Take the 2-minute tour ×  A standard example when learning to
price American options is the perpetual American put. This is a
put that has no expiry (or you can consider T = infinity). The
standard solution prices this using the basic Black-Scholes
assumptions (including no restrictions on short-selling) and
ends up concluding the optimal exercise strategy is finding the
right lower barrier that effectively knocks out your put with a
rebate.  But my question is this: doesn't the idea of such a
perpetual option contradict the Black-Scholes assumptions?  In
order for this pricing exercise (and yes, I'm aware it's just an
exercise and not a real pricing problem) to be valid, you must
be able to delta-hedge, correct? And since this will involve
shorting the stock, this would seem to allow shorting the stock
for arbitrary lengths of time. But doesn't this allow arbitrage?
After all, someone (who doesn't care about the put) can just
short the stock indefinitely and never close it out.  Isn't this
an arbitrage? If n