Exercising an American call option early
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?
Basically you'd make more money by selling the option than exercising. I am surprised no one actually gave a proof here...
Cases where exercising a call early makes sense:
There is a dividend payment the next day that is >= the interest on the strike price + a put with the same strike and expiration. Exercise the call, buy the put, and sell a zero-coupon bond for (strike price + put price - dividend). The transaction has a net 0 cashflow at time 0 and at expiry you will be long a share of stock and have to pay the bond off. If the stock is less than the strike price, you can exercise your put to get the strike price. As long as the maturity value of the bond is <= the strike price, you are in a better or equal position to having held the option to maturity.
You cannot short the stock or sell the option and believe it has peaked. If you believe that the stock has reached it's highest point before the expiration date and you are unable to hedge by shorting the stock (including using a synthetic short), you should exercise early. Since you are giving up the implicit put (right to not buy if it is below strike), you must believe that the difference between the current price and the price at expiry is larger than the price to buy this put + the interest on the strike price. Note: this would also apply if you cannot short the stock or sell the option, or borrow at a good rate and are in dire need of money.
The market is not efficient and you have an opportunity for arbitrage. There are a couple different ways this could happen, but if you can construct a series of purchases that give you risk free profit, congratulations, you have found a rare opportunity for arbitrage and should take advantage of it.
You will lose the option. If the option is conditional on some external factor (eg. employment at a company), you should obviously exercise it before losing it, as long as it is in the money.
The risk-free interest rate is negative. If for some strange reason, interest rates are negative, then the interest on the strike price is negative. If this is larger than the implicit put, buy the put and exercise the option.
You wish to be (subtly) charitable to the seller of the call. If exercising early is disadvantageous to you and if it is a zero-sum game, then your loss will be their gain. You could subtly donate to them by exercising non-optimally. They can simply sell another call, which will more than pay for the money they owe you (or else you are in situation 3 above) and they have made money.
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.
But my example describes a profit that wouldn't be possible if the option was European, why isn't that a benefit?
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.
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?
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.
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.
Right, that's another good argument. If you believe the stock will continue to climb, just wait since the cash for your margin deposit can accumulate interest. If you believe the stock will fall and you don't want to pay the short-sell interest, then just sell the option, as @Alexey states.
If the stock peaks at \$40, then drops to \$15, why would it not be better to exercise and get the \$20 from the up leg, as well as short it and get \$25 from the down?
At the time of exercise, you don't know what the final expiry stock value is.
Consider the portfolio consisting of the option and $K$ zero coupon bonds worth $B_t \leq 1.$ At expiry its value is
$$ \max(S,K) \geq S $$ since can you exercise and get the stock if $S>K$ and have $K$ 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.)
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).
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.
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.
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.
I think this is a common problem for a beginner in QF. The holder has two ways to treat the stock if he exercises the call option earlier:
- the holder holds the stock until maturity.
- the holder think the stock is overpriced, and sell the stock immediately.
In the first case, the holder lost the interest from the cash and also the stock price may fall.
The second case is a little tricky. Suppose the stock price is 50, and the strike price is 40. Selling the stock will give you 10. However, a better choice is to short the stock (borrow and sell the stock gives you 50) and close the short position at maturity use money equal or less than 40 (because you have the call option, you at most pay 40 for the stock). Then you can make a profit more than 10.
So you see in either case, we have a better solution than just exercising it. For more details, please check Hull's book: section 9.5 Early exercise: calls on a non-dividend-paying stock.
For this, let us consider two portfolios:
A:One American Call (c) and Kexp(-r(T-t)) in money mkt and
B: The stock S.
Now, let us say we decide to exercise the Call early,
Value of portfolio A (by exercising at t1): (S(t1) - K) + Kexp(-r(T-t1)) < S = B
Now if we consider the portfolio A at expiry (time T): Value of portfolio A (by exercising at T): max(S-K,0) + K = max(S,K) which is always greater than or equal to S (=B). Hence early exercise is not feasible for American Call.
This is not true for American puts, to see this let us consider a scenario in which Stock price falls to 0, by not early exercising now, we cannot gain much (stock won't go below this). But, if we exercise now, we can gain interest on the amount received. Hence, early exercising of American Put is feasible in this scenario.