# Early exercise of American options

I know this question is considered basic and has been asked millions of times, but I have done my research and there are some points that I just can't understand.

1. For an American call, many resources say that it's possible to early exercise right before a dividend if the dividend is worth more than the remaining time value. But if you exercise the call before the dividend and acquire the stock to capture the dividend, then right after the dividend payment, the stock price will drop by the exact amount of the dividend payment and your overall payoff will still be $S_t - K$ where $S_t$ is the stock price right before the dividend payment. So from my understanding it will still be better to sell the call option before the dividend payment as the payoff is $S_t - K$ plus the time value. Does it mean that this argument is invalid?

2. Many say that if an American put is deep-in-the-money, it might be optimal to early exercise even in absence of dividend. What confuses me is that in any case where you decide to early exercise, your payoff will only be the intrinsic value, will it not always be better to sell the put option directly?

It is easiest to just think about volatility dropping to near zero in each of these cases, and also to assume that you will immediately trade out of the stock position. Note the following principles apply when effective interest rate $$r>0$$.
1. For the call, the profit of an early exercise is $$S - K$$. The expected profit of future exercise is $$e^{rt}(S-D)-K$$ which has present value $$S - D - e^{-rt}K$$. We can rewrite that as $$S - K - (D - (1-e^{-rt})K)$$, which shows where the (zero-vol) threshold of dividend size must be to make the proposition attractive.
2. For the put, the exercise value now is $$K-S$$, and the expected profit of future exercise is $$K-e^{rt}S$$ which has present value $$e^{-rt}K-S$$. It is always better to exercise the put option when volatility is zero and $$r>0$$.
• Thanks for your explanation. Maybe I didn't understand you completely but my confusion is between selling the option versus early exercise instead of early exercise versus late exercise. For example in your point 2, the exercise value now is $K-S$, which is better than late exercise, but what about selling the put right now? As long as the time value of the put option is 0 or positive, we will always be better-off selling the option right? Unless the time value for deep-in-the-money American put is negative, which I recall is not the case. – Yilie Ma Jan 6 '18 at 0:05