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

A desk uses the Vasicek-type model dr = k(θ − r)dt + σ dw with k = 0.40, θ = 6%, σ = 1.0% per year, and current r = 4%. Under the model, what are the expected short rate and the standard deviation of the short rate change over a small horizon dt = 0.25 year?

The expected change is 0.40 times the 2% gap times 0.25, which is 0.20%. The standard deviation is 1.0% times the square root of 0.25, which is 0.50%. The other options omit dt in the drift or scale volatility linearly with time.

  1. AMean change +0.20%; standard deviation 0.50%Correct
  2. BMean change +0.80%; standard deviation 0.25%
  3. CMean change +0.20%; standard deviation 0.25%
  4. DMean change +0.80%; standard deviation 0.50%

Explanation

Expected change = k(θ − r)dt = 0.40 × 2% × 0.25 = 0.20%. Standard deviation = σ√dt = 1.0% × 0.5 = 0.50%. Using 0.25 as the volatility multiplier is the linear-time error, and omitting dt gives 0.80%.

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