FRM Exam Part II · Arbitrage Pricing with Term Structure Models

Time-Dependent Volatility and Cox-Ingersoll-Ross Models

Q: What is the difference between CIR and Vasicek?

Both mean-revert. In Vasicek the volatility term is constant (σ), so rates are normal and can be negative. In CIR it is σ√r, so volatility falls as the rate falls and the rate cannot go negative.

Q: Can the CIR model produce negative interest rates?

No. As r approaches zero, the volatility term σ√r goes to zero, and the mean-reversion drift is positive at zero when θ is positive. The Feller condition 2kθ ≥ σ² decides whether the rate can touch zero.

Q: What does the Hull-White model add to Vasicek?

It makes parameters time-dependent. A time-dependent drift θ(t) lets the model fit today's yield curve exactly, and σ(t) lets it fit the volatility term structure. It remains a normal model, so negative rates are possible.

Q: Why is Black-Karasinski called a lognormal model?

It models the natural log of the short rate as a mean-reverting normal process, so the rate itself is lognormal and always positive. The cost is that bond and option prices generally need numerical methods such as trees.

Updated 11 October 2026 · Fact-checked

Time-dependent volatility lets the rate volatility σ change with the date, so a model can match the market's volatility term structure. Hull-White adds this to Vasicek. CIR and lognormal models make basis-point volatility depend on the rate level, which keeps rates from going negative. To solve questions, identify the volatility term and drift.

Understand Time-Dependent Volatility and Cox-Ingersoll-Ross Models

Start with the simplest short-rate model: dr = σ dw. Here r is the short rate, σ is volatility in rate units (for example 100 bps a year) and dw is a random shock. Volatility is constant, and the rate can go negative. Real markets show something different. Short-dated and long-dated rate volatilities differ, and that pattern is the volatility term structure.

Time-dependent volatility fixes the first problem. Replace σ with σ(t), a function of calendar time. You can then choose σ(t) so the model reproduces observed volatilities, such as those implied by caps or swaptions. The Hull-White model is the standard example. It is a Vasicek-type model with a time-dependent drift θ(t) and a time-dependent volatility σ(t): dr = (θ(t) − a·r) dt + σ(t) dw. The drift θ(t) fits today's yield curve. σ(t) fits the volatility curve. The mean-reversion speed a pulls the rate back, which lowers the volatility of long-horizon rates compared with short-horizon rates. Textbook versions differ slightly in how they write the drift, so read the form given in the question.

The second problem is negative rates. In normal models (Ho-Lee, Vasicek, Hull-White), the rate is normally distributed, so negative rates always have some probability. The fix is to make volatility depend on the level of the rate. In the Cox-Ingersoll-Ross (CIR) model, dr = k(θ − r) dt + σ√r dw. The basis-point volatility is σ√r. As r falls toward zero, volatility shrinks, and the rate cannot go below zero. If 2kθ ≥ σ² (the Feller condition), r never even reaches zero.

The lognormal family goes further. Basis-point volatility is proportional to r, so volatility is a percentage of the rate. The Salomon Brothers-style model is dr = a·r dt + σ·r dw. Black-Karasinski adds mean reversion and time-dependent parameters to the log of the rate: d(ln r) = k(t)(ln θ(t) − ln r) dt + σ(t) dw. Because ln r is normal, r is lognormal and always positive.

The trade-off: normal models are tractable and often give closed-form bond and option prices (Vasicek and Hull-White do). CIR keeps some tractability. Lognormal models like Black-Karasinski usually need numerical methods such as trees. Negative rates have been observed in EUR and other markets, so a model that forbids them is not automatically better.

Key formulas to remember

Model with time-dependent volatility: dr = λ(t) dt + σ(t) dw; λ(t) is a time-dependent drift fitted to the yield curve. σ(t) is chosen to fit the volatility term structure. Normal distribution, so negative rates are possible.
Hull-White (extended Vasicek): dr = (θ(t) − a·r) dt + σ(t) dw; Mean reversion at speed a, with θ(t) fitting today's curve and σ(t) fitting volatilities. Rates are normal, so negative rates remain possible.
CIR model: dr = k(θ − r) dt + σ√r dw; k is the speed of mean reversion, θ is the long-run level. Basis-point volatility = σ√r. Rate cannot go negative.
Feller condition (CIR): 2kθ ≥ σ²; If true, the rate stays strictly above zero. If false, it can touch zero but still cannot go negative.
Lognormal model (no mean reversion): dr = a·r dt + σ·r dw; Basis-point volatility = σ × r. σ is a percentage volatility, for example 20%.
Black-Karasinski: d(ln r) = k(t)(ln θ(t) − ln r) dt + σ(t) dw; Mean-reverting lognormal model with time-dependent parameters. Rates always positive. Usually needs a numerical method.
Basis-point volatility by model: Normal: σ | CIR: σ√r | Lognormal: σ·r; Convert to the same unit (decimal or bps) before comparing.

How to solve Time-Dependent Volatility and Cox-Ingersoll-Ross Models questions

Use this routine for any question on time-dependent volatility, CIR or lognormal models.

1Read the equation and name the model: look at the dw term. σ means normal, σ(t) means time-dependent, σ√r means CIR, σ·r means lognormal.
2Check the drift: is there a mean-reversion term such as k(θ − r) or a·r? Constant, time-dependent or mean-reverting drift changes the answer to questions about long-horizon volatility.
3Ask whether the rate can be negative. Normal models (including Hull-White): yes. CIR and lognormal models: no.
4If a volatility number is needed, compute basis-point volatility at the stated rate level: σ, σ√r or σ·r. Work in decimals, then convert to bps.
5For CIR with parameters given, test the Feller condition 2kθ ≥ σ². Compare the two numbers directly.
6For time-dependent volatility, link σ(t) to the volatility term structure: it is chosen so model volatilities match market-implied volatilities at each horizon.
7State the interpretation in one line: what the model allows or prevents, and what the cost is (for example, no closed form, or negative rates possible).
8Sanity-check units and size. A 5% short rate with 20% lognormal volatility is 100 bps a year, not 20 bps.

Quickest way: Read the dw term, then test rate level

When to use it: Use it for multiple-choice questions where you are asked which model fits, what volatility results, or whether rates can turn negative.

Find the dw coefficient: σ, σ(t), σ√r or σ·r.
Normal (σ or σ(t)): negative rates possible. CIR and lognormal: not possible.
For volatility at a given r, multiply: σ√r (CIR) or σ·r (lognormal). Use decimals.
For CIR, compute 2kθ and σ² and compare. Larger or equal 2kθ means strictly positive.
Eliminate options that say CIR can go negative or that Hull-White prevents negative rates.

Common mistakes in Time-Dependent Volatility and Cox-Ingersoll-Ross Models

Saying Hull-White prevents negative rates because it has mean reversion and time-dependent terms.
Mean reversion pulls rates toward a level, so students assume it keeps them positive.
Fix: Hull-White is a normal model. Mean reversion changes where rates tend to go but not the normal distribution, so negative rates have positive probability.
Using σ as the basis-point volatility in CIR.
In Vasicek the σ is the volatility directly, and students carry that habit over.
Fix: In CIR, basis-point volatility is σ√r. Compute the square root of the rate in decimals first.
Confusing the Feller condition with the no-negative-rates result.
Both relate to the zero boundary.
Fix: CIR never produces negative rates. Feller (2kθ ≥ σ²) decides whether the rate can even reach zero. Failing it means touching zero is possible, not going below.
Reading lognormal volatility of 20% as 20 bps.
Mixing percentage volatility of the rate with volatility in rate units.
Fix: Basis-point volatility = σ × r. At r = 5% and σ = 20%, that is 1% = 100 bps a year.
Treating σ(t) as an extra source of randomness.
The notation looks like a new variable.
Fix: σ(t) is a deterministic function of time, chosen to fit the observed volatility term structure. There is still one shock, dw.
Assuming lognormal models are always preferred because rates stay positive.
Positivity is seen as the only quality of a model.
Fix: Lognormal models often lack closed-form prices and need trees or simulation. Normal models are easier, and negative rates do occur in practice. Pick based on the question's criterion.

Worked examples

Example 1

A risk analyst uses a CIR model with dr = k(θ − r) dt + σ√r dw, where σ = 0.05. The current short rate is 4%. What is the annual basis-point volatility of the short rate? Options: A) 20 bps; B) 50 bps; C) 100 bps; D) 200 bps.

Show the solution

Identify the model: CIR, so basis-point volatility = σ√r.
Convert the rate to a decimal: r = 0.04.
Compute √r = √0.04 = 0.20.
Multiply: 0.05 × 0.20 = 0.01.
Convert to bps: 0.01 = 1% = 100 bps a year.

Answer: C) 100 bps a year. If r fell to 1%, volatility would drop to 0.05 × 0.10 = 0.5% = 50 bps, which shows how CIR volatility shrinks as rates fall.

Example 2

A CIR model has k = 0.2, θ = 3% and σ = 0.15. Which statement is correct? A) The rate can become negative; B) The rate can never reach zero; C) The rate can touch zero but can never become negative; D) The Feller condition holds, so the rate is strictly positive.

Show the solution

Compute 2kθ = 2 × 0.2 × 0.03 = 0.012.
Compute σ² = 0.15² = 0.0225.
Compare: 0.012 < 0.0225, so the Feller condition 2kθ ≥ σ² fails.
Failing Feller means the rate can reach zero. A is wrong, because CIR volatility σ√r vanishes at zero and the rate cannot go below it.
So B and D are wrong. C is correct.

Answer: C) The rate can touch zero but can never become negative.

Exam tips

Always start by reading the dw coefficient. It tells you the model and the answer to most conceptual questions.
Convert percentages to decimals before taking √r or multiplying by r. Most numerical errors come from unit slips.
Be ready to say what time-dependent volatility is for: matching the volatility term structure, with θ(t) fitting the yield curve in Hull-White.
Expect trade-off questions. Link no-negative-rates to CIR and lognormal, and link closed-form tractability to the normal models.
For comparison questions, compute bp volatility for each model at the same rate level, then rank.

Practice questions from Arbitrage Pricing with Term Structure Models