  Suppose you have one unit of stock, currently worth 1, which you must sell
before time $T$. The Optional Sampling Theorem tells us that whatever stopping
time we choose to sell, the expected discounted value we get when we sell will
be 1. Suppose however that we are able to see $a$ units of time into the
future, and base our stopping rule on that; we should be able to do better than
expected value 1. But how much better can we do? And how would we exploit the
additional information? The optimal solution to this problem will never be
found, but in this paper we establish remarkably close bounds on the value of
the problem, and we derive a fairly simple exercise rule that manages to
extract most of the value of foresight.
