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

A quant has a Ho-Lee model with constant σ = 1.20%. Over a two-period tree, the drifts are λ1 and λ2. She wants to know how the recombining property affects the tree. Which result is correct about the rates at date 2 (dt = 1 year) given the rate r0?

The tree recombines into three nodes at date 2, centered on r0 plus both drifts, with outer nodes 2σ above and below. Because the drift depends only on time, an up-then-down path equals a down-then-up path, so the shocks cancel but the drifts accumulate.

  1. AThree nodes exist: r0 + λ1 + λ2 + 2σ, r0 + λ1 + λ2, and r0 + λ1 + λ2 − 2σCorrect
  2. BFour nodes exist because the drift differs between periods, preventing recombination
  3. CThree nodes exist, and the middle node equals r0 because up and down moves cancel the drifts
  4. DThree nodes exist: r0 + λ1 + λ2 + σ, r0 + λ1 + λ2, and r0 + λ1 + λ2 − σ

Explanation

Drifts depend on time only, not on the rate, so up-down equals down-up and the tree recombines. Two up shocks add 2σ, one each cancel, and two downs subtract 2σ, all on top of the cumulative drift λ1 + λ2. Drifts do not cancel.

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