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

A short-rate model is dr = λ(t)dt + σe^(-αt)dw with σ = 1.00% per year and α = 0.50. The rate shocks accumulate over time, and the drift is deterministic. What is the standard deviation of the short rate at T = 2 years, from the changes in r over that period? (e^(-2) ≈ 0.1353)

With exponentially decaying volatility, the variance at T is σ²(1 - e^(-2αT))/(2α) = 0.0001 × 0.8647. Its square root times is about 0.0093, so the standard deviation is roughly 93 basis points, below the 141 bp that constant volatility would give.

  1. A52.0 bp
  2. B86.5 bp
  3. C93.0 bpCorrect
  4. D141.4 bp

Explanation

Variance = σ²(1 - e^(-2αT))/(2α) = 0.0001 × (1 - 0.1353)/1 = 0.0001 × 0.8647. The standard deviation is 0.01 × √0.8647 ≈ 0.01 × 0.9299 = 93.0 bp. The constant-volatility answer σ√T gives 141.4 bp. Using the e^(-αT) scaling gives 52.0 bp, and forgetting the square root gives 86.5 bp.

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