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

In Model 1 (the normal model with constant volatility) of Tuckman's term structure treatment, the short-term rate follows dr = λ dt + σ dw. Which statement about the distribution of the rate change over a horizon is correct?

Under Model 1 the rate change is normally distributed with variance σ²T, so it is unbounded below and negative rates have positive probability. Lognormal or level-dependent volatility belongs to other models, and variance grows rather than shrinks with the horizon.

  1. AThe change in the rate is normally distributed, so negative rates have positive probabilityCorrect
  2. BThe change in the rate is lognormally distributed, so rates cannot become negative
  3. CThe change in the rate is normal with variance that falls as the horizon lengthens
  4. DThe change in the rate is normal with a variance proportional to the current rate level

Explanation

With constant σ and drift λ, r over horizon T is normal with mean r0+λT and variance σ²T. Because a normal variable has unbounded support, negative rates are possible. Lognormality and level-dependent variance belong to other models, and variance grows with T, not falls.

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