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

In Model 1 (Ho-Lee-type normal model with constant volatility), the short rate follows dr = λ(t)dt + σdw. Which statement about the distribution of the short rate at a future date is correct?

Model 1 produces normally distributed short rates because shocks are additive with constant volatility. Since the normal distribution covers all real values, negative rates have positive probability. It has no proportional volatility and no mean reversion, which belong to other models.

  1. ARates are normally distributed, so negative rates have positive probabilityCorrect
  2. BRates are lognormally distributed, so rates can never be negative
  3. CRates are normally distributed, and volatility rises proportionally with the rate level
  4. DRates follow a mean-reverting process with a stationary long-run distribution

Explanation

With constant σ and additive normal shocks, r at any future date is normally distributed around its drift-implied mean. A normal distribution has support over all real numbers, so negative rates can occur. Lognormal behaviour belongs to proportional-volatility models, and mean reversion is not in Model 1.

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