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

A desk calibrates a Ho-Lee model at σ = 0.90% and prices a 2-year option on a zero-coupon bond. Management wants to compare Ho-Lee with a Vasicek model with the same σ. Which conclusion is correct about the Ho-Lee model's implications for long-horizon rate volatility and its risk management use?

Ho-Lee has no mean reversion, so the variance of the short rate equals σ² times T and grows without limit. A mean-reverting model like Vasicek caps long-horizon variance, so Ho-Lee can overstate long-horizon uncertainty. Its rates are normal and can be negative.

  1. ARate variance keeps growing without bound as the horizon lengthens, so long-horizon rate uncertainty may be overstated relative to a mean-reverting modelCorrect
  2. BRate variance converges to a finite limit because λ(t) pulls the rate back toward the forward curve
  3. CRate volatility falls with horizon because the drift offsets the shocks
  4. DHo-Lee rates are lognormal, so the model avoids negative rates at long horizons

Explanation

With no mean reversion, Var[r(T)] = σ²T grows without bound, so long-horizon dispersion exceeds that of Vasicek, whose variance converges to σ²/(2a). λ(t) is deterministic and does not affect dispersion. Ho-Lee rates are normal and can be negative.

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