American options' early exercise feature has value when receiving the intrinsic value immediately dominates holding the option. For calls on non-dividend-paying stocks, early exercise is never optimal (time value exceeds intrinsic value). For puts, deep in-the-money puts may warrant early exercise. Dividend-paying stocks complicate decisions: large upcoming dividends can make early call exercise optimal.
Use binomial trees to solve the optimal exercise boundary and compare American and European option values under different scenarios.
Your prerequisite on American versus European options introduced the key asymmetry: an American option grants the right to exercise at any point up to expiration, not just at maturity. Your study of intrinsic and time value established that an option's total market value equals intrinsic value plus time value, where time value is always non-negative because optionality has value. These two facts together produce a surprisingly elegant result for calls — and a more nuanced story for puts.
Start with a call option on a non-dividend-paying stock. At any moment before expiration, the option trades at its intrinsic value plus some positive time value. If you exercise early, you receive only the intrinsic value — you permanently give up the remaining time value. Since you can always sell the option in the market for at least intrinsic value (and typically more), early exercise destroys value compared to selling. Therefore, early exercise of an American call on a non-dividend-paying stock is never optimal. The American and European call are worth exactly the same in this case. The early exercise right is valuable in principle, but you should never actually use it.
Puts are different. For a deep in-the-money put — say, the right to sell a stock at $50 when it currently trades near $2 — the intrinsic value is $48 and the maximum possible future payoff is capped at $50 (since the stock cannot fall below zero). Holding the option now means waiting to eventually receive something close to $48–$50 anyway. Meanwhile, you forgo the interest you could earn on $48 if you exercised today and invested the proceeds. When the interest forgone exceeds the remaining time value of keeping the put alive, early exercise is optimal. Deep in-the-money American puts can therefore trade above their European counterparts — the early exercise right has positive value.
Dividends complicate the call analysis. When a stock goes ex-dividend, its price drops by approximately the dividend amount, shrinking the call's intrinsic value. Exercising the call just before the ex-dividend date captures the pre-dividend stock price. The trade-off: early exercise forgoes the remaining time value but avoids the drop. The decision rule is that early exercise is worthwhile if the dividend exceeds the forgone time value from surrendering the option early. This is why American call options on high-dividend stocks trade above their European counterparts.
The optimal exercise boundary formalizes these comparisons as a function of time to expiration and, for dividend-paying stocks, the dividend schedule. Binomial trees compute this boundary numerically by backward induction: at each terminal node, the payoff is max(S − K, 0). At each earlier node, you compare the immediate exercise value (intrinsic value) to the continuation value (the discounted expected value of keeping the option alive). The optimal decision is always to take the higher of the two. This backward sweep — comparing exercise versus continuation at every node — is the fundamental algorithm underlying American option valuation.
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