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

Ho-Lee Model with Time-Dependent Drift Explained

Updated 11 October 2026 · Fact-checked

The Ho-Lee model sets the short rate change as dr = λ(t)dt + σdw. Volatility σ is constant. The drift λ(t) changes with time and is chosen so model bond prices match today's term structure. In continuous form, λ(t) = slope of the forward curve + σ²t. It has no mean reversion.

Understand Ho-Lee Model with Time-Dependent Drift

A term structure model describes how the short rate moves. A model is useful to a trader only if it prices today's bonds correctly. A model with a constant drift cannot do this for an arbitrary curve. It gives you a curve of one fixed shape.

Ho-Lee fixes this by letting the drift depend on time. The rate process is dr = λ(t)dt + σdw under the risk-neutral measure. Each date gets its own drift. You pick the drifts one by one so the model reproduces the observed zero-coupon bond prices or forward rates. This is calibration to the initial term structure, and it makes the model arbitrage-free relative to today's curve.

The drift has two parts. The first follows the slope of the forward curve: if forwards rise with maturity, the expected short rate must rise. The second is a convexity term, σ²t. It exists because bond prices are convex in rates. Higher volatility raises the value of convexity, so the expected short rate must drift up more to keep bond prices consistent with today's curve. Even with a flat curve, the Ho-Lee drift is positive because of this term.

Volatility is constant and there is no mean reversion. Rate variance grows in proportion to time, so far-future rate distributions are very wide. Rates are normally distributed, so they can go negative. Compare Vasicek: it has a constant volatility too, but it pulls the rate towards a long-run level, which reduces long-horizon variance. Vasicek's parameters are few, so it usually cannot fit the whole initial curve exactly. Ho-Lee fits it exactly but with no mean reversion.

In a binomial tree, Ho-Lee means each step moves the rate by the drift for that date plus or minus σ√dt with equal risk-neutral probabilities. The drift per step is chosen to fit the bond price for the next maturity.

Key formulas to remember

Ho-Lee short rate process
dr = λ(t)dt + σdw
σ is constant. λ(t) varies with time. Risk-neutral dynamics. Rates are normal, so negative rates are possible.
Continuous-time drift (calibration result)
λ(t) = ∂f(0,t)/∂t + σ²t
f(0,t) is today's instantaneous forward rate for date t. The first term is the forward curve slope. The second is the convexity term.
Expected short rate
E[r(t)] = f(0,t) + σ²t²/2
Risk-neutral expectation, equal to r(0) plus the integral of λ from 0 to t when r(0) = f(0,0).
Short rate variance
Var[r(t)] = σ²t
Grows without limit as t increases because there is no mean reversion.
Binomial tree step
r(next) = r + λ(t)dt ± σ√dt
Up and down moves have risk-neutral probability 0.5 each. The drift per step is set to fit the bond price.

How to solve Ho-Lee Model with Time-Dependent Drift questions

Use this order for most Ho-Lee questions, whether they ask for drift, expected rate or a model comparison.

  1. 1Identify the model. Ho-Lee means constant σ, time-dependent drift λ(t), and no mean reversion.
  2. 2List what is given: forward curve or bond prices, σ, and the time t you need.
  3. 3If asked for the drift, take the slope of the forward curve at t. Use the same units for time (years) and rate (decimals).
  4. 4Add the convexity term σ²t. Convert σ to a decimal before squaring (1% = 0.01).
  5. 5If asked for the expected short rate, use f(0,t) + σ²t²/2, or sum the drifts from 0 to t.
  6. 6For a tree, add the drift times dt, then add or subtract σ√dt for the up and down nodes.
  7. 7Interpret the result: the drift fits the initial curve, and the convexity term is positive even when the curve is flat.
  8. 8For comparison questions, check mean reversion, curve fit and negative rates against the stated limitations.

Quickest way: Drift = forward slope + σ²t

When to use it: Use when the question gives a forward curve (or its slope) and σ, and asks for the Ho-Lee drift at a date.

  1. Write the slope of the forward curve per year as a decimal.
  2. Compute σ² × t with σ as a decimal.
  3. Add the two. This is λ(t) per year.
  4. Sanity check: a flat curve should leave only the σ²t term.
  5. Eliminate options that show zero drift on a flat curve or that omit t.

Common mistakes in Ho-Lee Model with Time-Dependent Drift

  • Saying the drift is zero when the forward curve is flat

    Students forget the convexity term and think a flat curve means no expected change.

    Fix: Always add σ²t. A flat curve gives λ(t) = σ²t, which is positive.

  • Using σ in percent when squaring

    Mixing units, e.g. squaring 1 instead of 0.01.

    Fix: Convert to decimals first: σ = 1% = 0.01, so σ² = 0.0001.

  • Thinking Ho-Lee has mean reversion

    It is taught next to Vasicek, and both have constant volatility.

    Fix: Ho-Lee has no mean-reverting term. Time-dependent drift is not mean reversion. Only Vasicek pulls rates to a long-run level.

  • Believing the time-dependent drift is a forecast of future rates

    Drift sounds like a prediction.

    Fix: It is a risk-neutral drift set to fit today's prices. It is not a real-world forecast.

  • Claiming Ho-Lee cannot produce negative rates

    Confusing it with lognormal or CIR-type models.

    Fix: Ho-Lee rates are normally distributed. Negative rates have positive probability.

  • Saying Vasicek fits any initial curve exactly while Ho-Lee does not

    Reversing the comparison.

    Fix: Ho-Lee fits the initial curve by construction. Basic Vasicek, with constant parameters, generally does not.

Worked examples

Example 1

In a Ho-Lee model, σ = 1% per year. Today's instantaneous forward curve is f(0,t) = 3% + 0.2% × t, with t in years. Find the drift λ(4) and the risk-neutral expected short rate at t = 4, assuming r(0) = 3%.

Show the solution
  1. Forward curve slope = 0.2% per year = 0.002.
  2. σ = 0.01, so σ² = 0.0001.
  3. Convexity term at t = 4: 0.0001 × 4 = 0.0004.
  4. λ(4) = 0.002 + 0.0004 = 0.0024, or 0.24% per year.
  5. Expected short rate: f(0,4) = 3% + 0.2% × 4 = 3.8%.
  6. Convexity adjustment = σ²t²/2 = 0.0001 × 16 ÷ 2 = 0.0008 = 0.08%.
  7. E[r(4)] = 3.8% + 0.08% = 3.88%.
  8. Check by integrating drift: 0.002 × 4 + 0.0001 × 16 ÷ 2 = 0.008 + 0.0008 = 0.0088. Adding r(0) = 3% gives 3.88%.

Answer: λ(4) = 0.24% per year and E[r(4)] = 3.88%.

Example 2

Today's forward curve is flat at 4%. A Ho-Lee model has σ = 1.5% per year. What is the drift λ(2)?
A) 0%
B) 0.045%
C) 0.09%
D) 0.45%

Show the solution
  1. A flat forward curve has zero slope, so the first term is 0.
  2. σ = 0.015, so σ² = 0.000225.
  3. Convexity term = σ²t = 0.000225 × 2 = 0.00045.
  4. λ(2) = 0 + 0.00045 = 0.045% per year.
  5. Option A ignores convexity. Option C doubles the correct value. Option D is a decimal-place slip.

Answer: B) 0.045%

Exam tips

  • Expect conceptual comparisons: Ho-Lee vs Vasicek on mean reversion, curve fit and negative rates. Learn the limitations as a short list.
  • For numeric drift questions, the usual trap is omitting σ²t. Check it on every flat-curve question.
  • Watch units. Percent versus decimal errors produce the distractor options.
  • Know that time-dependent drift fits the initial curve but does not change the volatility structure. Volatility is still constant and the variance grows with time.
  • If a question asks which feature lets a model match today's bond prices, the answer is the time-dependent drift.

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

Ho-Lee Model with Time-Dependent Drift: frequently asked questions

What is the Ho-Lee model in simple terms?

It is a short-rate model where the rate moves by a time-varying drift plus constant normal volatility. The drift is chosen date by date so model bond prices equal market prices today. It has no mean reversion.

How do you calibrate the Ho-Lee model to the term structure?

Choose each drift λ(t) so that the model's zero-coupon prices match the observed curve. In continuous time this gives λ(t) = slope of the forward curve + σ²t. In a tree you solve for the drift at each step in turn.

What is the difference between Ho-Lee and Vasicek?

Both have constant volatility and normally distributed rates. Vasicek has mean reversion to a long-run level, and its constant parameters generally cannot fit the whole curve. Ho-Lee has time-dependent drift that fits the curve exactly, but no mean reversion.

What are the main limitations of Ho-Lee?

Rates can be negative. There is no mean reversion, so long-horizon rate variance grows with time. Volatility is the same at all dates, and shocks are the same size for all maturities. It fits only today's curve, not future volatility.