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

A CIR model has k = 0.30, θ = 5%, and σ = 0.20 (decimal rates). A risk manager checks the Feller condition, 2kθ ≥ σ². What is the result and its meaning?

Twice k times theta is 2 × 0.30 × 0.05 = 0.030, while sigma squared is 0.04. Because 0.030 is below 0.04, the Feller condition fails, so the short rate can reach zero. It is reflected there and does not become negative.

  1. A2kθ = 0.030 and σ² = 0.040, so the condition fails and the short rate can reach zeroCorrect
  2. B2kθ = 0.030 and σ² = 0.040, so the condition holds and the rate never reaches zero
  3. C2kθ = 0.060 and σ² = 0.040, so the condition holds and the rate never reaches zero
  4. D2kθ = 0.015 and σ² = 0.200, so the condition fails and rates can turn negative

Explanation

2kθ = 2 × 0.30 × 0.05 = 0.030. σ² = 0.04. Since 0.030 < 0.040, the Feller condition fails, so the rate can touch zero (it is reflected there and stays non-negative in the CIR model). A common error is computing 2kθ without the factor 2 or mistaking failure for negative rates; CIR never produces negative rates.

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