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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 specified as d(ln r) = a(t)[ln θ(t) − ln r]dt + σ(t)dw. Which feature distinguishes it from the simple lognormal model with time-dependent drift (Ho-Lee analog, d ln r = θ(t)dt + σdw)?

Black-Karasinski adds mean reversion to the logarithm of the short rate. This keeps rates positive while pulling them toward a target level. Unlike Vasicek, it lacks closed-form bond prices and is implemented numerically, and its basis-point volatility is not constant.

  1. AIt allows negative rates through the mean-reversion term
  2. BIt assumes constant basis-point volatility
  3. CIt incorporates mean reversion in the log of the short rate, so the rate stays positive and is pulled toward a targetCorrect
  4. DIt has closed-form bond price solutions like Vasicek

Explanation

Black-Karasinski applies Vasicek-style mean reversion to ln r, so r = exp(x) is always positive and reverts toward a time-varying target. Unlike Vasicek, it has no closed-form bond prices and requires numerical trees; it does not give constant basis-point volatility.

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