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

Ho-Lee Model and Time-Dependent Drift Explained

Updated 11 October 2026 · Fact-checked

The Ho-Lee model says the short rate follows dr = λ(t)dt + σdw. The drift λ(t) changes with time and is chosen so the model prices today's zero-coupon bonds exactly. Volatility σ is constant and there is no mean reversion. To solve questions, build the tree and set each period's drift to match market prices.

Understand Ho-Lee Model and Time-Dependent Drift

A term structure model describes how the short-term interest rate moves. The simplest model has a constant drift and constant volatility. It cannot reproduce the market's zero-coupon bond prices, because one drift number cannot fit a whole curve of maturities.

The Ho-Lee model fixes this by letting the drift depend on time: dr = λ(t)dt + σdw. Here λ(t) is a drift that varies by date, σ is a constant normal volatility, and dw is a random shock. You pick λ(t) so that the model's bond prices equal market prices. This is calibration, and the model is said to fit the term structure exactly.

On a binomial tree, each step moves the rate up or down with risk-neutral probability 0.5 each. Up: r + λ(t)dt + σ√dt. Down: r + λ(t)dt − σ√dt. The tree recombines, so an up-then-down path ends at the same node as a down-then-up path. The gap between neighbouring nodes in a period is 2σ√dt. The drift moves the whole set of nodes together, and σ sets how far apart they are.

In continuous time, the drift is linked to the market forward curve: λ(t) = ∂f(0,t)/∂t + σ²t, where f(0,t) is the instantaneous forward rate. The first term follows the slope of the forward curve. The second term, σ²t, is a convexity adjustment that grows with time. Because the drift is a function of time and not of the rate level, there is no mean reversion.

Compare this with Vasicek: dr = k(θ − r)dt + σdw. Vasicek has mean reversion, so the drift pulls the rate toward a long-run level θ at speed k. Vasicek has few parameters and does not fit the curve exactly. Ho-Lee fits exactly but has no mean reversion, so the rate can drift far from any anchor and can turn negative. Ho-Lee also gives the same volatility at all horizons, with no decay in the volatility of long rates.

Key formulas to remember

Ho-Lee short-rate process: dr = λ(t)dt + σdw; λ(t) is the time-dependent drift. σ is constant. There is no mean reversion.
Binomial tree step: r(up) = r + λ(t)dt + σ√dt; r(down) = r + λ(t)dt − σ√dt; Each branch has risk-neutral probability 0.5. The tree recombines.
Node spacing: Gap between adjacent nodes = 2σ√dt; Drift shifts all nodes together and does not change the gap.
Zero-coupon bond price by backward induction: P(node) = 0.5 × [P(up) + P(down)] ÷ (1 + r × dt); Use the rate at the node where you are discounting. Start from the par value at maturity.
Continuous-time drift fitted to the forward curve: λ(t) = ∂f(0,t)/∂t + σ²t; The first term is the slope of the instantaneous forward curve. The second is a convexity term.
Vasicek process (for comparison): dr = k(θ − r)dt + σdw; Mean reversion at speed k toward θ. Does not fit the curve exactly.

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

Use this order for any Ho-Lee tree, calibration or comparison question.

1Write the model: dr = λ(t)dt + σdw. Note σ, dt and the starting rate r0.
2Compute the step size σ√dt. It is the up or down shock around the drifted rate.
3Build the tree one period at a time. At each date, the rate is the previous rate plus λ(t)dt, then plus or minus σ√dt.
4If the drift is given, discount backward from maturity. At each node, average the two next-period values with 0.5 weights and divide by (1 + r × dt).
5If you must calibrate, treat the drift for the first period as the unknown. Set the model price of the zero-coupon bond for that maturity equal to the market price and solve for λ.
6Repeat period by period. Each later maturity fixes the next drift, so the tree matches all market zero prices.
7If the question asks about the continuous formula, use λ(t) = slope of the forward curve + σ²t.
8State the interpretation: exact fit to the curve, constant volatility, no mean reversion, negative rates possible.

Quickest way: Shortcut for exam time pressure

When to use it: Use it when a question asks you to price a bond on a given Ho-Lee tree, or to compare drift choices, and options are numerical.

Compute the rates at each node first. Write them in a small grid before discounting anything.
Discount backward only once. Do not recompute the earlier steps.
For a higher drift, expect lower bond prices. For a larger σ, expect a slightly higher bond price because of convexity.
For conceptual questions, eliminate options that mention mean reversion or a rate-dependent drift. Those describe Vasicek, not Ho-Lee.
For the forward-curve formula, add the forward slope and σ²t. Check that σ is in decimals (1% = 0.01) before squaring.

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

Adding a mean-reversion term to Ho-Lee.
Students mix up Ho-Lee with Vasicek because both have a drift.
Fix: In Ho-Lee, the drift depends only on time, never on the current rate. Mean reversion appears only in Vasicek.
Applying the drift only to the up branch.
Students think the drift is a separate third outcome.
Fix: The drift λ(t)dt is added to both branches. Only the σ√dt shock has opposite signs.
Forgetting that calibration works one maturity at a time.
Students try to solve one drift for all dates at once.
Fix: Fit the 2-period bond to get the first drift, then the 3-period bond for the next, and so on.
Dropping the σ²t convexity term in the continuous drift formula.
Students assume the drift equals the slope of the forward curve.
Fix: The correct formula is λ(t) = ∂f(0,t)/∂t + σ²t. The extra term accounts for convexity.
Using percentages directly when squaring σ.
Students enter σ = 1 and get σ² = 1 instead of 0.0001.
Fix: Convert to decimals first. σ = 1% = 0.01, so σ² = 0.0001.
Saying Ho-Lee prevents negative rates.
Students link a fitted tree to a realistic tree.
Fix: Ho-Lee uses normally distributed rate changes, so rates can be negative. Fitting prices does not remove this.

Worked examples

Example 1

A one-year Ho-Lee tree has r0 = 4% (annual compounding, dt = 1 year), σ = 1% and a first-period drift λ(1)dt = 0.5%. Probabilities are 0.5 each. Price a 2-year zero-coupon bond with par ₹100.

Show the solution

Rate at time 0 is 4%.
At time 1, the up rate = 4% + 0.5% + 1% = 5.5%. The down rate = 4% + 0.5% − 1% = 3.5%.
Value at time 1 per ₹1 par: up = 1 ÷ 1.055 = 0.947867. Down = 1 ÷ 1.035 = 0.966184.
Expected value at time 1 = 0.5 × (0.947867 + 0.966184) = 0.957026.
Discount one year at 4%: 0.957026 ÷ 1.04 = 0.920217.
Multiply by ₹100 par.

Answer: The 2-year zero-coupon bond price is about ₹92.02.

Example 2

A Ho-Lee model uses σ = 1% per year. The instantaneous forward curve in continuous compounding is f(0,t) = 3% + 0.2% × t. What is the continuous-time drift λ(t) at t = 5 years?

Show the solution

Take the slope of the forward curve: ∂f(0,t)/∂t = 0.2% = 0.002 per year.
Write σ in decimals: σ = 0.01, so σ² = 0.0001.
Compute the convexity term: σ²t = 0.0001 × 5 = 0.0005.
Add the terms: λ(5) = 0.002 + 0.0005 = 0.0025.

Answer: λ(5) = 0.25% per year (0.0025). Slope contributes 0.20% and convexity 0.05%.

Exam tips

Know the one-line difference: Ho-Lee has a time-dependent drift and no mean reversion. Vasicek has a mean-reverting drift and does not fit the curve exactly.
Expect conceptual questions on what 'fit the term structure exactly' means: the model reproduces today's market zero-coupon prices.
In tree questions, write down the node rates before discounting. Most errors come from wrong rates, not wrong discounting.
Check units. Convert percentages to decimals when you square σ or multiply by dt.
If an option says rates cannot go negative or that volatility decays with maturity, it is not describing Ho-Lee.

Practice questions from The Art of Term Structure Models: Drift

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

What is the Ho-Lee model in simple words?

It is a short-rate model where the rate changes by a time-dependent drift plus a random shock with constant volatility. The drift is chosen so the model matches today's market bond prices. It has no mean reversion.

How do you calibrate the Ho-Lee model to market prices?

Set the model price of each zero-coupon bond equal to its market price, one maturity at a time. The first bond fixes the first-period drift. The next maturity fixes the next drift, and so on until the tree reproduces the whole curve.

What is the difference between Ho-Lee and Vasicek?

Ho-Lee has a drift that depends on time and no mean reversion, and it fits the curve exactly. Vasicek has a drift that pulls the rate toward a long-run level, so the rate is mean-reverting, but it fits the market curve only approximately with few parameters.

Can Ho-Lee produce negative interest rates?

Yes. Rate changes are normally distributed with constant volatility, so the rate can fall below zero in some paths. Fitting the market curve does not remove this feature.