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IAI Actuarial Core Principles · Economic Modelling · Models of the term structure of interest rates

A Cox-Ingersoll-Ross model is calibrated with a = 0.4, μ = 6% and σ = 0.10 (volatility term σ√r). A colleague wants to check whether negative rates are excluded. Which condition is the standard one that guarantees the short rate stays strictly positive?

The Feller condition 2aμ ≥ σ² ensures CIR rates remain strictly positive. With a = 0.4, μ = 0.06 and σ = 0.10, 2aμ = 0.048 exceeds σ² = 0.01, so the condition is satisfied and the short rate cannot reach zero.

  1. A2aμ ≥ σ², here 0.048 ≥ 0.01, so positivity holdsCorrect
  2. Baμ ≥ 2σ², here 0.024 ≥ 0.02, so positivity holds
  3. C2aμ ≤ σ², here 0.048 ≤ 0.01, so positivity holds
  4. Dσ ≥ a, here 0.10 ≥ 0.4, so positivity fails
  5. μ ≥ 2σ, here 0.06 ≥ 0.20, so positivity fails

Explanation

The Feller condition for CIR is 2aμ ≥ σ². Here 2(0.4)(0.06) = 0.048 and σ² = 0.01, so 0.048 ≥ 0.01 holds and the rate stays strictly positive. The other conditions are not the Feller condition.

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