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IAI Actuarial Core Principles · Economic Modelling · Models of the term structure of interest rates

In the Heath-Jarrow-Morton (HJM) framework, the forward rate f(t,T) follows df(t,T) = α(t,T)dt + σ(t,T)dW(t) under one factor. To avoid arbitrage under the risk-neutral measure, what must the drift α(t,T) equal?

Under the risk-neutral measure, the HJM drift must equal the forward rate volatility multiplied by the integral of that volatility over maturities from t to T. Drift is thus fully determined by volatility, which removes arbitrage; it is not zero and not simply sigma squared.

  1. Aσ(t,T) × ∫ from t to T of σ(t,s) dsCorrect
  2. Bσ(t,T)²
  3. CZero for all maturities
  4. DThe short rate r(t) minus f(t,T)
  5. −σ(t,T) × ∫ from t to T of σ(t,s) ds

Explanation

The HJM no-arbitrage condition ties the drift to the volatility: α(t,T) = σ(t,T)∫_t^T σ(t,s)ds. Option B omits the integral over maturities. Option C would hold only if volatility were zero, and option E has the wrong sign.

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