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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 Ho-Lee style model, where short-rate volatility is a constant sigma, with the Cox-Ingersoll-Ross (CIR) model, where the dynamics are dr = k(theta - r)dt + sigma*sqrt(r)dW. Which statement best describes the key volatility feature of the CIR model?

In the CIR model the instantaneous basis-point volatility equals sigma times the square root of the short rate, so it rises with the rate level and shrinks as rates approach zero. This differs from constant-volatility models and from lognormal models, where volatility is proportional to the rate.

  1. ABasis-point volatility is independent of the rate level
  2. BBasis-point volatility increases with the square root of the rate level, so volatility falls as rates approach zeroCorrect
  3. CBasis-point volatility decreases as the rate level rises
  4. DBasis-point volatility is proportional to the rate level, so percentage volatility is constant

Explanation

In CIR the diffusion term is sigma*sqrt(r), so the standard deviation of rate changes scales with sqrt(r). As r falls toward zero, volatility shrinks. Option D describes the lognormal (Black-Karasinski/Dothan style) case, where volatility is proportional to r.

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