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

A desk calibrates a model with time-dependent volatility σ(t) = σ0·e^(−λt) to the term structure of volatility, as in the Ho-Lee model with decaying volatility. Relative to a constant-volatility Ho-Lee model with the same initial σ0, what is the effect on long-maturity rate uncertainty?

Decaying volatility makes later shocks smaller, so cumulative variance grows more slowly than the linear growth under constant volatility. Long-maturity rate uncertainty is therefore lower than in a constant-volatility Ho-Lee model with the same initial volatility.

  1. ALong-maturity rate uncertainty is greater because volatility compounds over time
  2. BLong-maturity rate uncertainty is lower because shocks to the short rate become progressively smaller, so long-horizon variance grows more slowlyCorrect
  3. CThere is no difference because the drift offsets the volatility decay
  4. DShort-rate variance becomes constant at long horizons only for the lognormal case

Explanation

With decaying σ(t), later shocks are smaller, so cumulative variance ∫σ(s)²ds grows more slowly than σ0²t and converges to a limit. Constant volatility yields linearly growing variance, so uncertainty is higher for constant volatility.

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