CFA Level II Exam · The Term Structure and Interest Rate Dynamics

Vasicek, CIR and Ho-Lee Term Structure Models

Updated 7 October 2026 · Fact-checked

Term structure models describe how the short rate moves over time. Vasicek and CIR are equilibrium models with mean reversion; CIR also stops rates going negative. Ho-Lee is arbitrage-free: it is fitted to today's curve, has no mean reversion, and uses time-dependent drift. Identify the model from its equation, then apply its features.

Understand Modern Term Structure Models

A term structure model is a set of assumptions about how interest rates change randomly over time. You use it to value bonds and interest rate derivatives, and to see how a yield curve might evolve.

There are two families. Equilibrium models start from assumptions about the economy and let the yield curve be an output. The model's curve may not match today's market curve. Arbitrage-free models start from today's observed market curve and fit to it, so the model prices current bonds correctly. They then add assumptions about how rates move.

On the exam, the equilibrium models are Cox-Ingersoll-Ross (CIR) and Vasicek. Both are single-factor models: one random factor, the short-term rate, drives everything. Both show mean reversion. The short rate is pulled toward a long-run mean, b, at a speed set by a. When the rate is above b, the drift is negative. When it is below b, the drift is positive.

The difference is in volatility. In Vasicek the random term is a constant σ times the random shock, so the short rate can become negative. In CIR the random term is σ times √r times the shock, so volatility rises with the rate level and falls toward zero as r nears zero. With the right parameters, the rate cannot go negative.

Ho-Lee is arbitrage-free. Its drift is a time-dependent term θt, chosen so the model matches the market curve, and its volatility is a constant σ. It has no mean reversion. Rates can go negative. Because it is calibrated to market prices, it can be used to value derivatives consistently with the market.

Key formulas to remember

Vasicek model: dr = a(b − r)dt + σ dZ; Equilibrium. Mean reversion to b at speed a. Constant volatility σ, so negative rates are possible.
Cox-Ingersoll-Ross (CIR) model: dr = a(b − r)dt + σ√r dZ; Equilibrium. Mean reversion like Vasicek, but volatility scales with √r, so volatility rises with the rate and rates stay non-negative under the right parameters.
Ho-Lee model: dr = θt dt + σ dZ; Arbitrage-free. Time-dependent drift θt fits the market curve. Constant volatility. No mean reversion.
Drift direction in mean-reverting models: Drift = a(b − r): positive if r < b, negative if r > b; Use this to say which way the expected short-rate change points. The random shock can still move the rate either way.
Classification: Equilibrium: Vasicek, CIR. Arbitrage-free: Ho-Lee; Vasicek and CIR are single-factor models. Ho-Lee is also a one-factor, short-rate model.

How to solve Modern Term Structure Models questions

Most questions give you a model equation or a verbal description and ask you to classify it, compare it or interpret a parameter. Work in this order.

1Read the vignette and write down the model equation or the described features: drift, volatility and whether a long-run mean is mentioned.
2Check the drift term. If it is a(b − r), the model has mean reversion. If it is θt, it is time-dependent and fitted to the market curve.
3Check the volatility term. Constant σ means rates can go negative. σ√r means volatility rises with the rate level and rates stay non-negative under the right parameters.
4Name the model: a(b − r) with constant σ is Vasicek, a(b − r) with σ√r is CIR, θt with constant σ is Ho-Lee.
5Classify it: Vasicek and CIR are equilibrium, Ho-Lee is arbitrage-free.
6If asked about drift, compute a(b − r) with the given numbers and state the sign and direction.
7Match your answer to the question: calibration, negative rates, mean reversion or fit to the market curve. Eliminate options that contradict the model's features.

Quickest way: Three-check model identification

When to use it: Use this when a question asks which model fits a description or which statement is correct.

Is there a mean (b)? Yes means Vasicek or CIR. No means Ho-Lee.
Is there a √r in the volatility? Yes means CIR. No means Vasicek or Ho-Lee.
Is it fitted to today's market curve? Yes means Ho-Lee (arbitrage-free). Otherwise it is equilibrium.
Then pick the option that matches: mean reversion, non-negative rates, or market fit.

Common mistakes in Modern Term Structure Models

Saying Ho-Lee has mean reversion.
Students assume every short-rate model pulls rates back to a long-run level.
Fix: Ho-Lee drift is θt, a time-dependent term. There is no a(b − r) term, so no mean reversion.
Saying Vasicek rates cannot go negative.
Mean reversion sounds like it keeps rates in a safe range.
Fix: Vasicek has constant volatility, so a shock can push the rate below zero. Only CIR, with σ√r, avoids negative rates under the right parameters.
Calling CIR an arbitrage-free model.
Students associate CIR with valuation and assume it fits the market curve.
Fix: CIR and Vasicek are equilibrium models. Only Ho-Lee is arbitrage-free in this topic.
Getting the drift sign wrong.
Students compute a(r − b) instead of a(b − r).
Fix: Write the formula as a(b − r). If r is below b, the drift is positive.
Believing the rate must move toward b in every period.
Drift is confused with the actual change.
Fix: Drift is the expected change. The random shock term can overwhelm it, so the actual move can go the other way.
Saying equilibrium models match today's market curve exactly.
Students forget why arbitrage-free models were built.
Fix: Equilibrium models generate the curve from their assumptions, so it may differ from the market curve. Arbitrage-free models are calibrated to fit it.

Worked examples

Example 1

An analyst models the short rate with dr = 0.30(0.04 − r)dt + 0.01√r dZ. (1) Name the model and classify it. (2) The current short rate is 2.5%. What is the drift per unit time and its direction? (3) Can the model produce negative rates?

Show the solution

(1) The drift a(b − r) shows mean reversion with a = 0.30 and b = 4%. The volatility has √r, so this is the Cox-Ingersoll-Ross model. It is an equilibrium model.
(2) Drift = 0.30 × (0.04 − 0.025) = 0.30 × 0.015 = 0.0045, or 0.45% per unit time. It is positive because r is below b, so the expected move is upward toward 4%.
(3) Volatility is σ√r, which shrinks toward zero as r approaches zero. With suitable parameters the rate stays non-negative.

Answer: (1) CIR, equilibrium. (2) Drift +0.45% per unit time, upward toward 4%. (3) No, rates remain non-negative under the right parameters.

Example 2

A vignette says: 'Model X is calibrated so that it reproduces the current market spot curve. Its drift varies with time and volatility is constant. It has no long-run mean.' (1) Which model is described? (2) Is it equilibrium or arbitrage-free? (3) A colleague says it prevents negative rates. Is that correct?

Show the solution

(1) Time-varying drift θt, constant volatility and no long-run mean describe the Ho-Lee model.
(2) Calibration to the current market curve is the defining feature of an arbitrage-free model, so it is arbitrage-free.
(3) Volatility is a constant σ, not scaled by √r, and there is no mean reversion. Rates can fall below zero, so the colleague is wrong.

Answer: (1) Ho-Lee. (2) Arbitrage-free. (3) Incorrect, because Ho-Lee can produce negative rates.

Exam tips

Match the model to its equation features first. Most questions are solved by checking for a(b − r), σ√r and θt.
Memorise the pairings: Vasicek has mean reversion and negative rates possible, CIR has mean reversion and non-negative rates, Ho-Lee has market fit and no mean reversion.
When a drift calculation appears, use a(b − r) and state the direction as well as the number.
Watch for answer options that swap features between models. Eliminate any option that gives Ho-Lee mean reversion or Vasicek non-negative rates.
The vignette may describe the model in words only. Translate the words into the three checks before looking at the options.

Modern Term Structure Models in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

FRM Part IIVasicek Model: Bond Prices and Term Structure

Modern Term Structure Models: frequently asked questions

What is the main difference between Vasicek and CIR?

Both have mean reversion to a long-run level. Vasicek has constant volatility, so rates can turn negative. CIR scales volatility with √r, so volatility rises with the rate and rates stay non-negative under the right parameters.

Why is Ho-Lee called arbitrage-free?

Its time-dependent drift is chosen so the model reproduces the observed market curve. Bonds priced with it agree with current market prices, so it does not allow arbitrage against those prices.

Do equilibrium models have mean reversion?

Vasicek and CIR do. Mean reversion pulls the short rate toward a long-run mean b at speed a. Ho-Lee, which is arbitrage-free, does not have it.

Which model should I use to price derivatives consistently with the market?

An arbitrage-free model such as Ho-Lee, because it is fitted to the current market curve. Equilibrium models may produce a curve that differs from the market.