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I have seen the rationale behind why it is never optimal to exercise an American call option early, but have a question about it.

If the option strike price is $E=\$20$ and it expires at $T=1yr$, if the share price is $S=\$25$ at expiry, the profit will be $\$5$ per share. However, what if at $T=0.5yr$ the share price was $S=\$40$ following a jump in the price. Exercising at this point would yield a profit of $\$20$ per share.

Would it not be more optimal to exercise early, in this case?

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Why is this downvoted. A genuine question isn't it? – Mew Oct 1 '13 at 13:37
Basically you'd make more money by selling the option than exercising. I am surprised no one actually gave a proof here... – Lost1 Apr 16 '14 at 17:40
Loving the answers here learning new information about stuff I thought I already knew! – Brian Topping May 6 '15 at 6:22

If you think the stock is going to continue going up, just wait. If you think the stock has reached its peak, then short it in the open market. If the shorted stock continues to climb, you can always cover with your call option. If however the stock falls below your strike price, then let the option expire and cover at the market price.

It's this very ability to choose the buying price (strike vs market) when covering a short sale that makes the option valuable. But of course, you can do the same thing with a European option! That's why the American option gives you no added value over the European option. There is no benefit to exercising an American call option early.

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But my example describes a profit that wouldn't be possible if the option was European, why isn't that a benefit? – dplanet May 13 '12 at 19:31
What are you talking about?! If you exercise now, you pay \$20 and sell at \$40. But the only reason to exercise now is if you believe \$40 is the highest the stock will get. It's foolish to exercise now if you believe the stock will go higher. If you believe the stock will max-out at \$40, then short it now and wait until the expiration to see what the market price is before buying it back. The market price could be lower than the exercise price, which is why you'd be giving-up money to exercise now if you believe it's going to fall further. – chrisaycock May 13 '12 at 19:35
Okay, so say that the stock is going to go down to $15, lower than the exercise price - the option will be worthless, I'll make a loss. But had I exercised when the stock price was higher than the exercise price (not necessarily highest) I'd have made a profit. Isn't that more optimal than the loss that has been made by holding onto the option until expiry? – dplanet May 13 '12 at 19:38
Do you know what short selling is? I've mentioned it twice because it's the classic textbook argument against early exercise. If the stock is at \$40 now and you believe it will drop to \$15 by expiration, short the stock now, let your option expire, and then buy back the stock ("cover your short" in industry jargon) on the regular market. – chrisaycock May 13 '12 at 19:40
Great, thanks. The rationale I read only gave the reason that you'll lose interest on the cash you spend to exercise the call option. – dplanet May 13 '12 at 19:45

If you can dynamically hedge then you can monetize the value of your option without prematurely exercising it. Before writing about Randomness and Black Swans, Taleb wrote a book on the topic. The short version of the story is find the DV01 of your position, and take an opposite position with the same DV01. (& if you want, line up the rest of the greeks)

If you can't dynamically hedge for whatever reason, then the "Don't exercise early" rule is not so black and white. For example, executives have shorting restrictions on their companies. In these cases, they may be restricted to exercising options early rather than dynamically hedging.

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Because you would make a higher profit if you sold the option on the open market at that point in time, rather than exercising it at that point in time due to the time value of money.

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This is already stated in a previous answer. – chrisaycock Sep 30 '13 at 2:05

Because if you sell, you will get a higher value than 20USD per share. You can think of the reason behind this added value is that having a deep ITM option is better than having a stock: your downside is limited. Therefore your option is worth more on the market than it's exercise value. This is why you are better off by selling it in your case (if you know that the price will drop to 25USD).

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at the time of exercise, you don't know what the final expiry stock value is.

Consider the portfolio consisting of the option and $E$ zero coupon bonds worth $B_t \leq 1.$ At expiry its value is

$$ \max(S,E) > S. $$ since can you exercise and get the stock if $S>E$ and have $E$ otherwise. So at all times previously, $$ C_t + K B_t > S_t $$ since it dominates at expiry. So $$ C_t > S_t - KB_t > S_t - K. $$ So the European option is always worth more than its intrinsic before expiry.

And the American is worth at least as much as the intrinsic so it's never sensible to early exercise.

Note that this does not require dynamic hedging.

(I discuss this question at length in my book Concepts etc.)

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The theoretical answer is: If you feel you should exercise at $t<T$ then you can get $S_t-K$ (obviously $S_t>K$). But price of the option at any$t<T$ is $C_t>S_t-K$ always. So rather than exercising just sell the option and let someone else hold it. But then practically there are transaction costs etc.

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You can only make a decision on whether to early exercise an American call option on an equity underlying with current information. In your question you anticipate a movement in the underlying higher in the future. This has no economic bearing on whether to exercise early. Given your inputs of $S=25$, $E=20$ and $T=1.0$ and assuming interest rate of $0$ and no dividends all you can say is that the option must be worth at least $5$ in the market. There will be no early exercise. Now what if $T = 0.02$ or approximately 1 week and tomorrow the stock goes ex-dividend for $2.00$? Back before the OCC starting adjusting strike prices for large dividends you would exercise early to capture the dividend. Tomorrow the stock will be $23$ and the intrinsic value will go from $5$ to $3$. If you did not exercise early then you would lose out on $\$2$ by holding call option to expiration.

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