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FRM Part II · FRM Exam Part II · The Art of Term Structure Models: Drift

A risk manager is calibrating a CIR model with k = 0.25 and long-run mean θ = 4%. The risk premium term is λ√r, and the risk-neutral process used for pricing is dr = [kθ − (k + λ)r]dt + σ√r dw, with λ = 0.05. What are the risk-neutral long-run mean rate and the risk-neutral speed of mean reversion?

Rewriting the risk-neutral drift as (k+λ)[kθ/(k+λ) − r] gives a speed of 0.30 and a long-run mean of 0.25 × 4% / 0.30 = 3.33%. The risk premium raises the speed and lowers the mean.

  1. AMean 3.33%, speed 0.30Correct
  2. BMean 4.00%, speed 0.30
  3. CMean 3.33%, speed 0.25
  4. DMean 5.00%, speed 0.20

Explanation

The risk-neutral drift is kθ − (k+λ)r = (k+λ)[kθ/(k+λ) − r]. Speed = 0.25 + 0.05 = 0.30. Mean = 0.25 × 4% / 0.30 = 1%/0.30 = 3.33%. Keeping 4% ignores that the risk premium lowers the mean, and keeping 0.25 ignores the risk-premium effect on the speed.

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