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Build the Ho-Lee short-rate tree, in which drift is a function of time.

FRM® Part II · Market Risk Measurement and Management · The Art of Term Structure Models: Drift · FRM2.MR.14.d

The idea

The Ho-Lee model keeps normally distributed rates and constant volatility. The drift depends on time, so the rate change is dr = λ(t) × dt + σ × dw. In the tree, a node with rate r moves to r + λ(t) × dt plus or minus σ times the square root of dt, each with probability one half. Because the drift depends only on the date and not on the current rate, an up move followed by a down move gives the same rate as the reverse. So the tree recombines. The rate at date n with j up moves is r0 plus the sum of the drift terms plus (2j − n) × σ times the square root of dt. The drifts are chosen date by date so the tree reprices the observed zero-coupon bonds. In continuous time, λ(t) is roughly the forward-curve slope plus σ² × t. Exam point: the drift shifts the whole tree, while volatility sets its spread.

Formulas

Ho-Lee rate at a node

Rate = r0 + Σλ × dt + (2j − n) × σ × √dt

r0
today's short rate
Σλ
the sum of the per-year drifts for the steps taken so far
dt
the step length in years
j
the number of up moves
n
the number of steps to the date
σ
the annual volatility

Holds because the drift depends only on time, so the tree recombines. The drift moves the centre; the volatility term sets the spread.

Ho-Lee drift that fits the market in continuous time

Drift = Slope + σ² × t

Slope
the slope of the forward-rate curve at time t, per year
σ
the annual volatility
t
the date in years

An approximation: the σ² × t term offsets convexity, so the drift exceeds the forward slope by more at later dates.

A practice question

From the question bank, with the reason each wrong option is wrong.

Why does the Ho-Lee short-rate tree recombine?

  1. Each move has probability one half, which guarantees recombination whatever the drift.Why not: Equal probabilities are not enough: a drift that depends on the rate breaks recombination even with 50/50 moves.
  2. The drift depends only on the date and not on the current rate, so an up move then a down move reaches the same rate as down then up. - correct
  3. Mean reversion pulls rising and falling paths back to the same node.Why not: Ho-Lee has no mean reversion; recombination comes from a rate-independent drift.
  4. The drift is set equal to the forward rate at each date, which forces paths to meet.Why not: Fitting the drift to forwards determines its level, not the path-independence property.

Up-then-down gives r + (drift1 + drift2) x dt; down-then-up gives the same, because the drift does not depend on the node. That is why only n + 1 nodes exist at date n.

A harder one

A Ho-Lee tree uses semi-annual steps (dt = 0.5), an initial rate of 3.00% and annual volatility of 1.00%. The drift is 0.40% per year for the first step and 0.60% per year for the second. The up-up rate at date 2 is closest to:

  1. 5.41%Why not: Adds the annual drifts without multiplying by dt.
  2. 4.21%Why not: Applies only one sigma x sqrt(dt) move instead of two.
  3. 4.91% - correct
  4. 4.50%Why not: Scales volatility by dt instead of the square root of dt.

Drift: (0.40 + 0.60) x 0.5 = 0.50%. Volatility per step: 1.00 x sqrt(0.5) = 0.707%, two up moves = 1.414%. Rate = 3.00 + 0.50 + 1.414 = 4.91%.

The hardest tier

A Ho-Lee tree has semi-annual steps and annual volatility of 1.00% (so sigma x sqrt(dt) = 0.707%). The 6-month rate is 3.00%, and the market 1-year spot rate is 3.30% (semi-annual compounding), so the 1-year zero price is 0.96780. The date-1 rates are r0 + lambda1 x dt plus or minus 0.707%, each with probability one half, and each date-1 rate is a 6-month rate that discounts one period. The drift lambda1, in basis points per year, that makes the tree reprice the 1-year zero is closest to:

  1. 60Why not: Reports the per-step shift in the rate as if it were the annual drift, forgetting to divide by dt.
  2. 121 - correct
  3. 30Why not: Uses the 1-year spot rate minus the 6-month rate and treats it as the drift, instead of the forward rate and annualising.
  4. -21Why not: Sets the up node equal to the forward rate rather than setting the expected price to match the market.

The 6m6m forward rate is 2 x (0.98522 / 0.96780 - 1) = 3.60%, so the expected date-1 rate is about 3.60%, a shift of about 0.60% in one step, i.e. lambda1 of about 1.2% per year. Solving the price equation exactly gives 120.6 bp per year; the small excess over 120 reflects convexity.

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