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

A risk manager compares two Vasicek calibrations that share the same θ and σ. Calibration A has k = 0.10 and Calibration B has k = 0.80. Which conclusion is correct about the short rate's behavior?

Calibration B has the faster mean reversion and therefore the lower long-horizon variance. Stationary variance equals σ² divided by 2k, so raising k from 0.10 to 0.80 shrinks it from 5σ² to 0.625σ².

  1. AB's short rate reverts faster, so its long-horizon variance σ²/(2k) is lower than A'sCorrect
  2. BA's short rate reverts faster, so its long-horizon variance is lower than B's
  3. CBoth have the same long-horizon variance because σ and θ are equal
  4. DB's long-horizon variance is higher because faster reversion increases the drift

Explanation

Stationary variance is σ²/(2k), which falls as k rises. A: σ²/0.2 = 5σ²; B: σ²/1.6 = 0.625σ². Higher k means faster reversion and a tighter long-run distribution, so B is lower.

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