FRM Part II · FRM Exam Part II · The Vasicek and Gauss+ Models
A risk analyst calibrates a Vasicek model with risk-neutral parameters k = 0.20, θ = 5.00% and σ = 1.20%. As maturity goes to infinity, toward what continuously compounded spot rate does the model's term structure converge?
The long-maturity spot rate converges to 4.82%. In the Vasicek model it equals θ minus σ²/(2k²), which is 5.00% minus 0.18%. The convexity effect from rate volatility pulls the long end below the risk-neutral long-run mean of the short rate.
- A4.82%Correct
- B5.00%
- C4.64%
- D5.18%
Explanation
The long-maturity yield is θ − σ²/(2k²). Here σ² = 0.000144 and 2k² = 0.08, so the convexity term is 0.0018 = 0.18%. The limit is 5.00% − 0.18% = 4.82%. The 4.64% option divides by 2k instead of 2k², and 5.00% ignores convexity.
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