Payoff at expiration and exercise value before it

In short

• At expiration a European call pays $\max(0, S_T - X)$ and a put pays $\max(0, X - S_T)$. The buyer exercises only when the payoff is positive.
• If an option expires unexercised, the buyer's loss is the premium paid.
• Before expiration (t < T), the exercise value compares the spot price $S_t$ with the present value of the exercise price, $PV(X) = X(1+r)^{-(T-t)}$.
• Call exercise value = $\max(0, S_t - PV(X))$. Put exercise value = $\max(0, PV(X) - S_t)$. Neither can be negative.
• With X set equal to the forward price, a call's exercise value matches a long forward's value whenever $S_t > PV(X)$; a put's exercise value matches a short forward only on the downside.