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

A risk analyst compares two one-factor short-rate models calibrated to the same curve. Model A assumes the change in the short rate over dt is dr = λ(t)dt + σ dw with constant σ. Model B assumes dr = λ(t)dt + σ r dw. Which statement about the distribution of the short rate is correct?

Constant basis-point volatility gives normally distributed rates that can turn negative, whereas volatility proportional to the rate level gives lognormally distributed, strictly positive rates. Hence the first statement is correct, and the other options reverse the properties of the normal and lognormal models.

  1. AModel A permits negative rates with normally distributed rates, while Model B with constant proportional volatility keeps rates positive with a lognormal distributionCorrect
  2. BModel A produces lognormally distributed rates and cannot generate negative rates
  3. CModel B produces normally distributed rates and rate volatility independent of the rate level
  4. DBoth models imply that the basis-point volatility of rates is independent of the rate level

Explanation

With constant σ (Model A, Ho-Lee type) the rate change is normal, so rates are normally distributed and can be negative. With σr (Model B, lognormal) the rate cannot cross zero and basis-point volatility rises with the rate level. The other options reverse these properties.

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