Although I understand the arguments by which an American and a European call (when underlying pays no dividends) must have the same price, I haven't built up any intuition around the idea. I'm wondering if anyone here can give me a way to think about it. Let me just mention some of the issues that puzzle me:* The argument seems to rely on the fact that when you short a stock, you're getting the money now but don't have to pay it back until the future (i.e. expiration date of call). This seems fishy to me, like getting something for nothing. I intellectually realize that it's _not_ fishy, but as I said, I don't have a good intuitive way to think about it; my initial hunches are often wrong when it comes to this area.* When you first tell someone that it would be a mistake to exercise an American call early, the initial reaction would probably be, "Oh, because the stock might rise even further!" Yet it's the exact opposite, right? I.e. you don't want to exercise early because the stock might drop a lot (below the exercise price). Did that initially surprise others on the forum, the first time you read the argument?* When trying to understand how dividends upset this result, am I right to think that it's because if you exercise early, you are entitled to the flow of dividends, and so the decision to postpone exercise carries an additional cost in this scenario? Or is this not really what's driving the difference?Thanks for any help.

There are two reasons why it's not optimal to exercise an American call early: (a) the strike price is not discounted back to today so the later you pay it the better and (b) you own the "optionality" which must be worth something and you can enjoy this ownership until expiry. Giving it up early without some corresponding gain must be bad.If you want to liquidate your position it's more valuable to sell the option rather than exercise it and sell the share. You don't lose out paying the strike in full now (someone else can pay it at expiration) and someone buying the option from you is paying for both the intrinsic value (its in the moneyness) AND the optionality (or time value). If you exercise it and sell the share you're getting hit by the fact that the strike is more expensive now and you've given up your time-value which someone else would pay for.It's got nothing with do with the fact the the underlying might go up or might go down. Thinking along those lines will mislead you. You can hedge against that.

QuoteOriginally posted by: MattFThere are two reasons why it's not optimal to exercise an American call early: (a) the strike price is not discounted back to today so the later you pay it the better and (b) you own the "optionality" which must be worth something and you can enjoy this ownership until expiry. Giving it up early without some corresponding gain must be bad.If you want to liquidate your position it's more valuable to sell the option rather than exercise it and sell the share.Thanks for the response. If you'll bear with me for one more pass, can you elaborate on this? Strictly speaking, it seems as if your second reason can't be quite right. Otherwise you would never want to exercise an American put early since you'd be losing out on the optionality, and your (b) argument would "prove" that it's always better to sell the American put rather than exercise it.So it seems to me that really the brunt of the causality (if we want to speak in these terms) falls on reason (a), the one that I initially found suspicious but now seems far more reasonable.

Best to think of the dividends case with an exaggerated example. Consider S=15, K=10, time to expiry = 1 week, and there is a $10 dividend in a couple days. You obviously want to exercise before the dividend.

As I said in my initial post on this thread, there were a couple reasons that the short selling argument (which proves that you shouldn't exercise an American call early) seemed fishy to me. After thinking them through, I believe I've figured out why my reductio ad absurdum arguments were indeed valid, it's just that the "absurd" conclusion was true after all. If anyone cares to endorse or object to the following, I'd appreciate the feedback. Thanks.Initial objection #1: "The argument for holding an American call until expiration relies on the fact that the call allows you to short sell now and then buy back the stock at most for the PDV of the strike price. So that would imply that the further in the future the expiration, you could basically short the stock now and then buy it back for zero present dollars!"Answer: That's exactly right. Looking at the Black-Sholes formula, if you let T-->infinity, then the price of the call becomes S.Initial objection #2: "Something has to be fishy with the short-selling argument, because we can just forget about the call altogether. Suppose I short sell a stock and plan on paying it back in a thousand years. Does that mean I get S(t) now and only have to pay back a ridiculously discounted expected S(t+1000)?"Answer: That's exactly right. However, this isn't an arbitrage opportunity. The expected S(t+1000) (assuming no dividends) will be so large that it will exactly offset the discounting, if the stock is riskless. And to the extent that the stock is riskier than government bonds, yes its expected growth rate will be higher. But as a risk neutral investor, you can always earn this differential by borrowing money at the risk-free rate and investing it in equities that have a higher volatility. There's no unique opportunity presented by the short-selling of a stock and buying it back far in the future.

My second reason is quite right and is actually the important one. As interest rates tend to zero the strike discounting issue vanishes.Exercising a put early does indeed give up the time-value early however here it's not for nothing. You receive the strike payment earlier too! If the time-value of the option is very small then the benefit of getting the payment earlier can outweigh its loss.

QuoteOriginally posted by: MattFMy second reason is quite right and is actually the important one. As interest rates tend to zero the strike discounting issue vanishes.Exercising a put early does indeed give up the time-value early however here it's not for nothing. You receive the strike payment earlier too! If the time-value of the option is very small then the benefit of getting the payment earlier can outweigh its loss.Ah okay, thanks! Does it then follow that, as interest rates go to zero, you wouldn't exercise the American put early, either? (Since you're losing the exercise value but not getting much benefit in earlier strike price.)Also--and I realize I'm reneging on my earlier request for just "one more pass" on this thread--were my responses to my #1 and #2 problems okay? Or am I missing the big picture?

QuoteOriginally posted by: BobMurphyAh okay, thanks! Does it then follow that, as interest rates go to zero, you wouldn't exercise the American put early, either? (Since you're losing the exercise value but not getting much benefit in earlier strike price.)Consider an American put with S=.1, K=10, r=0.0, time to expiry = 10 years. Do you exercise early?

QuoteOriginally posted by: kwsConsider an American put with S=.1, K=10, r=0.0, time to expiry = 10 years. Do you exercise early?Well it seems as if this is either supposed to be self-evident (i.e. to clarify my thinking because the answer is obvious) or a trick question. The layman would probably say, "Of course you exercise early, because you lock in $9.90 now when the best you could possibly do is $10, and you might do a lot worse."However, in light of MattF's posts, this instinct is presumably wrong. By analogy with the call case, I suppose it goes like this: No, you don't exercise the put early, because you would do better by buying the stock now for 10 cents. In ten years when the put expires, you can sell the stock for max ($10, S(t+10)). Since the interest rate is zero, the present value of this strategy is max($10,S(t+10))-.1, which is >= $9.90.So in the unrealistic case of zero interest rates, you never exercise a put early, proving MattF's contention.Did I pass? :/