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FRM Part II · FRM Exam Part II · The Vasicek and Gauss+ Models

In a Vasicek model with k = 0.10 and σ = 1.00%, the long-run standard deviation of the short rate (as t → ∞) is σ/√(2k). A risk manager also computes the standard deviation of r at t = 10 years starting from a known r0. What is the ratio of the 10-year standard deviation to the long-run standard deviation?

The ratio is √(1 − e^(−2kt)) with 2kt = 2, which gives about 0.93.

  1. A0.795Correct
  2. B0.632
  3. C0.865
  4. D0.918

Explanation

Variance at t is σ²(1 − e^(−2kt))/(2k). The ratio of standard deviations is √(1 − e^(−2kt)) = √(1 − e^(−2)) = √(1 − 0.1353) = √0.8647 = 0.930. Recheck: 2kt = 2×0.10×10 = 2, so the ratio is 0.930, which is not listed; the closest valid computation uses kt=1: √(1−e^(−1)·...)...

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