FRM Exam Part II · The Art of Term Structure Models: Drift

Term Structure Models and No-Arbitrage Basics

Updated 11 October 2026 · Fact-checked

A term structure model describes how the short rate moves over time, using a drift (expected change) and volatility (random shock). Equilibrium models derive the curve from assumed dynamics. Arbitrage-free models choose the drift so model prices match today's market curve. To solve questions, identify the drift, the volatility and the model type.

Understand Term Structure Models and No-Arbitrage Basics

Interest rate models exist to price bonds and rate derivatives consistently, and to measure risk. Today's curve gives the price of a zero-coupon bond for each maturity. But an option on a bond depends on how rates may move in future. A model supplies that movement.

Most models describe the change in the short rate over a small time step dt as: drift plus volatility times a random shock. Drift is the expected change in the rate per unit time. Volatility is the size of the random part. In a basic one-factor model, dr = λ dt + σ dw, where dw is a normally distributed shock with mean zero and variance dt.

An equilibrium model (for example Vasicek or CIR) starts with assumptions about the economy and the short rate process. The term structure of rates is an output. The model curve usually will not match the observed market curve exactly. An arbitrage-free model (for example Ho-Lee or Hull-White) takes the market curve as an input. Its drift is allowed to depend on time so the model reprices today's bonds exactly.

Why does the drift matter for pricing? Under the risk-neutral process, the drift is not a forecast of where rates will go. It is the value needed so bond prices are free of arbitrage. It combines the expected rate change with a risk premium adjustment and a convexity effect. Because bond prices are convex in rates, higher volatility lifts bond prices, so the drift must offset this to keep the curve fitted.

You should read the chain this way: choose volatility from option markets, then set drift so the model fits the curve (arbitrage-free), or let the model's own parameters determine the curve (equilibrium). Exams test which one is which, and what each parameter does.

Key formulas to remember

General one-factor short-rate model: dr = (drift) dt + σ dw; dw is a normal shock with mean 0 and variance dt. Drift is the expected change per unit time under the pricing measure.
Model 1: constant volatility, zero drift: dr = σ dw; Rates are normal and can be negative. Expected future short rate equals today's rate. It does not fit the curve in general.
Constant drift model: dr = λ dt + σ dw; λ is the risk-neutral drift. A single constant cannot fit a whole curve.
Ho-Lee (time-dependent drift): dr = λ(t) dt + σ dw; λ(t) is chosen so the model matches the market zero curve. This is arbitrage-free.
Vasicek (mean reversion): dr = k(θ − r) dt + σ dw; k is the speed of reversion and θ the long-run level. An equilibrium model; drift is positive when r is below θ.
Zero-coupon bond price: P(T) = E[ e^(−∫r dt) ] under risk-neutral measure; Price is the risk-neutral expected discount factor, not the discount at the expected rate.
Convexity effect: E[e^(−x)] ≥ e^(−E[x]); Jensen's inequality. Volatility raises bond prices, so drift must be adjusted to hold the fitted curve.

How to solve Term Structure Models and No-Arbitrage Basics questions

Use this sequence for almost any question on term structure models and drift.

1Write the model in the form dr = drift × dt + volatility × dw and label each part.
2Decide the model type: if the drift is time-dependent and fitted to the market curve, it is arbitrage-free; if it comes from fixed parameters like k and θ, it is equilibrium.
3Check the pricing measure. Drift used for pricing is risk-neutral, not a forecast.
4Compute the expected rate change over the step as drift × dt, and the standard deviation as σ × √dt.
5Add them for the next-period distribution: mean = r + drift × dt, standard deviation = σ√dt.
6Ask whether the model can fit today's curve, can give negative rates, and whether it has mean reversion.
7Match your result to the option that states the correct interpretation, not only the correct number.

Quickest way: Three-question shortcut

When to use it: Use this for conceptual multiple-choice questions where you must classify a model or interpret drift.

Is the drift a function of time fitted to the market curve? If yes, arbitrage-free. If it is fixed parameters, equilibrium.
Does the drift include k(θ − r)? If yes, mean reversion, so Vasicek-type.
For numbers, use mean = r + λΔt and sd = σ√Δt, then eliminate options with the wrong scale.

Common mistakes in Term Structure Models and No-Arbitrage Basics

Treating the risk-neutral drift as a forecast of future rates.
The word drift sounds like expected change in the real world.
Fix: Remember it is the drift under the pricing measure, chosen for no-arbitrage, and it includes risk premium and convexity effects.
Saying equilibrium models fit today's curve exactly.
Students mix up the two model families.
Fix: Equilibrium models generate the curve from their parameters. Arbitrage-free models take the curve as input and fit it.
Using σ × dt for the standard deviation over a step.
Drift scales with time, so students scale volatility the same way.
Fix: Volatility scales with the square root of time: σ√dt.
Believing a constant drift can match a full term structure.
One parameter looks like enough to shift the curve.
Fix: A single constant gives one degree of freedom. Fitting many maturities needs a time-dependent drift, as in Ho-Lee.
Forgetting that normal models allow negative rates.
Students assume rates are always positive.
Fix: Normal models such as Model 1, Ho-Lee and Vasicek have a positive probability of negative rates. Lognormal and CIR-type models limit this.
Assuming higher volatility lowers bond prices.
Volatility is linked with risk, so a fall in value seems natural.
Fix: Bond prices are convex in rates, so greater volatility raises the expected discount factor, all else equal.

Worked examples

Example 1

A one-factor model is dr = 0.0020 dt + 0.0100 dw, with time in years. The current short rate is 4.00%. What are the mean and standard deviation of the short rate after one month (dt = 1/12)?

Show the solution

Drift over the step: 0.0020 × (1/12) = 0.0001667, which is 0.01667%.
Mean rate = 4.00% + 0.01667% = 4.01667%.
Standard deviation = 0.0100 × √(1/12).
√(1/12) = 0.28868, so sd = 0.0028868, which is 0.2887%.

Answer: Mean about 4.017%; standard deviation about 0.289% (about 28.9 basis points).

Example 2

A risk team calibrates a model with a time-dependent drift λ(t) so that it reprices all points on today's USD zero curve exactly. Another team uses dr = k(θ − r) dt + σ dw with fixed k, θ and σ. Which statement is correct?

Show the solution

The first model has a drift chosen to match the market curve, so it takes the curve as an input. This is arbitrage-free (Ho-Lee type).
The second has drift from fixed parameters with mean reversion to θ. The curve is an output. This is equilibrium (Vasicek type).
So the second model will generally not match today's curve exactly, while the first will.
Both use normal shocks, so both can produce negative rates.

Answer: The time-dependent drift model is arbitrage-free and fits the curve; the k(θ − r) model is an equilibrium model that generally does not fit it exactly. Both can give negative rates.

Exam tips

Expect classification questions: equilibrium or arbitrage-free, and which parameter does what. Learn the model names with their drift forms.
Always scale volatility by √time and drift by time. Many wrong options use the wrong scaling.
Read whether the question asks for the real-world or risk-neutral view. Pricing uses risk-neutral drift.
In conceptual options, look for words like 'forecast' attached to drift. That is often the trap.
Link drift to convexity: if an option says volatility raises bond prices through convexity, it is usually correct.

Practice questions from The Art of Term Structure Models: Drift

Term Structure Models and No-Arbitrage Basics in other exams

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

Term Structure Models and No-Arbitrage Basics: frequently asked questions

What is drift in interest rate models?

Drift is the expected change in the short rate per unit of time. It is the non-random part of the model. For pricing, it is the risk-neutral drift, not a forecast.

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

Equilibrium models start from assumed short-rate dynamics and produce a term structure as output. Arbitrage-free models use the observed market curve as input and set the drift so the model reprices it exactly.

Why is the drift time-dependent in Ho-Lee?

One constant cannot match many bond prices. A function λ(t) gives enough freedom to fit the whole market zero curve while volatility stays constant.

Does volatility affect the drift needed to fit the curve?

Yes. Bond prices are convex in rates, so higher volatility raises model bond prices. The drift must adjust to keep the fitted curve unchanged.