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As we have noted, American options can be exercised early and in this section we specify cases in which early exercise can have value. Because early exercise is never mandatory, AND the right to exercise early may be worth something but could never hurt the option holder.
Consequently, the prices of American options must be no less than the prices of European options:
Recall that we already used this result in establishing the minimum price from the lower bounds and intrinsic value results. Now, however, our concern is understanding the conditions under which early exercise of an American option might occur. Suppose today, time 0, we are considering exercising early an in-the-money American call. If we exercise, we pay X and receive an asset worth So. But we already determined that a European call is worth at least So – Xl(1 + r)T-that is, the underlying price minus the present value of the exercise price, which is more than So – X. Because we just argued that the American call must be worth no less than the European call, it therefore must also be worth at least So – X/(1 + r)T. This means that the value we could obtain by selling it to someone else is more than the value we could obtain by exercising it. Thus, there is no reason to exercise the call early.
Some people fail to see the logic behind not exercising early. Exercising a call early simply gives the money to the call writer and throws away the right to decide at expiration if you want the underlying. It is like renewing a magazine subscription before the current subscription expires. Not only do you lose the interest on the money, you also lose the right to decide later if you want to renew. Without offering an early exercise incentive, the American call would have a price equal to the European call price. Thus, we must look at another case to see the value of the early exercise option.
If the underlying makes a cash payment, there may be reason to exercise early. If the underlying is a stock and pays a dividend, there may be sufficient reason to exercise just before the stock goes ex-dividend. By exercising, the option holder throws away the time value but captures the dividend. We shall skip the technical details of how this decision is made and conclude by stating that
When the underlying makes no cash payments, Co = co.
When the underlying makes cash payments during the life of the option, early exercise can be worthwhile and Co can thus be higher than co.
We emphasize the word can. It is possible that the dividend is not high enough to justify early exercise.
For puts, there is nearly always a possibility of early exercise. Consider the most obvious case, an investor holding an American put on a bankrupt company. The stock is worth zero. It cannot go any lower. Thus, the put holder would exercise immediately. As long as there is a possibility of bankruptcy, the American put will be worth more than the European put. But in fact, bankruptcy is not required for early exercise. The stock price must be very low, although we cannot say exactly how low without resorting to an analysis using option pricing models. Suffice it to say that the American put is nearly always worth more than the European put: Po > po.