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

A desk calibrates a Vasicek model dr = k(θ − r)dt + σ dW with k = 0.25, θ = 4%, and current short rate r0 = 2%. Using the expected-rate formula E[r(t)] = θ + (r0 − θ)e^(−kt), what is the expected short rate in 4 years? (e^(−1) = 0.3679)

The expected short rate is about 3.26%. With k×t equal to 1, the gap of minus 2% from the long-run mean shrinks by a factor of 0.3679, leaving minus 0.74%, so the expected rate is 4% minus 0.74%, which equals 3.26%.

  1. A2.74%
  2. B3.26%Correct
  3. C3.63%
  4. D3.37%

Explanation

kt = 0.25×4 = 1, so e^(−1)=0.3679. E[r] = 4% + (2% − 4%)(0.3679) = 4% − 0.7358% = 3.264%, about 3.26%. Option A (2.74%) adds the deviation to r0 wrongly or flips the sign, and option C ignores much of the reversion.

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