CFA Level II Exam · The Arbitrage-Free Valuation Framework

Vasicek, CIR and Ho-Lee Term Structure Models Explained

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 volatility scales with √r, so rates stay non-negative. Ho-Lee and Kalotay-Williams-Fabozzi are arbitrage-free models fitted to today's curve, with no mean reversion. Match each model to its stated features.

Understand Term Structure Models: Vasicek, CIR and Ho-Lee

A term structure model gives a rule for how the short-term interest rate changes through time. From that rule you can build a tree or simulate paths, then price bonds and interest rate options. The model is usually written as a small change in the rate, dr, over a small time step, dt. It has a drift part (the expected change) and a shock part (random noise, dz).

There are two families. Equilibrium models start from assumptions about the economy and the short rate. Bond prices and the whole yield curve come out of the model. Because of that, the model curve may not match the curve you see in the market today. Vasicek and Cox-Ingersoll-Ross (CIR) belong here. Both are one-factor models: the short rate is the only source of uncertainty.

Arbitrage-free models work the other way round. They take today's market prices or the observed curve as given and choose the model inputs so the model reproduces them exactly. Ho-Lee and Kalotay-Williams-Fabozzi (KWF) belong here. This is why they are the tools for pricing bonds with embedded options consistently with market prices.

The key feature of Vasicek and CIR is mean reversion. The rate is pulled toward a long-run level b at a speed a. If r is above b, the drift is negative. If r is below b, the drift is positive. The size of the pull is a × (b − r). Ho-Lee has no mean reversion. Its drift is a time-dependent term θ that is set so the model fits the observed curve.

The models differ in how they handle volatility. Vasicek uses a constant σ, so rates can go negative. CIR multiplies σ by √r, so volatility shrinks as rates fall and the rate cannot go negative. Ho-Lee uses constant volatility, so negative rates are possible. KWF models the log of the rate, so the rate stays positive and volatility is proportional to the rate level.

Key formulas to remember

Vasicek model: dr = a(b − r)dt + σ dz; Equilibrium, one factor. a = speed of mean reversion, b = long-run mean rate, σ = constant volatility. Rates can turn negative.
Cox-Ingersoll-Ross (CIR) model: dr = a(b − r)dt + σ√r dz; Equilibrium, one factor. Same mean reversion as Vasicek but volatility is σ√r, so it falls when rates are low. Rates do not go negative.
Ho-Lee model: dr = θ dt + σ dz; Arbitrage-free. θ is a time-dependent drift chosen to fit the observed curve. No mean reversion. Constant volatility, so negative rates are possible.
Kalotay-Williams-Fabozzi (KWF) model: d ln(r) = θ dt + σ dz; Arbitrage-free, lognormal version of Ho-Lee. Rates stay positive. No mean reversion.
Expected change from mean reversion: Expected change in r ≈ a(b − r) × dt; Positive if r < b, negative if r > b. This ignores the random shock, whose expected value is zero.
CIR shock size: Volatility of rate change = σ√r per √(unit time); If r falls to one quarter of its level, the volatility halves.

How to solve Term Structure Models: Vasicek, CIR and Ho-Lee questions

Most questions ask you to identify a model from its features, compare two models, or do a small calculation with the drift or volatility term. Use this routine.

1Read the vignette and underline the model name, or the equation if one is given. Note the values of a, b, σ and the current rate r.
2Classify the model: equilibrium (Vasicek, CIR) or arbitrage-free (Ho-Lee, KWF). Ask whether it is fitted to today's curve or derived from assumptions.
3Check for mean reversion. If the drift contains a(b − r), the model has it. If the drift is θ alone, it does not.
4Look at the volatility term. A constant σ means negative rates are possible. σ√r means volatility depends on the level and rates stay non-negative. A log rate means rates stay positive.
5If a calculation is asked, plug in the numbers. Compute a(b − r) × dt for drift, or σ√r for CIR volatility. Keep rates in decimals, such as 0.02 for 2%.
6Check the sign and direction. If r is above b the drift must be negative. Make sure the time step matches the unit of a and σ.
7Choose the answer that matches the model's exact features, not a general idea of interest rate models.

Quickest way: Feature-matching grid

When to use it: Use it for any qualitative or comparison question. It saves time because most wrong options mix up one feature.

Vasicek: equilibrium, mean reversion, constant σ, negative rates possible.
CIR: equilibrium, mean reversion, σ√r, no negative rates.
Ho-Lee: arbitrage-free, no mean reversion, constant σ, negative rates possible, fits the curve.
KWF: arbitrage-free, no mean reversion, lognormal, rates positive, fits the curve.
For a drift question, compute a(b − r) × dt and check its sign first. Above b means down, below b means up.

Common mistakes in Term Structure Models: Vasicek, CIR and Ho-Lee

Saying Ho-Lee has mean reversion.
Students remember that interest rates revert over time and assume every model includes it.
Fix: Ho-Lee and KWF have a drift θ but no a(b − r) term. Only Vasicek and CIR have mean reversion in this topic.
Saying Vasicek rates cannot be negative.
Mean reversion toward a positive b feels like a floor on the rate.
Fix: Vasicek shocks are normally distributed with constant σ, so the rate can fall below zero. Only the √r in CIR, or the log in KWF, stops negative rates.
Calling CIR an arbitrage-free model.
It is a well-known model used for bond pricing, so students assume it fits the market.
Fix: CIR and Vasicek are equilibrium models. They may not reproduce the observed curve exactly. Arbitrage-free models are Ho-Lee and KWF.
Getting the sign of the drift wrong.
Students write (r − b) instead of (b − r), or ignore which side of b the rate is on.
Fix: The drift is a(b − r). If r is above b the result is negative, so the rate is pulled down.
Using the CIR volatility as σ instead of σ√r.
Students copy σ from the vignette without applying the √r term.
Fix: Multiply σ by the square root of the current rate in decimals. For r = 4%, use √0.04 = 0.20.
Treating the speed a as a percentage drop in the rate.
Students misread a as the amount of reversion rather than a multiplier on the gap.
Fix: a scales the gap b − r. With a = 0.30 and a gap of 2%, the drift is 0.6% per year, not 30%.

Worked examples

Example 1

Vignette: An analyst models the short rate with a Vasicek model: a = 0.30, b = 4.0%, σ = 1.0%. The current short rate is 2.0%. She also considers a CIR model with the same a and b and σ = 0.04. Q1: What is the expected change in the short rate over half a year in the Vasicek model, ignoring the random shock? Q2: In the CIR model, what is the volatility term σ√r at r = 2.0% and at r = 0.5%? Q3: If the short rate were 6.0% instead, in which direction does the drift push the rate?

Show the solution

Q1: Drift = a(b − r) × dt = 0.30 × (0.04 − 0.02) × 0.5.
0.30 × 0.02 = 0.006 per year. Multiply by 0.5 to get 0.003, or 0.30%.
Q2: At r = 0.02, √0.02 = 0.14142. σ√r = 0.04 × 0.14142 = 0.005657, about 0.57%.
At r = 0.005, √0.005 = 0.07071. σ√r = 0.04 × 0.07071 = 0.002828, about 0.28%. This is half the first value because the rate fell to one quarter.
Q3: At r = 0.06, a(b − r) = 0.30 × (0.04 − 0.06) = −0.006 per year. The drift is negative.

Answer: Q1: +0.30% expected change over half a year (from 2.0% toward 2.30%). Q2: about 0.57% at r = 2.0% and about 0.28% at r = 0.5%, so volatility falls as the rate falls. Q3: The drift is negative (−0.6% per year), so the rate is pulled down toward 4.0%.

Example 2

Vignette: A fixed-income team needs a model to value callable bonds consistently with today's market prices. They compare three one-factor models: Vasicek, Ho-Lee and KWF. Q1: Which of the three are arbitrage-free? Q2: Which of the three can produce negative interest rates? Q3: Which of the three includes mean reversion?

Show the solution

Q1: Arbitrage-free models are calibrated to reproduce the observed curve. Ho-Lee uses a time-dependent drift θ for this. KWF does the same in log form. Vasicek is an equilibrium model.
Q2: Vasicek and Ho-Lee have normally distributed shocks with constant σ, so rates can go below zero. KWF models ln(r), so r is always positive.
Q3: Mean reversion needs a term a(b − r). Only Vasicek has it. Ho-Lee and KWF have only the drift θ.

Answer: Q1: Ho-Lee and KWF. Q2: Vasicek and Ho-Lee. Q3: Vasicek only.

Exam tips

Learn the grid of four models against five features: type, mean reversion, volatility form, negative rates, fit to the curve. Most conceptual questions test one cell.
When an equation appears in the vignette, identify the model from the equation first. Look for a(b − r), √r and ln(r).
Do drift calculations in decimals and check that the time step matches the units of a. Then check the sign before you pick an option.
Watch for distractors that say equilibrium models fit today's curve exactly or that Ho-Lee reverts to a mean. Both are false.
There is no penalty for wrong answers, so answer every question. If stuck, eliminate options that give a feature to the wrong model.

Term Structure Models: Vasicek, CIR and Ho-Lee 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 IITerm Structure Models with Drift: Ho-Lee and Vasicek

Term Structure Models: Vasicek, CIR and Ho-Lee: frequently asked questions

What is the main difference between Vasicek and CIR?

Both are equilibrium models with mean reversion. Vasicek has constant volatility σ, so rates can go negative. CIR uses σ√r, so volatility falls as rates fall and rates do not go negative.

What is the difference between arbitrage-free and equilibrium term structure models?

Equilibrium models derive the curve from assumptions about the short rate and the economy, so they may not match market prices exactly. Arbitrage-free models take the observed curve as an input and are fitted to reproduce it. That is why they are used to price bonds with embedded options consistently with the market.

Does the Ho-Lee model have mean reversion?

No. The Ho-Lee drift is a time-dependent term θ chosen to fit the observed curve, and volatility is constant. Because there is no mean reversion and the shocks are normal, rates can become negative.

How is the Kalotay-Williams-Fabozzi model different from Ho-Lee?

KWF is a lognormal arbitrage-free model, written as d ln(r) = θ dt + σ dz. Because it models the log of the rate, the rate stays positive. Like Ho-Lee, it has no mean reversion.