FRM Part II · FRM Exam Part II · The Vasicek and Gauss+ Models
A Vasicek model is calibrated under the risk-neutral measure with long-run short-rate level θ* = 6%, mean-reversion speed k = 0.3, and volatility σ = 1.2% per year. As maturity tends to infinity, to what level does the continuously compounded zero-coupon yield converge?
The long-maturity yield converges to about 5.92%. In Vasicek the asymptotic yield is the risk-neutral long-run rate minus a convexity term σ²/(2k²). That term is 0.000144/0.18 = 0.08%, so 6.00% − 0.08% = 5.92%.
- A5.92%Correct
- B6.00%
- C5.84%
- D5.98%
Explanation
The limiting yield is θ* − σ²/(2k²). σ² = 0.000144 and 2k² = 0.18, so the adjustment is 0.0008 = 0.08%, giving 5.92%. Ignoring convexity gives 6.00%. Using σ²/k² gives 5.84%. Using σ²/(2k) gives about 5.98%.
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