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

A Vasicek model has k = 0.40, θ = 5.00% and σ = 1.20%. Which of the following describes the short rate's distribution over a long horizon and the implication for modeling?

Over long horizons the short rate is normal with mean θ = 5% and variance σ²/(2k), which is 0.00018 here. Because the distribution is normal, negative rates remain possible, and variance stays bounded due to mean reversion.

  1. ANormally distributed with long-run mean 5.00% and variance σ²/(2k) = 0.00018, so negative rates are possibleCorrect
  2. BLognormally distributed with mean 5.00%, so rates cannot be negative
  3. CNormally distributed with variance that grows without bound as the horizon lengthens
  4. DNormally distributed with mean 5.00% and variance σ²/k, so rates cannot be negative

Explanation

The Vasicek rate is Gaussian with a stationary variance of σ²/(2k) = 0.000144/0.8 = 0.00018 (standard deviation about 1.34%). Because it is normal, negative rates have positive probability. Variance does not grow without bound because mean reversion bounds it; σ²/k omits the factor 2.

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