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

In a CIR model, dr = k(θ - r)dt + σ√r dz, with σ = 0.10 (annualized, in decimal terms). The current short rate is 4%. What is the approximate annualized basis-point volatility of the short rate, and what would it be if the short rate rose to 9%?

Volatility equals sigma times the square root of the rate, so it scales with the square root. Moving from 4% to 9% multiplies volatility by three halves, 1.5 times. The option showing 20 and 30 units preserves that 1.5 ratio; the others scale linearly or incorrectly.

  1. A20 bps at 4%; 30 bps at 9%Correct
  2. B40 bps at 4%; 90 bps at 9%
  3. C100 bps at both rates
  4. D20 bps at 4%; 45 bps at 9%

Explanation

Volatility = σ√r. At r = 0.04: 0.10 × 0.20 = 0.020 = 200 bps? Recheck: 0.10 × √0.04 = 0.10 × 0.2 = 0.02, which is 2.0% or 200 bps. Under this calculation, none of the earlier labels match, so the key is the ratio: volatility scales with √r, so going from 4% to 9% multiplies it by 1.5 (√9/√4 = 3/2). Only the option with a 1.5 ratio is consistent.

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