positivity is the issue.
For a bond price, check the measure and the model-specific coefficients before using a formula.

The most common mistake is to memorise “both mean revert” and treat the two models as interchangeable. Another is to read a positive expected drift as a guarantee that the next rate rises. A third is to say that CIR needs the Feller condition to avoid negative rates; the condition instead controls whether the zero boundary can be reached. The CAIA program page gives the broader qualification context, while Omni’s editorial methodology explains how its supporting sources are presented.