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

In a Vasicek model, the instantaneous short-rate volatility is σ = 1.2% per year and the mean reversion speed is k = 0.40. What is the long-horizon (stationary) standard deviation of the short rate, σ/√(2k)? (√0.8 = 0.8944)

The long-run standard deviation is about 1.34%. In Vasicek the stationary variance is sigma squared divided by twice the reversion speed, so 1.2% divided by the square root of 0.8 gives roughly 1.34%. Mean reversion therefore caps the dispersion of the short rate.

  1. A1.34%Correct
  2. B3.00%
  3. C1.50%
  4. D1.07%

Explanation

Stationary variance is σ²/(2k), so the standard deviation is σ/√(2k) = 1.2%/√0.8 = 1.2%/0.8944 = 1.342%. Option D multiplies by 0.8944 instead of dividing. Option B uses σ/k = 3.0%, omitting the square root of 2k.

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