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

Cox-Ingersoll-Ross and Lognormal Interest Rate Models

Updated 11 October 2026 · Fact-checked

The CIR model makes short-rate volatility proportional to the square root of the rate, so volatility shrinks as rates fall and rates stay non-negative. Lognormal models make volatility proportional to the rate itself. Normal models like Vasicek have constant volatility and can produce negative rates. To solve questions, compute the volatility at the given rate and compare.

Understand Cox-Ingersoll-Ross and Lognormal Models

Normal models such as Ho-Lee and Vasicek assume the change in the short rate has a fixed dollar (basis point) volatility. Because the rate is normally distributed, there is always some probability that it falls below zero. The model does not care how low the rate already is.

The fix is to let volatility depend on the level of the rate. If the rate is low, volatility is low too. Shocks then get smaller as the rate approaches zero, so the rate is less likely to cross it.

The CIR model sets volatility proportional to the square root of the rate: dr = k(θ − r)dt + σ√r dw. It keeps mean reversion, like Vasicek. Enter r as a decimal (4% = 0.04). Then σ√r is the annual rate volatility in decimal form, for example 0.018 = 180 bps. A higher rate means higher volatility, but not in proportion. Rates stay non-negative; in the standard form, they stay strictly positive when 2kθ ≥ σ² (the Feller condition).

The lognormal model sets volatility proportional to the rate: dr = a(t)·r dt + σ·r dw. This is the lognormal model with time-dependent drift, the lognormal analogue of Ho-Lee, where the drift is also proportional to the rate. Ho-Lee itself is a normal model with constant volatility. Here σ is a percentage volatility and the basis-point volatility is σr. For this specific form the rate follows a lognormal distribution, so it can never be negative. A further example is Black-Karasinski (lognormal with mean reversion in the log of the rate).

The trade-off is tractability and realism. Normal models give simple closed-form bond prices. Rate-dependent volatility is more realistic when rates are very low, but it makes the distribution skewed, and many lognormal models need trees or numerical methods. Also, real rates have turned negative in some markets, which these models cannot capture.

Key formulas to remember

Vasicek (normal, mean-reverting): dr = k(θ − r)dt + σ dw; Basis-point volatility is constant at σ. Rates can go negative.
CIR model: dr = k(θ − r)dt + σ√r dw; Enter r as a decimal (4% = 0.04). Then σ√r is the annual rate volatility in decimal form. For example σ = 0.10 and r = 0.04 gives 0.10 × 0.2 = 0.02 = 2% a year = 200 bps. Do not use √4.
Lognormal model with time-dependent drift (lognormal analogue of Ho-Lee): dr = a(t)·r dt + σ·r dw; Basis-point volatility = σ × r. For this specific form the rate is lognormal and stays positive. σ is percentage volatility. Ho-Lee itself is a normal model with constant volatility.
Black-Karasinski: d(ln r) = k(t)(ln θ(t) − ln r)dt + σ(t)dw; Lognormal with mean reversion in the log of the rate. Rate stays positive. Constant k, θ and σ give a simplified special case.
Feller condition (CIR): 2kθ ≥ σ²; If satisfied, the CIR rate cannot reach zero.
Rate volatility over horizon dt: Volatility over dt = (annual volatility function) × √dt; dt is in years. Scale by √dt when converting an annual figure to a shorter or longer horizon.

How to solve Cox-Ingersoll-Ross and Lognormal Models questions

Use this approach for any question comparing term structure models or computing rate volatility.

1Identify the model from the volatility term: constant σ is normal, σ√r is CIR, σr is lognormal.
2Check what unit σ is in. Is it a basis-point volatility or a percentage volatility?
3Plug the stated rate into the volatility term to get the basis-point volatility per year.
4Scale by √dt if the question uses a time step shorter or longer than one year.
5Compare the results at different rate levels to see how volatility changes as rates rise or fall.
6Answer conceptual parts: can the rate go negative, is mean reversion present, is a closed form likely?
7Check the answer for reasonableness: lognormal volatility moves proportionally with the rate, CIR moves with the square root.

Quickest way: Volatility-at-the-rate shortcut

When to use it: Use when the question gives a rate level and asks for volatility or compares models at different rates.

Read the volatility term and substitute r.
Normal: answer is σ. CIR: answer is σ√r. Lognormal: answer is σr.
If the rate doubles, normal vol is unchanged, lognormal vol doubles, CIR vol rises by about 41% (√2).
If the rate falls to zero, CIR and lognormal vol go to zero. Normal vol does not.
Eliminate options on negativity: only normal models allow negative rates.

Common mistakes in Cox-Ingersoll-Ross and Lognormal Models

Saying the CIR volatility is proportional to the rate.
Students blur CIR with the lognormal model because both have rate-dependent volatility.
Fix: CIR uses √r. Lognormal uses r. Write both next to each other in your notes.
Thinking CIR and lognormal models can never be wrong about negative rates because they forbid them.
Non-negativity is presented as a pure advantage.
Fix: It is a modelling choice. When market rates actually go negative, these models cannot fit them. Normal models can.
Forgetting that Vasicek has constant basis-point volatility.
Students think mean reversion changes volatility.
Fix: Mean reversion changes the drift, not the volatility. Vasicek volatility stays σ at every rate level.
Mixing units, such as using σ = 10% as 10 basis points.
Lognormal σ is a percentage of the rate, not an absolute amount.
Fix: For lognormal, multiply σ by r. For example 20% × 5% = 1% = 100 basis points.
Stating that CIR rates are guaranteed positive under any parameters.
Students remember the non-negative property and drop the condition.
Fix: Rates are always non-negative, but they stay strictly above zero only if 2kθ ≥ σ² holds.

Worked examples

Example 1

A lognormal short-rate model has σ = 20% a year. A normal model has σ = 90 bps a year. The current short rate is 5%. Compare the annual basis-point volatility of each, and repeat when the rate falls to 2%.

Show the solution

Lognormal at 5%: 0.20 × 5% = 1.00% = 100 bps.
Normal at 5%: 90 bps, unchanged by the rate level.
Lognormal at 2%: 0.20 × 2% = 0.40% = 40 bps.
Normal at 2%: still 90 bps.
At 2%, a 90 bps normal shock is large relative to the rate, so a negative rate is plausible. The lognormal model has only 40 bps volatility and cannot go below zero.

Answer: At 5%: lognormal 100 bps, normal 90 bps. At 2%: lognormal 40 bps, normal 90 bps. Only the normal model can produce negative rates.

Example 2

In a CIR model, the volatility term is σ√r with σ = 0.09. Compute the annual basis-point volatility when r = 4% and when r = 1%. Interpret the change.

Show the solution

Enter r as a decimal. At r = 4% = 0.04: √0.04 = 0.20. Volatility = 0.09 × 0.20 = 0.018 in decimal form = 1.8% = 180 bps.
At r = 1% = 0.01: √0.01 = 0.10. Volatility = 0.09 × 0.10 = 0.009 in decimal form = 0.9% = 90 bps.
Ratio: 90 ÷ 180 = 0.5. The rate fell by a factor of 4 and volatility fell by a factor of 2, which is the square root of 4.
Interpretation: volatility falls as rates fall, but less than proportionally. A lognormal model would have cut volatility by a factor of 4.

Answer: 180 bps at 4% and 90 bps at 1%. Volatility halves when the rate falls to a quarter, because it scales with √r.

Exam tips

Memorise the three volatility forms: σ, σ√r, σr. Most questions are solved by substituting the rate.
Watch the units in the question. Percentage volatility for lognormal, basis-point volatility for normal. For CIR, enter r as a decimal.
Expect a conceptual option on negative rates. Normal models allow them; CIR and lognormal do not.
Remember CIR keeps mean reversion. Be ready to link it to Vasicek as the same drift with a different volatility term.
Know Black-Karasinski as a lognormal model, and that lognormal models usually lack simple closed-form prices.

Practice questions from The Art of Term Structure Models: Drift

Cox-Ingersoll-Ross and Lognormal Models: frequently asked questions

What is the main difference between the CIR model and Vasicek?

Both have the same mean-reverting drift. Vasicek has constant basis-point volatility σ, so rates can go negative. CIR volatility is σ√r, which shrinks as rates fall and keeps rates non-negative.

Why do some short rate models allow negative rates?

Normal models assume rate changes are normally distributed with constant volatility. A normal distribution has probability on both sides of zero, so the rate can fall below zero. Using a normal model is a modelling choice that trades realism at low rates for tractability and the ability to fit markets with negative rates.

Is Black-Karasinski a normal or lognormal model?

It is lognormal. The log of the short rate follows a mean-reverting process, so the rate itself is always positive. Volatility in rate units is proportional to the rate.

Which model fits negative interest rate markets?

Normal models such as Ho-Lee and Vasicek can fit them because they allow negative rates. CIR and lognormal models cannot, which is a limit when markets trade below zero.