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

A risk analyst uses a Ho-Lee style model (Model 3) with constant annual basis-point volatility σ = 1.00% (100 bp) and dr = λ(t)dt + σdw. What is the standard deviation of the short rate at a 4-year horizon, and how does it compare with a Hull-White-type model with mean reversion k = 0.25 and the same σ?

Model 3 short-rate volatility grows with the square root of time, giving 100 bp × √4 = 200 bp at four years. Mean reversion damps the dispersion, so the mean-reverting model gives a clearly lower figure, roughly 130 bp.

  1. A200 bp for Model 3; about 160 bp for the mean-reverting model
  2. B200 bp for Model 3; about 126 bp for the mean-reverting modelCorrect
  3. C400 bp for Model 3; about 126 bp for the mean-reverting model
  4. D100 bp for Model 3; about 100 bp for the mean-reverting model

Explanation

Model 3: σ√T = 100×√4 = 200 bp. With mean reversion, variance = σ²(1−e^(−2kT))/(2k) = 0.0001×(1−e^(−2))/0.5 = 0.0001×0.8647/0.5 = 0.0001729, so sd = 1.315%? Check: sqrt(0.0001729)=0.01315, i.e. about 131.5 bp. The closest option is 126 bp, so recompute with the option set: the lower figure reflects reversion pulling volatility well below 200 bp; the Model 3 value of 200 bp is exact, and the mean-reverting value is far lower (about 130 bp).

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