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

The Black-Karasinski model is commonly written d(ln r) = a(t)[ln θ(t) − ln r]dt + σ(t)dw. Which description is correct?

In Black-Karasinski the natural log of the short rate follows a mean-reverting normal process with time-dependent parameters. That makes the short rate lognormally distributed and always positive, and the mean reversion operates on the log rate toward a target.

  1. AThe rate r is normally distributed and can be negative, with mean reversion in r itself
  2. BThe log of the short rate is normally distributed and mean-reverts toward a time-dependent target, so the rate is always positiveCorrect
  3. CVolatility is a constant basis-point amount and the drift is zero
  4. DThe model is a lognormal model without mean reversion, like Ho-Lee with proportional volatility

Explanation

Black-Karasinski applies a Vasicek/Hull-White style mean-reverting process to ln r, with time-dependent parameters. Hence r is lognormal and positive. Option A describes a normal model; the others ignore mean reversion in the log rate.

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