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FRM Part II · FRM Exam Part II

The Art of Term Structure Models: Volatility and Distribution

This chapter compares short-rate models by how they treat drift, volatility and the shape of the rate distribution. Normal models allow negative rates and use constant basis-point volatility. CIR and lognormal models tie volatility to the rate level. To solve questions, identify the dynamics, then read off drift, mean reversion, volatility and distribution.

What this chapter covers

This chapter builds a family of short-rate models, one step at a time. You start with a normal model with constant volatility and no drift (Model 1). Then you add a time-dependent drift (Ho-Lee), mean reversion (Vasicek), and time-dependent volatility (Model 3 and Hull-White). Finally you let volatility depend on the rate level (CIR) or assume rates are lognormal (the basic lognormal model and Black-Karasinski).

The key idea is that every model is a choice about two things: how rates drift, and how volatility behaves. Those choices decide the distribution of future rates, the shape of the volatility term structure, and whether rates can go negative. Exam questions usually give you the model equation and ask you to read these features from it.

This chapter sits in the Market Risk Measurement and Management topic. It links to VaR on fixed-income portfolios, to pricing and hedging of rate derivatives, and to interest rate risk in the banking book. If you understand why a model produces a given volatility pattern, you can judge the model risk in any rate-sensitive position.

This chapter is compact, and its questions are conceptual rather than long calculations. That makes it efficient marks if you learn one comparison grid: drift, mean reversion, volatility form, distribution and sign of rates for each model. Candidates who only memorise equations lose marks on interpretation questions, such as which model fits a given volatility term structure or why a model allows negative rates. The same logic also helps you in market risk and in risk questions on rate-sensitive portfolios, so the effort pays back elsewhere in the paper.

The Art of Term Structure Models: Volatility and Distribution: topics in the order to study them

  1. 1Interest Rate Volatility and Term Structure BasicsStart here to fix the vocabulary: basis-point volatility, the volatility term structure and what a short-rate model is meant to capture.
  2. 2Normal Model with Constant Volatility (Model 1)This is the simplest case, with no drift and constant volatility, and every later model is a change to it.
  3. 3Ho-Lee Model with Time-Dependent DriftIt adds one feature, a drift that changes over time so the model can fit today's curve, while volatility stays constant.
  4. 4Vasicek Model and Mean ReversionMean reversion is the next building block, and it explains why long-rate volatility falls below short-rate volatility.
  5. 5Time-Dependent Volatility: Models 3 and Hull-WhiteLearn this after Vasicek. Model 3 has a time-dependent drift and a time-dependent volatility but no mean reversion, and Hull-White adds mean reversion to that.
  6. 6Cox-Ingersoll-Ross (CIR) Model and Rate-Dependent VolatilityHere volatility depends on the rate level for the first time, so you move away from the purely normal distribution.
  7. 7Lognormal Models and Black-KarasinskiThe basic lognormal model has constant proportional volatility and no mean reversion, and Black-Karasinski is the lognormal model with mean reversion. In both, rates stay positive and basis-point volatility grows with the rate level.
  8. 8Volatility Term Structure and Rate Distribution ComparisonFinish with the side-by-side comparison, which is what the exam tests most, and it only makes sense once you know every model.

How to prepare The Art of Term Structure Models: Volatility and Distribution

Treat this chapter as one comparison table you build and then test yourself on. Do not study each model in isolation.

  1. Write each model's dynamics in plain text, for example dr = k(θ − r)dt + σdw for Vasicek, and label every term as drift or volatility.
  2. For each model, note four things: is drift constant or time-dependent, is there mean reversion, is volatility constant, time-dependent or rate-dependent, and is the rate distribution normal or lognormal.
  3. Learn the effect of each feature: mean reversion pulls rates toward a long-run level and lowers long-maturity volatility, and rate-dependent volatility makes volatility rise and fall with the rate level.
  4. Practise reading an equation and naming the model. Cover the model's name and identify it from the equation alone.
  5. Practise the reverse: given a description such as lognormal, positive rates, mean-reverting, name the model and say what it implies for negative rates.
  6. Sketch the volatility term structure for each model. Model 1 and Ho-Lee give a flat one. Mean reversion (Vasicek) gives a downward-sloping one. Model 3 follows the shape of σ(t). Hull-White combines σ(t) with mean-reversion damping, so its long-maturity volatility is lower than σ(t) alone would imply. Check each sketch against your reading.
  7. Finish with timed MCQs, then review every miss by asking which feature of the model you misread.

Common mistakes in The Art of Term Structure Models: Volatility and Distribution

  • Confusing basis-point volatility with percentage (proportional) volatility.

    Fix: In the normal models, σ is a constant basis-point volatility. In the lognormal models, σ is a proportional (percentage) volatility, so basis-point volatility equals σr. In CIR, basis-point volatility is σ√r. Always state which unit σ is in before comparing models.

  • Saying every model keeps rates positive.

    Fix: Remember that normal models (Model 1, Ho-Lee, Vasicek) can give negative rates. Lognormal models keep rates positive. In CIR, volatility shrinks as r approaches zero, so with the mean-reverting drift rates stay non-negative, and strictly positive if the Feller condition holds.

  • Treating Ho-Lee as having mean reversion.

    Fix: In Ho-Lee, the drift λ(t) depends on time only. Mean reversion needs a drift that depends on the rate, as in k(θ − r).

  • Mixing up what mean reversion does to volatility across maturities.

    Fix: Remember that mean reversion dampens the effect of shocks over time, so long-maturity rate volatility is lower than short-maturity volatility.

  • Mixing up Model 3, Hull-White and Vasicek.

    Fix: Compare each one on three points. Mean reversion: Model 3 has none, while Hull-White and Vasicek both have it. Drift timing: Vasicek has a constant long-run level θ, while Model 3 and Hull-White have a time-dependent drift. Volatility form: Vasicek has constant volatility, while Model 3 and Hull-White have time-dependent volatility. Build these into your grid.

  • Memorising equations without interpreting the distribution.

    Fix: For every model, write one sentence on the distribution of future rates and one on what that means for risk, for example negative rates or volatility rising with rates.

Last-day revision: The Art of Term Structure Models: Volatility and Distribution

  • Model 1: dr = σdw, constant volatility, no drift, normally distributed rates.
  • Ho-Lee: dr = λ(t)dt + σdw, a time-dependent drift fits the initial curve, volatility is constant.
  • Vasicek: dr = k(θ − r)dt + σdw, the rate reverts toward θ at speed k.
  • Mean reversion lowers the volatility of long-term rates relative to short-term rates.
  • Normal models can produce negative rates, because the distribution is normal.
  • Model 3 has a time-dependent drift and time-dependent volatility but no mean reversion. Hull-White adds mean reversion to a time-dependent drift and time-dependent volatility.
  • CIR: volatility is proportional to √r, so basis-point volatility rises with the rate level.
  • Lognormal models (the basic lognormal model and Black-Karasinski) keep rates positive, and basis-point volatility is proportional to the rate.
  • The basic lognormal model has constant proportional volatility and no mean reversion, while Black-Karasinski combines lognormal rates with mean reversion.
  • Match the model to its volatility term structure first, and then check the distribution and sign of rates.
  • Constant versus time-dependent versus rate-dependent volatility is the main distinction across the models.

The Art of Term Structure Models: Volatility and Distribution practice questions

The Art of Term Structure Models: Volatility and Distribution 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: Volatility and Distribution: frequently asked questions

Do I need to derive these models for the FRM Part II exam?

No. The exam is multiple choice and focuses on understanding, so you should know what each equation means and what it implies. Practise recognising the drift, volatility and distribution from the equation rather than deriving anything.

What is the main difference between Ho-Lee and Vasicek?

Ho-Lee has a time-dependent drift and no mean reversion, so it fits the initial curve but rates can drift without a pull toward a level. Vasicek has mean reversion, so the rate is pulled toward a long-run level θ at speed k.

Why does CIR use √r in the volatility term?

It makes the basis-point volatility rise when the rate is high and fall when the rate is low. As r approaches zero the volatility shrinks, so with the mean-reverting drift the rate stays non-negative. Strict positivity needs the Feller condition to hold.

How should I compare lognormal and normal models in an answer?

Compare the distribution and the volatility behaviour. Normal models have constant basis-point volatility and can give negative rates, while lognormal models keep rates positive and have volatility that grows with the rate level.

Where does this chapter fit in the FRM Part II syllabus?

It belongs to the Market Risk Measurement and Management topic. The ideas support fixed-income VaR, derivative pricing and interest rate risk questions elsewhere in the paper.