# 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 $$r>0$$.

Note that you already own the option(s) in question, so we can just compare the profit of the various exercise strategies.

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
• I think in cases where we should early exercise an American put, the option price will be the same as the intrinsic value, so we prefer early exercise to selling due to market friction (like tax, trading cost etc), am I right? – Yilie Ma Jan 6 '18 at 0:28
• Generally you would be able to sell the option for just a little bit less money than the best market makers can realize by exercising the option themselves. They will take into account a bit of risk along with their stock trading costs. If you end goal is to hold cash, selling will probably be a good idea, while if your end goal is to have a stock position you will just want to exercise. – Brian B Jan 6 '18 at 2:19
• I'm pretty sure that there's a typo on equation S - K - e^{-rt}K, I think you meant S - D - e^{-rt}K ? – Wilmer E. Henao Feb 7 '19 at 17:59

Your understanding is correct. It makes no sense to exercise a call (as well as a put) when there is time premium remaining because you are throwing away that time premium by doing so. Sell the call and buy the stock if you want to own it in order to capture the dividend.

Buying the stock to capture the dividend makes no sense to me if it's a non sheltered account because since share price is reduced by the exact amount of the dividend on the ex-div date, you're incurring a taxable event with zero total return from the process and therefore you are effectively going to pay taxes for the privilege of receiving some of your own money from your equity position.

OTOH, if the bid of an ITM option is less than its intrinsic value then it makes sense to exercise it in order to avoid the haircut. Take the opposing equity position first and then exercising, thereby avoiding leg out risk.

• Does anything change when risk-free interest rates are negative? – Dimitri Vulis Oct 2 '20 at 13:42
• I've utilized options for over 30 years as a retail trader and I have never seen negative interest rates so I have never thought about that. Perhaps someone who knows more about theoretical pricing can answer that. If I came across that situation, I'd just crunch the numbers and let them dictate you what the best course of action is. – Bob Baerker Oct 2 '20 at 13:52
• USD hasn't has material negative rates yet (Matt King), but in some European currencies they are the new normal. You might like this paper papers.ssrn.com/sol3/papers.cfm?abstract_id=2738098 . Also Peter Carr wrote about <0 rates and American options, but I can't find it this minute. – Dimitri Vulis Oct 2 '20 at 15:16