There are many "interpretations" of what no-arbitrage means in mathematical finance, the most well known is no free lunch with vanishing risk: If $S=\left(S_{t}\right)_{t=0}^{T}$ is a semimartingale with values in $\mathbb{R}^{d}$ then $S$ does not allow for a free lunch with vanishing risk if and only if there exists an equivalent martingale measure $\mathbb{Q}$ such that $S$ is a sigma-martingale under $\mathbb{Q}$.

But regardless of the interpretation, it seems that all these notions agree that we should have no arbitrage everywhere on the timeline we are considering. But if arbitrage can happen on a subset of the real line that has Lebesgue measure $0$, is it really arbitrage ? Because no human/machine could be able to exploit it, i.e if a tree falls in a forest and no one is around to hear it, does it make a sound ?

Are you aware of an interpretation of no-arbitrage on the real line that would allow arbitrage on sets of Lebesgue measure 0 ? Does it even make sense to you ? Is there literature on this ?

Here is an example: If $S_t$ is a semimartingale that is a (sigma-)martingale on $[0,T/2)$ and on $(T/2,T]$ but not in $T/2$, is there a notion of no-arbitrage that would accept $S_t$ as a no arbitrage semimartingale ?

  • $\begingroup$ very interesting perspective. Im inclined to practically say "no". Consider a scenario where this 'might' exist, at the precise moment a contract settles, say, based on a exchange delivery settlement price being calculated at the moment the exchange closes, and simultaneously the contract might not have the equivalent value. The arbitrage is practically impossible since it cannot be traded, and my conclusion is it does not exist. I have never seen in literature, that was just an interested response! $\endgroup$
    – Attack68
    Apr 14 '21 at 20:40

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