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

A desk calibrates the Hull-White model dr = k[θ(t) − r]dt + σ(t)dw. Compared with the Ho-Lee model with the same time-dependent drift, which statement about the rate distribution is correct?

Hull-White's mean reversion keeps long-horizon rate variance bounded, while Ho-Lee's variance keeps growing linearly with the horizon. Both models produce normally distributed rates, so neither rules out negative rates, and only Hull-White has the mean-reverting parameter k.

  1. AHull-White has mean reversion, so long-horizon rate variance is bounded, whereas Ho-Lee variance grows without boundCorrect
  2. BHull-White rates are lognormal, so they cannot be negative, whereas Ho-Lee rates are normal
  3. CHo-Lee has mean reversion, so its long-horizon variance is bounded
  4. DBoth models have rate variance that rises linearly with horizon when σ is constant

Explanation

With k > 0 the Hull-White model pulls rates toward θ(t), so variance converges to a finite level, about σ²/(2k) for constant σ. Ho-Lee has no mean reversion and variance σ²T grows without limit. Both models are normal, so negative rates are possible in each.

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