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FRM Part II · FRM Exam Part II · The Art of Term Structure Models: Volatility and Distribution

In a Hull-White model dr = (θ(t) − a r)dt + σ dw with a = 0.10 and σ = 1.20%, the volatility of the 10-year spot rate relative to the short-rate volatility is best described as:

Ten-year spot rate volatility is lower than short-rate volatility. Mean reversion makes long rates load on the short rate by (1−e^(−aT))/(aT), about 0.63 here, giving roughly 0.76% versus 1.20%. Higher duration affects bond price volatility, not the volatility of the rate itself.

  1. AEqual to σ, because spot rates share the short-rate volatility
  2. BLower than σ, because the sensitivity of long rates to the short rate declines with maturity through mean reversion, approximately (1−e^(−aT))/(aT)Correct
  3. CHigher than σ, because long-maturity bonds have higher duration
  4. DZero, because the drift θ(t) hedges the volatility

Explanation

Under mean reversion, the spot rate's loading on the short-rate shock is B(T)/T = (1−e^(−aT))/(aT). For a=0.10, T=10 this is (1−0.3679)/1 = 0.632, so the 10-year rate volatility is about 0.76%, below 1.20%. Duration raises price volatility, not rate volatility.

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