FRM Exam Part II · Arbitrage Pricing with Term Structure Models

Ho-Lee and Vasicek Models: Drift and Mean Reversion

Updated 11 October 2026 · Fact-checked

Ho-Lee and Vasicek are normal short-rate models. Ho-Lee uses a time-dependent drift λ(t) to fit today's curve, with constant volatility and no mean reversion. Vasicek uses a drift k(θ − r) that pulls the rate toward a long-run level θ. To solve questions, identify the drift, then compute the expected rate and variance.

Understand Term Structure Models with Drift: Ho-Lee and Vasicek

A short-rate model describes how the instantaneous interest rate moves randomly over time. The simplest version, Model 1, has zero drift: dr = σ dw. Rates then wander with no direction. Such a model cannot match today's term structure, and it has no view on where rates should settle.

The Ho-Lee model adds a drift that depends on time: dr = λ(t) dt + σ dw. You choose λ(t) so that model bond prices equal market bond prices today. That is calibration. Volatility σ is constant. Rates are normally distributed, so they can go negative. There is no pull back toward any level, so the variance of the rate grows as σ²t without limit.

The Vasicek model gives the drift a different job: dr = k(θ − r) dt + σ dw. When r is above θ, the drift is negative. When r is below θ, it is positive. This is mean reversion. The parameter k is the speed of reversion and θ is the long-run mean (in pricing, the risk-neutral long-run mean). Rates are still normal, so negative rates remain possible.

The key difference shows up in volatility. In Ho-Lee, the standard deviation of the rate grows like σ√t. In Vasicek, mean reversion limits the spread, and the standard deviation of the rate levels off at σ/√(2k). So Vasicek implies that long-horizon rate uncertainty is smaller than Ho-Lee implies, and that long-term yields are less volatile than short-term yields.

In exams, treat the drift as the tool. Ho-Lee's drift fits the curve. Vasicek's drift describes behaviour. Basic Vasicek has constant parameters, so it fits the initial curve only approximately. Remember the convexity effect: under Jensen's inequality, long forward rates sit below the expected future short rate. In Ho-Lee, which has zero mean reversion and constant σ, the gap is σ²t²/2. In Vasicek, mean reversion makes the convexity term smaller, and it depends on k, so do not use σ²t²/2 for Vasicek.

Key formulas to remember

Ho-Lee dynamics: dr = λ(t) dt + σ dw; λ(t) is chosen to match the market term structure. σ is constant. Rates are normal.
Vasicek dynamics: dr = k(θ − r) dt + σ dw; k = speed of mean reversion, θ = long-run mean rate, σ = constant volatility of the rate.
Ho-Lee distribution of r(t): Mean = r0 + ∫λ(s) ds from 0 to t; Variance = σ² t; Std dev = σ√t; Variance grows without bound as t increases.
Vasicek expected rate: E[r(t)] = θ + (r0 − θ) e^(−kt); The gap to θ shrinks by the factor e^(−kt).
Vasicek variance of r(t): Var[r(t)] = σ² (1 − e^(−2kt)) ÷ (2k); As t → ∞, the variance tends to σ² ÷ (2k), so the long-run std dev is σ ÷ √(2k).
Half-life of a shock: Half-life = ln(2) ÷ k; Time for the expected gap between r and θ to halve.
Forward rate and convexity (Ho-Lee: constant σ, no mean reversion): f(t) = E[r(t)] − σ² t² ÷ 2, so λ(t) = f′(t) + σ² t; Applies to Ho-Lee only. Use the second form to back out the Ho-Lee drift from the slope of the forward curve. f′(t) is the slope of the instantaneous forward curve. In Vasicek the convexity term is smaller and depends on k.

How to solve Term Structure Models with Drift: Ho-Lee and Vasicek questions

Use this order for any Ho-Lee or Vasicek question, whether it asks for an expected rate, a volatility, a drift or a comparison.

1Identify the model from the drift: λ(t) means Ho-Lee, k(θ − r) means Vasicek.
2List the inputs with units: r0, σ, and either λ(t) or k and θ. Convert percentages and basis points to decimals or stay consistent.
3Note the horizon t and whether the question uses the risk-neutral process (pricing) or a real-world view.
4Compute the expected rate: add the accumulated drift for Ho-Lee, or use θ + (r0 − θ)e^(−kt) for Vasicek.
5Compute the spread: σ√t for Ho-Lee, or √[σ²(1 − e^(−2kt)) ÷ (2k)] for Vasicek.
6If asked about calibration, use λ(t) = f′(t) + σ² t, or the convexity term σ²t²÷2 between expected rate and forward rate.
7Interpret: state whether negative rates are possible, whether volatility is bounded, and how the answer would change as t grows.
8Sanity check: Vasicek variance must be below σ²t and below σ²÷(2k).

Quickest way: Drift test and bound check

When to use it: Use this for conceptual or comparison MCQs, and to eliminate options fast in numerical ones.

Ask first: is there a k(θ − r) term? If yes, expect mean reversion and bounded long-run volatility. If no, expect σ√t growth.
Eliminate any option that says Vasicek or Ho-Lee rates cannot be negative. Both are normal models.
For Vasicek, the expected rate always lies between r0 and θ. Remove options outside that range.
For Vasicek, the standard deviation is always less than σ√t and less than σ÷√(2k). Remove larger values.
For Ho-Lee calibration, use λ(t) = f′(t) + σ² t. If the forward curve is flat, λ(t) is just σ² t, not zero.

Common mistakes in Term Structure Models with Drift: Ho-Lee and Vasicek

Saying Ho-Lee has mean reversion because it has a drift.
Both models have drift terms, so they blur together.
Fix: Ho-Lee's drift depends only on time. It does not depend on the current rate r, so nothing pulls r back. Mean reversion needs a drift that depends on r.
Using σ√t as the Vasicek standard deviation.
Students carry over the Ho-Lee or Model 1 formula.
Fix: Use σ²(1 − e^(−2kt)) ÷ (2k) for the variance. It is always below σ²t.
Claiming Vasicek rates are always positive because of mean reversion to a positive θ.
Mean reversion feels like it prevents extremes.
Fix: Vasicek rates are normally distributed, so any value is possible, including negative. Only CIR-type models with rate-dependent volatility avoid negative rates.
Setting the Ho-Lee drift equal to the slope of the forward curve only.
Students forget the convexity term.
Fix: Use λ(t) = f′(t) + σ² t. The σ²t part offsets the Jensen's inequality effect.
Treating θ as the rate the market expects in the real world.
θ is called the long-run mean.
Fix: In pricing, θ is a risk-neutral parameter and can include a risk premium. Do not read it as a real-world forecast unless the question says so.
Mixing the speed k with the half-life.
Both describe how fast reversion occurs.
Fix: Half-life = ln(2) ÷ k. A larger k gives a shorter half-life and a lower long-run volatility.

Worked examples

Example 1

Under a Vasicek model, r0 = 2%, θ = 5%, k = 0.40 per year and σ = 1.2% per year. Find the expected short rate and its standard deviation after 2 years. Compare the standard deviation with the Ho-Lee value for the same σ.

Show the solution

Expected rate: E[r(2)] = θ + (r0 − θ)e^(−kt) = 5% + (2% − 5%) × e^(−0.8).
e^(−0.8) ≈ 0.4493, so (−3%) × 0.4493 ≈ −1.348%. E[r(2)] ≈ 5% − 1.348% = 3.652%.
Variance: σ²(1 − e^(−2kt)) ÷ (2k). Here 2kt = 1.6 and e^(−1.6) ≈ 0.2019, so 1 − 0.2019 = 0.7981.
σ² = 1.44 (in %²). Divide by 2k = 0.8 to get 1.8. Then 1.8 × 0.7981 ≈ 1.4366 (%²).
Standard deviation = √1.4366 ≈ 1.199%, about 1.20%.
Ho-Lee comparison: σ√t = 1.2% × √2 ≈ 1.697%.
Vasicek's 1.20% is lower than Ho-Lee's 1.70%, because mean reversion limits dispersion.

Answer: Expected rate ≈ 3.65%; standard deviation ≈ 1.20%. The Ho-Lee standard deviation would be about 1.70%, so Vasicek shows less rate uncertainty.

Example 2

A Ho-Lee model has σ = 1% per year. The market instantaneous forward curve is f(t) = 3% + 0.20% × t (t in years). Find the drift λ(2) and the gap between the expected short rate and the forward rate at t = 5.

Show the solution

Use λ(t) = f′(t) + σ² t.
Slope of the forward curve: f′(t) = 0.20% per year = 0.0020.
σ² = (0.01)² = 0.0001. At t = 2, σ² t = 0.0002.
λ(2) = 0.0020 + 0.0002 = 0.0022, or 0.22% per year.
Convexity gap = σ² t² ÷ 2 = 0.0001 × 25 ÷ 2 = 0.00125, or 12.5 bps.
The expected short rate at t = 5 exceeds the forward rate by 12.5 bps, because f(t) = E[r(t)] − σ²t²÷2.

Answer: λ(2) = 0.22% per year. At t = 5, the expected short rate is 12.5 bps above the forward rate.

Exam tips

Look at the drift term first. It tells you the model, and most conceptual MCQs depend on whether it contains r.
Expect questions that ask which model bounds long-run volatility. The answer is Vasicek, with long-run standard deviation σ ÷ √(2k).
Remember that both models allow negative rates. This is a favourite trap option.
For calibration questions, include the σ² t term. Forgetting convexity is the most common lost mark.
Check units: σ in percent per year and t in years. Do a quick bound check before choosing an option.

Practice questions from Arbitrage Pricing with Term Structure Models

Term Structure Models with Drift: Ho-Lee and Vasicek in other exams

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

CFA Level IITerm Structure Models: Vasicek, CIR and Ho-Lee

Term Structure Models with Drift: Ho-Lee and Vasicek: frequently asked questions

What is the main difference between the Ho-Lee and Vasicek models?

Ho-Lee has a time-dependent drift and no mean reversion, so the rate's variance grows as σ²t. Vasicek has a drift that pulls the rate toward a long-run level θ, so variance levels off. Ho-Lee fits today's curve exactly, while basic Vasicek has constant parameters and fits only approximately.

Can the Vasicek model produce negative interest rates?

Yes. The short rate in Vasicek is normally distributed, so any value has some probability, including negative ones. This is a known limitation compared with models whose volatility depends on the rate level, such as CIR.

How does mean reversion affect volatility of long-term rates?

Mean reversion makes shocks fade, so the spread of future short rates stops growing and approaches σ ÷ √(2k). Long-term yields therefore move less than short-term yields in Vasicek. In Ho-Lee, all rate shocks are permanent.

How do I calibrate the Ho-Lee drift?

Choose λ(t) so that model bond prices match market prices. With constant σ, this gives λ(t) = f′(t) + σ² t, where f′(t) is the slope of the instantaneous forward curve. The σ² t term corrects for convexity.