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

A risk manager calibrates a Vasicek model to a set of at-the-money caplet volatilities. Holding σ constant, she increases the mean reversion speed k. What is the most likely effect on the model-implied volatility of the price of a long-dated zero-coupon bond option?

Raising mean reversion speed k while holding σ fixed lowers the model-implied volatility of long-dated bond options, because stronger pull toward the long-run mean dampens the accumulation of shocks, reducing both long-horizon rate variance and the bond price sensitivity factor.

  1. AImplied volatility increases because rates drift more quickly
  2. BImplied volatility decreases because higher k dampens the cumulative effect of shocks over longer horizonsCorrect
  3. CImplied volatility is unchanged because it depends only on σ
  4. DImplied volatility becomes negative

Explanation

Higher k pulls rates back to θ faster, so long-horizon rate variance, σ²(1−e^{−2kt})/(2k), shrinks and long-maturity bond volatility factor (1−e^{−kτ})/k also falls. Thus long-dated option volatilities fall. The claim that only σ matters ignores k.

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