Furthermore, I suspect it requires that each measurement is independent and identically-distributed.

If your measurements are correlated (as prices generally are -- if they were lower than average yesterday, they are more likely to be lower than average today), the secretary-problem approach will not work.

Yes, exactly. I was looking to solve the second problem, not the former. Following the secretary rule is good strategy for a lot of things, but I think we actually have more information here.