It is often explained, that the rule of thumb for exercising American options is to check when the benefit from the interest rate (sell the stock earlier, get the cash, put in the bank) is higher than the time value of the option. This is all clear in the positive interest rate environment, but the question is then - would we exercise some put options in case $r = 0$, and why would we do that? I thought, that there's no reason for such exercise, however according to this paper it is even sometimes optimal to early exercise American puts when $r<0$. What am I missing?
The classic result is never early exercise an American call if $r \geq 0, d \leq 0.$ If we think in terms of FX, calls and puts are really the same thing and by switching currency, we get never early exercise an American put if $ r \leq 0, d \geq 0.$ If one of these is violated it may be worth early exercising.
- Let $r=0$:
The maximum payoff ever from the put is when $S=0$ so $P= K$. So one would always exercise at this maximum because you cant get any better in the future and dont forego any interest. Based on @MarkJoshi 's comment, we have to assume that $S=0$ never recovers, so if $r=0$ you are essentially indifferent between exercising then or later because you gain/loose no interest and the payoff will never decrease. So it holds $P_t=p_t$.
If $S>0$ you can theoretically always have an infinitesimal increase in payoff when $S\to 0$ and wait, however I would expect that there is an exercise boundary at which point you would exercise because the probability of ending OTM vs. being close to maximum payoff already is too high and hence the AM put dominates at this point and $P_t\geq p_t$.
- Let $r<0$:
If you exercise at the maximum payoff, your received cash will vanish over time through the negative interest. I expect it is then a question of the expected opportunity cost from not exercising at the maximum vs. the negative interest until maturity. If the volatility of the asset is very high, you have a high probability of moving away from the maximum in the future and would probably rather accept the negative interest. If the volatility is very low or even close to zero, you will likely not move away from the maximum much and rather exercise later to avoid negative interest. Hence the AM put fully dominates and $P_t>p_t$.