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

Hull-White Model and Time-Dependent Volatility Explained

Updated 11 October 2026 · Fact-checked

Model 3 lets short-rate volatility change with time, usually declining as σ(t) = σe^(−αt), with a time-dependent drift fitted to the curve. Hull-White adds mean reversion to a time-dependent drift: dr = (θ(t) − a·r)dt + σ·dw. To solve questions, identify the drift and volatility terms, then compute the horizon variance.

Understand Time-Dependent Volatility: Models 3 and Hull-White

Start with the simplest idea. In Model 1 and the Ho-Lee model, the short rate has constant volatility σ. Rate changes over T years have standard deviation σ√T, so uncertainty grows without limit. Real rate markets rarely look like that. Volatility of long-horizon rates is lower than a constant-volatility model implies.

Model 3 fixes this by making volatility a function of time. The usual form is dr = λ(t)dt + σ(t)dw with σ(t) = σe^(−αt) and α > 0. Volatility is highest today and decays exponentially. The drift λ(t) depends on time so the model can match today's term structure. The rate itself does not revert to any level. Only the size of the shocks shrinks.

The Hull-White model (the extended Vasicek) takes a different route. It writes dr = (θ(t) − a·r)dt + σ·dw. The term −a·r pulls the rate back toward a moving target, so there is mean reversion at speed a. The time-dependent θ(t) is chosen so the model prices today's bond curve exactly. Vasicek has a constant long-run level, so it cannot do this.

Here is the key link. With constant σ, the standard deviation of the short rate at horizon T in Hull-White is σ√((1 − e^(−2aT)) ÷ (2a)). In Model 3 it is σ√((1 − e^(−2αT)) ÷ (2α)). These have the same form. Both models give lower long-horizon uncertainty than constant volatility. They differ in mechanism. Model 3 has shrinking shocks and no pull-back. Hull-White has constant shocks and a pull-back. They also differ in how rates behave. Mean reversion produces a pull toward the target, while Model 3 simply has smaller moves later.

Both models are normal models, so rates can go negative. Both are no-arbitrage models because their time-dependent drift is calibrated to the current curve.

Key formulas to remember

Model 3 dynamics: dr = λ(t)dt + σ(t)dw, with σ(t) = σ·e^(−αt); Volatility of rate changes declines at exponential rate α > 0. λ(t) is fitted to the initial curve. No mean reversion.
Model 3 short-rate standard deviation at horizon T: s.d.(r_T) = σ·√[(1 − e^(−2αT)) ÷ (2α)]; Variance is the integral of σ²e^(−2αs) from 0 to T. As T grows it approaches σ ÷ √(2α).
Constant-volatility comparison: s.d.(r_T) = σ·√T; Ho-Lee and Model 1. Grows without bound, so it is larger than the Model 3 value for the same σ.
Hull-White dynamics: dr = (θ(t) − a·r)dt + σ·dw; a is the mean-reversion speed. θ(t) is time-dependent and fitted to today's curve. Vasicek is the case with constant θ.
Hull-White short-rate standard deviation: s.d.(r_T) = σ·√[(1 − e^(−2aT)) ÷ (2a)]; Same form as Model 3 with a in place of α. Long-run limit is σ ÷ √(2a).
Hull-White zero-coupon bond price: P(t,T) = A(t,T)·e^(−B(t,T)·r), B(t,T) = (1 − e^(−a(T−t))) ÷ a; B is the sensitivity of ln P to the short rate. For a given Δr, ΔP ÷ P ≈ −B·Δr. B is less than T−t when a > 0.

How to solve Time-Dependent Volatility: Models 3 and Hull-White questions

Use this order for any question on Model 3 or Hull-White. It keeps you from mixing up the two mechanisms.

1Read the dynamics. Look for a term −a·r (mean reversion, Hull-White or Vasicek) or a σ that depends on t (Model 3).
2Identify the drift. If it depends on t, the model is fitted to today's curve. If it is constant, it cannot fit an arbitrary curve.
3Identify the volatility. Constant σ gives σ√T growth. Declining σe^(−αt) or mean reversion gives σ√((1 − e^(−2kT)) ÷ (2k)).
4Pick the right speed parameter: α for Model 3 volatility decay, a for Hull-White mean reversion. Use the same formula shape for both.
5Compute the exponent first (2kT), then e^(−2kT), then the bracket, then the square root, then multiply by σ.
6For bond-price questions, compute B = (1 − e^(−a(T−t))) ÷ a and use ΔP ÷ P ≈ −B·Δr.
7Interpret. Compare with the constant-volatility answer σ√T and say the result is lower, and why.

Quickest way: Shortcut for horizon volatility and bond sensitivity

When to use it: Use when you have a numeric question with a given σ, speed parameter and horizon, and four close answer choices.

Compute x = 2kT and e^(−x). Know e^(−1) ≈ 0.3679 and e^(−2) ≈ 0.1353.
Bracket = (1 − e^(−x)) ÷ (2k). Take the square root and multiply by σ.
Sanity check: the answer must be below σ√T and below the limit σ ÷ √(2k). Drop any option that breaks either bound.
For bond moves, B = (1 − e^(−aT)) ÷ a. It must be below T, so the percent price move is smaller than a duration of T would suggest.
If the question is conceptual, remember: Model 3 shrinks shocks, Hull-White pulls the rate back, and both fit today's curve through a time-dependent drift.

Common mistakes in Time-Dependent Volatility: Models 3 and Hull-White

Saying Model 3 has mean reversion.
Both Model 3 and Hull-White give lower long-horizon variance, so they look alike.
Fix: Model 3 has no pull-back term. Only the volatility σ(t) declines. Mean reversion needs −a·r in the drift.
Using σ√T for the horizon standard deviation in Model 3 or Hull-White.
Students default to the Ho-Lee or Model 1 result.
Fix: Use σ√((1 − e^(−2kT)) ÷ (2k)). Check that your answer is below σ√T.
Thinking Hull-White and Vasicek differ in volatility.
Both have constant σ and mean reversion, so the difference is easy to miss.
Fix: The difference is the drift. Hull-White has time-dependent θ(t) that fits today's curve. Vasicek has constant long-run level and cannot match an arbitrary curve.
Forgetting the factor of 2 in e^(−2kT) and 2k.
The formula is for variance, and students recall the rate's decay e^(−kT) instead.
Fix: Variance integrates σ²e^(−2ks). Always use 2k in both places, then take the square root.
Treating B(t,T) as equal to T − t for a bond price move.
Duration intuition from a zero-coupon bond carries over.
Fix: With mean reversion, a short-rate shock fades, so B = (1 − e^(−a(T−t))) ÷ a is smaller than T − t. A short-rate shock moves long bonds less.
Claiming these models rule out negative rates.
Students mix them up with CIR or lognormal models.
Fix: Model 3 and Hull-White are normal models with constant or time-only volatility, so rates can be negative. Rate-dependent volatility, as in CIR, is a different feature.

Worked examples

Example 1

In Model 3, σ = 100 bp per year and α = 0.20. What is the standard deviation of the short rate at T = 5 years? (A) 37 bp (B) 100 bp (C) 147 bp (D) 224 bp

Show the solution

Use s.d. = σ·√[(1 − e^(−2αT)) ÷ (2α)].
2αT = 2 × 0.20 × 5 = 2. e^(−2) ≈ 0.135335.
1 − 0.135335 = 0.864665. Divide by 2α = 0.40: 2.16166.
Square root of 2.16166 ≈ 1.4703.
Multiply by σ = 100 bp: ≈ 147 bp.
Check: constant volatility would give 100 × √5 ≈ 224 bp (option D). The Model 3 value is lower, as expected. Option A is the instantaneous volatility at t = 5, 100 × e^(−1) ≈ 37 bp, which is a different quantity.

Answer: (C) about 147 bp

Example 2

In a Hull-White model with a = 0.10, the short rate rises by 50 bp. Using ΔP ÷ P ≈ −B·Δr, what is the approximate percentage price change of a 10-year zero-coupon bond? (A) −1.58% (B) −3.16% (C) −5.00% (D) −6.32%

Show the solution

B(0,10) = (1 − e^(−a×10)) ÷ a = (1 − e^(−1)) ÷ 0.10.
e^(−1) ≈ 0.367879, so 1 − 0.367879 = 0.632121.
B = 0.632121 ÷ 0.10 = 6.32121.
Δr = 0.005, so ΔP ÷ P ≈ −6.32121 × 0.005 = −0.03161, or about −3.16%.
Check: with no mean reversion B would be 10 and the move would be −5.00% (option C). Mean reversion shrinks it. Option D is B itself with Δr = 1% instead of 50 bp.

Answer: (B) about −3.16%

Exam tips

Expect conceptual comparisons: Model 3 versus Hull-White versus Vasicek versus Ho-Lee. Learn one line for each: what is time-dependent, and is there mean reversion.
For numeric questions, the bounds check (below σ√T, below σ ÷ √(2k)) often eliminates two options at once.
State what the time-dependent drift does. It fits today's curve, which is the no-arbitrage feature.
Be ready to name the effect of mean reversion on long-horizon uncertainty and on how much a short-rate shock moves long bonds.
Do not mix up the instantaneous volatility σe^(−αt) and the horizon standard deviation of the rate. They are different quantities.

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

Time-Dependent Volatility: Models 3 and Hull-White: frequently asked questions

What is the difference between Hull-White and Vasicek?

Both have mean reversion and constant volatility. Vasicek has a constant long-run level, while Hull-White uses a time-dependent θ(t). That lets Hull-White fit today's term structure exactly, which Vasicek generally cannot.

What is Model 3 in Tuckman's term structure models?

It is a normal short-rate model with time-dependent drift and exponentially declining volatility, σ(t) = σe^(−αt). It generalises Ho-Lee, which has constant volatility, and it has no mean reversion.

Does time-dependent volatility give the same result as mean reversion?

For the horizon variance of the short rate, the formulas have the same shape. The mechanism is different. In Model 3 shocks shrink over time, while in Hull-White shocks stay the same size but the rate is pulled back toward a target.

Can the Hull-White model produce negative interest rates?

Yes. It is a normal model with constant volatility, so rates are normally distributed and can fall below zero. Models with rate-dependent volatility, such as CIR, behave differently.