Why HJM Forward-Rate Drifts Differ Across Probability Measures
Summary
The discussion corrects a misunderstanding about the Heath–Jarrow–Morton framework. The familiar expression that determines the forward-rate drift from its volatility applies under the risk-neutral measure, where it is part of the no-arbitrage restriction. It is not a general rule that fixes the drift under every probability measure.
A change of measure can alter the finite-variation component of a stochastic process, including its drift, while leaving the diffusion coefficient unchanged under the stated setup. Therefore the risk-neutral and real-world drifts need not be equal, even when they share the same volatility specification. The answers also point out that drift behavior can differ under other measures, such as a forward measure. The note is conceptual and does not develop a specific calibration procedure or numerical example.
Key ideas
- The HJM drift restriction stated in the document applies under the risk-neutral measure.
- A measure change can alter a process's drift while preserving its diffusion term.
- The real-world drift is not determined by the risk-neutral HJM restriction alone.
- Drift expressions can also change under forward measures.
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# Heath–Jarrow–Morton under real-world measure # Heath–Jarrow–Morton under real-world measure In HJM model (framework), the drift of the forward is determined by its diffusion coefficient: $$ \mu(t,s) = \sigma(t,s)\int_t^s \sigma(t,v)^Tdv $$ My understanding, is that the change of measure under Grisanov theorem for continuous-time semi-martingales only affect the finite variation part (i.e. drift for HJM). Thus, if we start with an SDE under a risk-neutral-measure $Q$ $$ df(t,s) = \mu^Q(t,s)dt + \sigma(t,s)dW_t^Q $$ and the change to the real-world measure $P$ changes this to $$ df(t,s) = \mu^P(t,s)dt + \sigma(t,s)dW_t^P $$ does this then mean that $\mu^Q(t,s) = \mu^P(t,s)$ since they are both functions of $\sigma(t,s)$? ## Answer by user39119 (score 3) https://quant.stackexchange.com/a/58985 Your statement at the beginning of the question is not correct. That's why you have the "contradiction" later. It should say: In HJM model (framework), the drift of the forward under the risk-neutral measure Q is determined by its diffusion coefficient: $$ \mu^Q(t,s) = \sigma(t,s)\int_t^s \sigma(t,v)^Tdv. $$ That formula is not a general formula to obtain the drift under any probability measure, only applies to $Q$. Note that later under the forward measure $Q^T$ the drift is zero and the volatility is terms are not zero. ## Answer by Xiaohuolong (score 0) https://quant.stackexchange.com/a/54563 As you said, the change of measure only affects the finite variation part, which is the drift. It's not clear why this implies $\mu^Q(t,s)=\mu^P(t,s)$. These are the drifts under the two measures, so I don't think they have to be the same.
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