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

A risk analyst compares a normal (Ho-Lee type) short-rate model with a lognormal (Black-Karasinski type) short-rate model, both calibrated to a current short rate of 4%. Which statement best describes a key difference in the distribution of future short rates?

The normal model has constant basis-point volatility and can produce negative rates, while the lognormal model keeps rates positive and makes basis-point volatility proportional to the rate level, producing a right-skewed distribution of future rates.

  1. AThe normal model allows negative rates with constant absolute volatility, whereas the lognormal model keeps rates positive with volatility proportional to the rate levelCorrect
  2. BThe normal model keeps rates strictly positive, whereas the lognormal model permits negative rates
  3. CBoth models imply that basis-point volatility rises in proportion to the rate level
  4. DBoth models imply that the distribution of future rates is symmetric around the mean

Explanation

In a normal model dr = λdt + σdw, rate changes have constant basis-point volatility and rates can turn negative. In a lognormal model, d(ln r) is normal, so rates stay positive and basis-point volatility scales with r. The option reversing this is wrong, and the lognormal rate distribution is skewed, not symmetric.

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