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For example in the Black_scholes case the delta N(d1) does appear to be equal to the expectation (under the stock measure) of the delta at expiration, which is the expectation of I(S(T)>K).

Is there a fundamental reason to believe that the delta will always be a martingale under the stock numeraire?

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for a large class of models that is ones where $\log S_T - \log S_0$ has distribution independent of level, it is possible to show that the delta is $$ \mathbb{P}_S(S_T>K) $$ and this is a martingale in the stock measure. (For a proof see More Mathematical Finance by me)

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