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

Interest Rate Volatility and Term Structure Basics

Updated 11 October 2026 · Fact-checked

A short-rate model describes the change in the short rate as drift plus volatility times a random shock: dr = λ(t)dt + σ(t)dw. Drift sets the expected path; volatility sets the spread and, through convexity, lowers bond prices. To solve questions, identify drift, identify volatility structure, then compute the rate's mean and standard deviation.

Understand Interest Rate Volatility and Term Structure Basics

A short-rate model describes how the instantaneous interest rate r moves over time. Every model has two parts. The drift is the expected change in the rate over a small time step. The volatility is the size of the random shock around that expected change.

In the basic form, dr = λ(t)dt + σ(t)dw. Here λ(t) is the drift, σ is the annualised volatility of the rate in absolute terms (for example 1.00% per year), and dw is a normally distributed shock with mean 0 and variance dt. Over a short step of Δt years, the rate change has mean λΔt and standard deviation σ√Δt.

The volatility structure is the rule that says how σ behaves. It can be constant, change with time, or depend on the level of the rate. In a normal model, σ is a fixed number, so rates can go negative. In a lognormal model, volatility is proportional to the rate (σ × r), so rates stay positive and shocks are bigger when rates are high. In CIR, volatility is proportional to √r.

Why does volatility matter for prices? Bond prices are a convex function of rates. By Jensen's inequality, higher rate volatility raises the average of the discount factor in a way that lowers the spot rates implied for long maturities in a no-arbitrage model. The effect is the convexity effect, and it grows with maturity and with σ². Options are even more sensitive: higher volatility raises option values directly.

So the choice of volatility structure drives model prices. Two models with the same drift but different volatility structures give different rate distributions, different tail risks, and different option prices. Calibration therefore fits volatility to market data such as cap or swaption prices.

Key formulas to remember

Short-rate dynamics: dr = λ(t)dt + σ(t)dw; λ is drift, σ is volatility, dw is a normal shock with variance dt.
Rate change over a step: Δr ~ Normal(λΔt, σ²Δt); Standard deviation of the change is σ√Δt. Valid for normal (constant or given-σ) models.
Distribution of the rate in Model 1 (zero drift): r(T) ~ Normal(r₀, σ²T); Standard deviation grows with √T.
Constant drift normal model: r(T) ~ Normal(r₀ + λT, σ²T); Drift shifts the mean; volatility drives the spread.
Lognormal volatility: dr = λ(t)r dt + σ r dw; Absolute rate volatility equals σ × r, so it scales with the rate level.
CIR volatility: dr = k(θ − r)dt + σ√r dw; Absolute volatility is σ√r. Rate-dependent and keeps rates nonnegative under the usual conditions.
Annualising volatility: σ_annual = σ_daily × √252 (or σ_monthly × √12); Assumes independent changes. Convert basis points consistently.

How to solve Interest Rate Volatility and Term Structure Basics questions

Use this order for any question on drift, volatility and rate distributions.

1Identify the model: normal, lognormal, time-dependent or CIR. Read how σ is defined.
2Write down drift and volatility, and check units (percent, basis points, per year).
3Find the time horizon T in years and convert if given in months or days.
4Compute the mean: r₀ plus drift × T for a normal model with constant drift.
5Compute the standard deviation: σ√T for the normal model, or scale σ by the rate level for lognormal or CIR models.
6Apply the question: probability, percentile range, or price effect, using the normal distribution.
7Interpret: higher σ means wider rates, bigger convexity effect, lower long-term spot rates under no-arbitrage, and higher option values.
8Check that the answer matches the model's feature, such as possible negative rates.

Quickest way: Mean, √T and sigma scaling

When to use it: Use for numeric questions asking for a rate range or the effect of changing σ or T.

Convert σ to the same unit as the rate, for example 100 bp = 1.00%.
Multiply σ by √T to get the standard deviation.
Add drift × T to the starting rate for the mean.
Use ±1.645σ or ±1.96σ for 90% or 95% two-sided bands.
For lognormal or CIR, first compute the effective σ at the current rate (σr or σ√r).

Common mistakes in Interest Rate Volatility and Term Structure Basics

Scaling volatility by T instead of √T.
Drift scales linearly with time, so students apply the same rule to volatility.
Fix: Variance scales with T. Standard deviation scales with √T.
Mixing basis points and percent.
Volatility is quoted in bp, rates in percent.
Fix: Convert first: 120 bp = 1.20%. Write units beside every number.
Treating lognormal σ as an absolute volatility.
The symbol σ appears in every model.
Fix: In lognormal models absolute volatility is σ × r. At r = 4% and σ = 20%, it is 0.80% per year.
Saying higher volatility raises bond yields because bonds become riskier.
Confusing with credit risk.
Fix: In no-arbitrage rate models, volatility enters through convexity, which raises bond prices relative to the simple drift-based value and lowers implied long-maturity rates.
Assuming normal models can never give negative rates, or that lognormal can.
Mixing up which structure ties volatility to the rate level.
Fix: Normal models allow negative rates. Lognormal models keep rates positive. CIR also stays nonnegative.

Worked examples

Example 1

A normal model has dr = 0.10% × dt... precisely: the short rate is 4.00%, annual drift is 0.20%, and annual volatility is 1.00%. What is the mean and standard deviation of the rate after 4 years, and the 95% two-sided range using 1.96?

Show the solution

Mean = 4.00% + 0.20% × 4 = 4.80%.
Standard deviation = 1.00% × √4 = 2.00%.
Band = 4.80% ± 1.96 × 2.00% = 4.80% ± 3.92%.
Lower = 0.88%, upper = 8.72%.

Answer: Mean 4.80%, standard deviation 2.00%, 95% range 0.88% to 8.72%.

Example 2

In a lognormal model the short rate is 5.00% and σ = 20% per year. What is the one-year standard deviation of the rate change, approximately, and what happens if the rate falls to 2.50%?

Show the solution

Absolute volatility = σ × r = 20% × 5.00% = 1.00% per year.
Over one year, standard deviation ≈ 1.00% × √1 = 1.00%.
At r = 2.50%: 20% × 2.50% = 0.50% per year.
Volatility halves because it is proportional to the rate.

Answer: About 1.00% at a 5.00% rate, falling to 0.50% at a 2.50% rate.

Exam tips

Always check whether the question gives σ in absolute terms (normal) or proportional terms (lognormal).
Expect conceptual items on why volatility lowers long-term spot rates through convexity.
Convert time to years before using √T.
Link model choice to a feature: negative rates, rate-dependent volatility, or fit to the volatility term structure.
Eliminate options that confuse drift with volatility or scale volatility by T.

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

Interest Rate Volatility and Term Structure Basics in other exams

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

Interest Rate Volatility and Term Structure Basics: frequently asked questions

What is the difference between drift and volatility in a short-rate model?

Drift is the expected change in the rate per unit of time. Volatility is the size of the random shock around that expectation. Drift moves the centre of the distribution; volatility sets its width.

Why does volatility affect bond prices?

Bond prices are convex in rates. Higher volatility raises the convexity benefit, so in a no-arbitrage model it pushes up bond prices relative to a zero-volatility case and lowers implied long-term rates.

How do I annualise rate volatility?

Multiply daily volatility by √252 or monthly volatility by √12, assuming independent changes. Keep the unit, such as basis points, the same throughout.

Which volatility structure keeps rates positive?

Lognormal models do, because volatility is proportional to the rate. CIR also keeps rates nonnegative under its usual parameter conditions. A normal model with constant volatility can produce negative rates.