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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 Cox-Ingersoll-Ross (CIR) model. In the CIR model the instantaneous rate volatility is proportional to the square root of the short rate. Which statement about the CIR model is correct?

In the CIR model volatility is proportional to the square root of the short rate, so it rises with the rate and shrinks near zero. Rates therefore stay non-negative, and under the Feller condition they never hit zero. Constant volatility is a feature of normal models, not CIR.

  1. ARate volatility rises when the short rate rises, and the model avoids negative rates when its parameters satisfy the Feller conditionCorrect
  2. BRate volatility is constant across all levels of the short rate, and negative rates are always possible
  3. CRate volatility falls as the short rate rises, so rates stay mean-reverting only below the long-run level
  4. DRate changes are lognormally distributed, so volatility in basis points is proportional to the rate itself

Explanation

In CIR, dr = k(θ - r)dt + σ√r dz, so basis-point volatility increases with r and shrinks toward zero as r approaches zero. With 2kθ ≥ σ² (Feller condition) the rate cannot reach zero, so it stays non-negative. Constant volatility describes the normal models, and proportional-to-r volatility describes lognormal models, not CIR.

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