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FRM Part II · FRM Exam Part II

The Art of Term Structure Models: Drift for FRM Part II

Drift is the expected change in the short rate in a term structure model. You solve questions by naming the model, reading its drift term, and finding the expected rate path and the effect on bond prices. Zero drift, constant drift, Ho-Lee, Vasicek and CIR differ mainly in how drift is set.

What this chapter covers

This chapter asks one question: how should the short-rate drift be specified in an interest rate model? Every model here has the form dr = (drift) dt + (volatility) dw. The drift is the expected change in the rate. The volatility term is the random shock. You start with the simplest case, a normal model with zero drift, and add features step by step.

The progression is the core of the chapter. Model 1 has zero drift. Model 2 adds a constant drift that can be read as a risk premium (risk-neutral drift). Ho-Lee makes drift depend on time so the model fits today's market curve. Vasicek makes drift depend on the rate level, so rates revert to a long-run mean. CIR and the lognormal model change how volatility depends on the rate level, which keeps rates from going negative.

The chapter links to the rest of Market Risk Measurement and Management. Models like these price bonds and interest rate derivatives, and the same ideas feed hedging and risk measurement for rate-sensitive books. It also pairs with the companion chapter on volatility and distribution. Study that one after this one, because it builds on the same model family.

Questions on this chapter are applied and compact. You are given a model, a few parameters and a rate, and you must work out an expected rate, a spot rate effect or a property of the model. If you know the drift term of each model and what it implies, these are fast marks. The same facts also help you answer conceptual items on negative rates, convexity and mean reversion. Every question is worth the same as any other, so a chapter you can answer quickly is worth the effort.

The Art of Term Structure Models: Drift: topics in the order to study them

  1. 1Term Structure Models and No-Arbitrage BasicsYou need the vocabulary first: short rate, drift, volatility, and the difference between fitting today's curve and explaining it.
  2. 2Model 1: Normal Model with Zero DriftIt is the simplest model, and it introduces the convexity effect that every later model builds on.
  3. 3Model 2: Constant Drift and Risk-Neutral DriftIt adds one parameter to Model 1, so you can see exactly how drift moves the spot rate curve.
  4. 4Ho-Lee Model and Time-Dependent DriftIt extends constant drift to a drift that changes with time, and it shows how a model is made to match market prices.
  5. 5Vasicek Model and Mean ReversionHere drift depends on the rate level, which is a different idea from time dependence, so learn it after Ho-Lee.
  6. 6Cox-Ingersoll-Ross and Lognormal ModelsThese change the volatility specification, so you compare them with Vasicek and the normal models last.

How to prepare The Art of Term Structure Models: Drift

Treat this chapter as a family of models. Learn one template, then note what changes in each model. Aim to recall each drift term from memory and explain it in words.

  1. Write the template dr = (drift) dt + (volatility) dw. For each model, fill in the drift and the volatility from memory until you can do it without notes.
  2. Work out Model 1 and Model 2 yourself. In both, the spot rate is r0 + μT/2 − σ²T²/6, where μ is the drift (zero in Model 1) and σ is the volatility. The last term is the convexity effect, which grows with the square of maturity.
  3. Learn Ho-Lee as constant volatility plus a drift chosen for each date so the model matches market bond prices. Note that it has no mean reversion.
  4. Practise Vasicek with numbers. Use dr = k(θ − r) dt + σ dw, where k is the speed of reversion and θ is the long-run rate. Compute expected rates with E[r(t)] = θ + (r0 − θ)e^(−kt), and half-lives with ln 2 ÷ k.
  5. Make a comparison table for yourself. List each model by drift type, volatility type, whether rates can go negative, and whether it fits the market curve exactly.
  6. Finish with mixed questions. For each one, name the model first, then pick the property being tested, then calculate.

Common mistakes in The Art of Term Structure Models: Drift

  • Treating zero drift as meaning the spot rate curve is flat.

    Fix: Remember that convexity still lowers spot rates as maturity grows. In Model 1 the spot rate is r0 − σ²T²/6, so the curve slopes down.

  • Confusing time-dependent drift (Ho-Lee) with mean reversion (Vasicek).

    Fix: In Ho-Lee, drift depends on the date and has no pull toward a level. In Vasicek, drift depends on the current rate and pulls it toward θ.

  • Mixing up the long-run rate θ and the speed k in Vasicek and CIR.

    Fix: θ is where the rate is heading. k is how quickly it gets there. Check the expected rate formula and the half-life ln 2 ÷ k each time.

  • Saying all these models keep rates positive.

    Fix: Normal models, Ho-Lee and Vasicek can all give negative rates. CIR and lognormal models avoid this because volatility depends on the rate level.

  • Reading the constant drift in Model 2 only as a forecast of rate rises.

    Fix: In the risk-neutral setting it reflects a risk premium. Read it as the drift used for pricing, not necessarily a real-world forecast.

  • Skipping the model-naming step in a question.

    Fix: Identify the model from the drift and volatility terms first. Then decide which property or formula the question is testing.

Last-day revision: The Art of Term Structure Models: Drift

  • Every model here is dr = (drift) dt + (volatility) dw. The drift is the expected change in the rate.
  • Model 1 has zero drift and constant volatility: dr = σ dw. The expected future short rate equals today's rate.
  • Model 2 adds a constant drift: dr = λσ dt + σ dw. The drift can be read as a risk premium.
  • With constant drift μ, the spot rate is about r0 + μT/2 − σ²T²/6. The last term is the convexity effect.
  • Convexity lowers spot rates, and the effect grows with the square of maturity and with volatility squared.
  • Ho-Lee: dr = λ(t) dt + σ dw. Drift is time-dependent, volatility is constant, and there is no mean reversion.
  • Ho-Lee drift is chosen to fit market bond prices, so it matches today's curve exactly.
  • Vasicek: dr = k(θ − r) dt + σ dw. The rate is pulled toward θ at speed k.
  • Vasicek expected rate: E[r(t)] = θ + (r0 − θ)e^(−kt). Example: r0 = 4%, θ = 6%, k = 0.5, t = 2 gives 6% − 2% × e^(−1) ≈ 5.26%.
  • Half-life of the gap to the long-run rate is ln 2 ÷ k. If k = 0.25, it is about 2.77 years.
  • Normal models, including Ho-Lee and Vasicek, allow negative rates.
  • CIR: dr = k(θ − r) dt + σ√r dw. Volatility falls as the rate falls, which keeps rates from going negative. A lognormal model has volatility proportional to r and also keeps rates positive.

The Art of Term Structure Models: Drift practice questions

The Art of Term Structure Models: Drift in other exams

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

The Art of Term Structure Models: Drift: frequently asked questions

What is drift in a term structure model?

Drift is the expected change in the short rate over a small time step. It is the dt term in dr = (drift) dt + (volatility) dw. The models in this chapter differ mainly in how that term is specified.

What is the difference between Ho-Lee and Vasicek?

Ho-Lee has a drift that depends on time and is chosen to fit the market curve, with no mean reversion. Vasicek has a drift that pulls the rate toward a long-run level θ at speed k. Both use constant volatility, so both allow negative rates.

Why does the spot rate curve slope down in the zero-drift model?

Because of convexity. Uncertainty about future rates raises bond prices on average, which lowers the implied spot rates. The effect, σ²T²/6, grows quickly with maturity.

Do I need to memorise formulas for this chapter?

Yes, but only a short list. Know the drift and volatility terms of each model, the Vasicek expected rate formula, and the half-life. Practise using them in small numerical questions.

Which models avoid negative interest rates?

CIR and lognormal models, because their volatility falls as the rate falls. Normal models, Ho-Lee and Vasicek can produce negative rates.