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FRM Part II · FRM Exam Part II · Arbitrage Pricing with Term Structure Models

A trader uses a Vasicek model with mean-reversion speed k = 0.25 and volatility σ = 1.0% per year. Compared with a Ho-Lee model using the same σ, how does the standard deviation of the short rate at a long horizon behave?

Under Vasicek, mean reversion caps the long-horizon standard deviation at σ/√(2k), about 1.41% here. Ho-Lee has no mean reversion, so its standard deviation grows as σ√T and becomes unbounded as the horizon lengthens.

  1. AVasicek volatility converges to a finite level σ/√(2k) ≈ 1.41%, while Ho-Lee volatility grows as σ√T without boundCorrect
  2. BBoth converge to σ/√(2k)
  3. CVasicek volatility grows as σ√T while Ho-Lee volatility converges to a constant
  4. DBoth grow as σ√T because volatility is constant in both

Explanation

Vasicek's long-horizon variance tends to σ²/(2k) = 0.0001/0.5 = 0.0002, so the standard deviation is about 1.41%. Ho-Lee has no mean reversion so variance is σ²T, growing without bound.

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