So I've kind of formulated a possible way to use markets to predict quantiles. It seems quite flawed looking back on it two and a half weeks later, but I still think it might be an interesting line of inquiry.
You want options (as in, the financial market instruments called "options").
A sufficiently deep and wide options market basically provides most of the market-expected distribution of the future value of the underlying.
If it's worth saying, but not worth its own post (even in Discussion), then it goes here.
Duration set to six days to encourage Monday as first day.