Never Exercise the American Call

An American call on a non-dividend-paying stock may be exercised at any time before maturity. Show that it is never optimal to exercise it early, so its price matches the European call.

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#A call is worth more alive than exercised

Exercising early hands you the intrinsic value $S - K$ . But a call is worth at least its lower bound,

C \ge S - Ke^{-rT}, \tag{1}

which follows from put-call parity with $P \ge 0$ , or directly from a no-arbitrage comparison of holding the call against holding the stock financed by borrowing.

#The bound beats exercising

With $r > 0$ the discounted strike $Ke^{-rT}$ is smaller than $K$ , so

C \ge S - Ke^{-rT} > S - K. \tag{2}

The live call is worth strictly more than the $S - K$ you would collect by exercising. You should sell it, or simply hold it, rather than exercise.

The call is worth at least its lower bound, the price minus the discounted strike, which already exceeds the exercise value of price minus full strike. So the live call beats what early exercise would hand you, everywhere.

#Why it makes sense

Exercising early throws away two things, the interest you could still earn on the strike by paying it later, and the protection the call carries against the stock dropping below $K$ . With no dividend to capture by owning the share sooner, there is never a reason to surrender either, so the American call is worth exactly what the European one is.