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

A risk analyst compares a Ho-Lee style model, where dr = λ(t)dt + σ dw, with a Cox-Ingersoll-Ross (CIR) model, where dr = k(θ − r)dt + σ√r dw. Which statement correctly distinguishes how the two models treat the volatility of the short rate?

Ho-Lee has constant basis-point volatility, so rate changes are independent of the rate level. In CIR, volatility is σ√r, so it grows with the square root of the short rate and shrinks near zero, which helps keep rates non-negative.

  1. AIn both models the basis-point volatility of the short rate is constant and independent of the rate level
  2. BIn the Ho-Lee model basis-point volatility is constant, while in CIR it rises with the square root of the rate levelCorrect
  3. CIn the Ho-Lee model volatility is proportional to the rate level, while in CIR it is constant
  4. DIn CIR basis-point volatility falls as the short rate rises, which prevents negative rates

Explanation

Ho-Lee has a constant σ, so rate changes are normally distributed with the same basis-point volatility at any rate level. CIR has σ√r, so basis-point volatility increases with the rate and shrinks toward zero as r approaches zero. The option claiming volatility falls as rates rise reverses this relationship.

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