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

A risk manager compares two Vasicek calibrations with the same θ and σ, but k rises from 0.1 to 0.5. What is the most likely effect on the long-horizon short-rate distribution and on long-maturity rate volatility?

A higher k lowers the long-run variance of the short rate, since variance is σ²/(2k), and reduces long-maturity rate volatility because shocks die out faster. Strong mean reversion keeps rates closer to θ and dampens the long end.

  1. ALower long-run variance of the short rate and lower volatility of long-maturity ratesCorrect
  2. BHigher long-run variance of the short rate and higher volatility of long-maturity rates
  3. CNo effect on long-run variance, since only θ matters
  4. DLower long-run variance but higher volatility of long-maturity rates

Explanation

Stationary variance σ²/(2k) falls as k rises. Stronger reversion also dampens the effect of shocks on longer-term rates, since shocks decay faster. So long-maturity rate volatility declines.

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