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

In a Vasicek model, the short rate follows dr = k(θ − r)dt + σdw with k = 0.10, θ = 6.00% and a current short rate r0 = 4.00%. Using the expected (risk-neutral) path of the short rate, what is the expected short rate in 10 years?

The expected short rate in 10 years is 5.26%. The rate reverts toward 6% and the gap of 2% decays by e^(−0.1×10) = 0.368, leaving a gap of 0.74%. So the expected rate is 6.00% − 0.74% = 5.26%.

  1. A5.26%Correct
  2. B4.74%
  3. C6.00%
  4. D4.19%

Explanation

E[r(T)] = θ + (r0 − θ)e^(−kT) = 6% + (−2%)(e^−1) = 6% − 0.736% = 5.26%. The 4.74% option reverses the roles of r0 and θ, so it starts at 4% and adds the decayed gap. The 4.19% option uses T = 1 instead of T = 10.

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