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 not, what am I missing? Is there some adjustment one needs to make to the idea of arbitrage in this infinite time case?
Note my question is about the standard Black-Scholes theory. Someone asked about margin, but I don't think that's part of the standard theory.