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

In a Vasicek model with k = 0.50 and σ = 1.00% (annual), what is the standard deviation of the short rate at a horizon of 2 years, conditional on today's rate?

The standard deviation is about 0.93%. Vasicek's conditional variance is σ²(1 − e^(−2kT))/(2k), which equals 0.0001 × 0.8647 for these inputs. Mean reversion caps the variance below the σ²T of a random walk and keeps it below the 1.00% long-run level.

  1. A0.93%Correct
  2. B1.41%
  3. C1.00%
  4. D0.80%

Explanation

Variance = σ²(1 − e^(−2kT))/(2k) = 0.0001 × (1 − e^(−2))/1 = 0.0001 × 0.864665 = 8.64665e-5, so the standard deviation is 0.930%. The figure 1.41% is σ√T, which ignores mean reversion. The figure 1.00% is the long-run standard deviation σ/√(2k), which is only reached as T tends to infinity. The figure 0.80% wrongly uses e^(−kT) in place of e^(−2kT).

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