FRM Exam Part II · The Art of Term Structure Models: Volatility and Distribution

Volatility Term Structure and Rate Distribution Comparison

Updated 11 October 2026 · Fact-checked

The volatility term structure shows how rate volatility changes with maturity. Each term structure model implies its own shape and its own rate distribution: normal models allow negative rates, lognormal models do not. To solve questions, identify the model's volatility rule, derive the implied shape, then link it to option prices and calibration.

Understand Volatility Term Structure and Rate Distribution Comparison

Interest rate volatility is not the same at every maturity. The volatility term structure plots the volatility of rates, usually the standard deviation of rate changes, against maturity. Market data show short-maturity rates often swing more than long-maturity rates. Observed shapes are often humped or downward sloping.

Each model builds in a volatility rule. In the normal model with constant volatility (Model 1), the short rate changes by a fixed amount σ per year, whatever the rate level. Volatility term structure is flat in normal terms. Rates are normally distributed and can go negative. Ho-Lee adds a time-dependent drift but keeps constant volatility, so it cannot change the volatility shape. Vasicek adds mean reversion, so volatility of long rates falls with maturity: a downward sloping volatility term structure. Time-dependent volatility models (Model 3, Hull-White) let σ(t) vary, so they can fit a humped or any observed shape.

Rate-dependent volatility changes the distribution. In CIR, volatility is proportional to √r, so volatility falls as rates fall and rates stay non-negative. In the plain lognormal model, volatility is proportional to r, so rates stay positive and the distribution is skewed right. Black-Karasinski also has lognormal rates, but it adds mean reversion in the log of the rate (ln r). A rate-dependent model gives different volatility at different rate levels, so it produces skewed implied distributions.

This matters for pricing. Option prices depend on the distribution and on volatility at the relevant maturity. Calibration means choosing parameters so that model prices match market prices of liquid instruments such as caps and swaptions. A model with few parameters, such as Model 1 or Vasicek, cannot match a whole volatility curve. A model with σ(t) can match it, but may behave poorly outside the calibrated instruments.

Choosing a model is a trade-off. Normal models are simple and tractable and allow negative rates, which is relevant when rates are near zero or negative. Lognormal models respect positivity but are less tractable. They cannot produce negative rates, and they imply very small basis-point volatility when rates are very low. Richer models fit better but risk overfitting.

Key formulas to remember

Model 1 (normal, constant volatility): dr = λ dt + σ dw; Rates are normal, change volatility is σ at all rate levels and maturities. Flat volatility term structure.
Ho-Lee: dr = λ(t) dt + σ dw; Time-dependent drift fits the initial curve. Volatility stays constant.
Vasicek: dr = k(θ − r) dt + σ dw; Mean reversion at speed k. A higher k makes long-maturity rate volatility decay faster.
Vasicek volatility of the instantaneous forward rate at T: σ(T) = σ × e^(−kT); This is the volatility of the instantaneous forward rate at T, not of the T-year spot rate. It decays exponentially with maturity.
Vasicek volatility of the T-year spot rate: σ × (1 − e^(−kT)) ÷ (kT); Use this when the question asks for the volatility of the T-year rate. It also falls as T and k rise.
Time-dependent volatility (Model 3 / Hull-White): dr = λ(t) dt + σ(t) dw; σ(t) can be set to match a humped or sloping market volatility curve.
CIR: dr = k(θ − r) dt + σ√r dw; Volatility rises with √r. Rates stay non-negative under the standard condition on parameters.
Plain lognormal model: dr = a r dt + σ r dw; Percentage volatility is constant. Rates are positive and right-skewed. Black-Karasinski is a different model: lognormal rates with mean reversion in ln r.

How to solve Volatility Term Structure and Rate Distribution Comparison questions

Use this order for any question on volatility shape, rate distribution or model choice.

1Name the model and write down its volatility rule: constant, time-dependent, mean-reverting, √r or proportional to r.
2Decide the rate distribution: normal (negative rates possible) or lognormal/CIR (non-negative, skewed).
3Work out the volatility term structure: flat, downward from mean reversion, or free from σ(t).
4Check units. Normal volatility is in basis points or rate units. Lognormal volatility is a percentage of the rate.
5Link to pricing: higher volatility at the option's maturity raises option value, and a distribution with more mass in the tail near the strike changes out-of-the-money values.
6Link to calibration: count parameters against the market instruments to match, and note whether an exact fit is possible.
7State the trade-off or interpretation asked for, such as tractability versus realism.

Quickest way: Model to shape in 30 seconds

When to use it: When a multiple-choice question asks which model gives a given volatility shape or distribution.

Negative rates allowed? Normal models: Model 1, Ho-Lee, Vasicek, Hull-White.
Rates must stay positive? CIR or lognormal.
Volatility falling with maturity? Look for mean reversion.
Volatility must fit any humped curve? Look for σ(t).
Volatility depends on rate level? CIR (√r) or lognormal (r).
Constant volatility with no mean reversion means flat.

Common mistakes in Volatility Term Structure and Rate Distribution Comparison

Saying Ho-Lee can produce a declining volatility term structure.
Students remember that Ho-Lee has time-dependent drift and assume it is flexible.
Fix: Ho-Lee's drift fits the curve but its volatility is constant. Only mean reversion or σ(t) changes the volatility shape.
Claiming all normal models give a flat volatility term structure.
Confusing Model 1 with Vasicek.
Fix: Vasicek is normal but mean reversion makes long-rate volatility fall with maturity.
Saying CIR allows negative rates.
CIR looks like Vasicek with an extra term.
Fix: The √r volatility shrinks to zero near zero rates, keeping rates non-negative under the standard parameter condition.
Comparing normal and lognormal volatility numbers directly.
Both are called σ.
Fix: Normal σ is in rate units, such as 100 bps. Lognormal σ is a percentage of the rate. Convert using σ_normal ≈ σ_lognormal × r.
Thinking a better calibration fit always means a better model.
Matching market prices feels like proof.
Fix: More parameters can overfit and give unstable hedges. Judge fit, stability and economic sense together.

Worked examples

Example 1

A lognormal short-rate model has percentage volatility of 20%. Convert to approximate normal (basis point) volatility when the rate is 4% and when it is 1%.

Show the solution

Normal volatility ≈ lognormal volatility × rate.
At r = 4%: 0.20 × 4% = 0.80% = 80 bps per year.
At r = 1%: 0.20 × 1% = 0.20% = 20 bps per year.
So basis-point volatility falls as rates fall.

Answer: About 80 bps at a 4% rate and 20 bps at a 1% rate.

Example 2

In a Vasicek model, the volatility of the short rate is 1.20% and k = 0.10. Using σ(T) = σ × e^(−kT) for the volatility of the instantaneous forward rate, find it at T = 10 years. Then find the volatility of the 10-year spot rate. Is the forward-rate volatility term structure flat?

Show the solution

Forward rate: σ(10) = 1.20% × e^(−0.10 × 10) = 1.20% × e^(−1).
e^(−1) ≈ 0.3679.
1.20% × 0.3679 ≈ 0.4415%, about 44 bps.
Forward-rate volatility falls from 120 bps to about 44 bps, so it slopes downward, not flat.
Spot rate: kT = 0.10 × 10 = 1, so the volatility is 1.20% × (1 − e^(−1)) ÷ 1.
1 − 0.3679 = 0.6321, and 1.20% × 0.6321 ≈ 0.7585%, about 76 bps.

Answer: Forward-rate volatility at 10 years is about 0.44% (44 bps), so it slopes downward because of mean reversion. The 10-year spot rate volatility is about 0.76% (76 bps).

Exam tips

Match each model to its one defining feature first: constant σ, mean reversion, σ(t), √r or proportional to r.
Watch for the words 'negative rates' and 'non-negative'; they usually decide the answer.
Read whether volatility is quoted in basis points or percent before comparing.
For calibration questions, count free parameters against the number of market quotes.

Practice questions from The Art of Term Structure Models: Volatility and Distribution

Volatility Term Structure and Rate Distribution Comparison: frequently asked questions

What is the volatility term structure in term structure models?

It shows how rate volatility varies with maturity. Each model implies a shape: flat for Model 1 and Ho-Lee, downward sloping for Vasicek, and any shape for models with σ(t).

How does normal versus lognormal affect option pricing?

Normal models allow negative rates and give symmetric rate outcomes. Lognormal models keep rates positive and skew outcomes to the right, so values of out-of-the-money options differ, especially at low rates.

How do I choose between term structure models?

Weigh tractability, fit to market prices, and realism of rate behaviour. Use positive-rate models when negative rates are implausible, and richer models when you must match a full volatility curve.

Why does calibration matter?

Calibration sets parameters so the model reproduces prices of liquid instruments such as caps and swaptions. A model that cannot match them will misprice related derivatives.