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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 short-rate model in a low-rate environment. Which statement correctly describes a key property of the lognormal model?

The lognormal model keeps the short rate strictly positive, and its basis-point volatility is proportional to the rate level (σ times r). Constant basis-point volatility and possible negative rates are features of the normal model, not the lognormal model.

  1. AThe short rate can become negative with positive probability, and its basis-point volatility is constant
  2. BThe short rate is always positive, and its basis-point volatility rises in proportion to the rate levelCorrect
  3. CThe short rate is always positive, and its basis-point volatility is independent of the rate level
  4. DThe short rate can become negative, and its basis-point volatility falls as the rate rises

Explanation

In a lognormal model dr = a·r dt + σ·r dw, so the rate cannot go below zero, and the absolute (basis-point) volatility equals σ·r, which scales with the level of the rate. Constant basis-point volatility with possible negative rates describes the normal model.

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