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

In the Vasicek model dr = k(θ − r)dt + σdw, a risk analyst notes that the short rate is currently well above θ. Which statement best describes the expected behavior of the rate over the next small time step, ignoring the random shock?

The rate is expected to drift downward toward θ, with the speed proportional to the gap. In Vasicek the drift is k(θ − r), which is negative when r is above θ, so mean reversion pulls the rate back toward its long-run level.

  1. AThe rate is expected to drift downward toward θ at a speed proportional to the gapCorrect
  2. BThe rate is expected to drift upward because volatility is constant
  3. CThe rate is expected to stay unchanged because the drift is zero on average
  4. DThe rate is expected to fall by a fixed amount equal to k each period

Explanation

The drift term is k(θ − r). When r exceeds θ, this is negative and its size is proportional to the gap (r − θ). So the expected change is a pull down toward θ. A fixed fall of k ignores proportionality to the gap.

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