LessonThis is a lesson restated in new words. Any figure in it is an example from the source, not this desk's record.
A European option is a choice, not a duty, to buy or sell an agreed asset at an agreed strike on one future date. A call is the choice to buy. A put is the choice to sell. The payoff uses only the price at that date. The premium is cash paid or received at the start, and gain is payoff minus that premium for the buyer.
Why it matters
The contract has two numbers you set at the beginning. Capital T is the only exercise date. K is the strike, the price used on that date. Nothing in the definition says you must trade if the trade is a bad one. That “choice, not a duty” is why the payoff has a floor at zero for the holder, and why someone must be paid to stand on the other side.
The idea in plain words
Draw time from today, t0, to the expiry T. At T the share price is still unknown. Call it S of T. If that price is 150 and you hold the call, you may buy at 120 and the gap is 30. If the price is 90, buying at 120 would lose 30, so you do nothing and the option ends at zero. You do not “lose the 30” inside the payoff. You simply decline.
The compact form is the maximum of S of T minus K, and zero. At 150 the maximum is 30. At 90 the maximum of negative 30 and zero is zero. That expression is the call payoff. Gain subtracts the premium p paid at t0: the same maximum, minus p. The premium is negotiated today. It does not change the rule at expiry. It changes whether the 30, if you get it, still leaves you ahead.
A put reverses the subtraction: the maximum of K minus S of T, and zero. You want the share to be cheap, because you may sell it at K. The put payoff cannot exceed K, because the share price cannot help you by more than the full strike in this formula. Gain for the put holder is that maximum minus the put premium.
Plots of payoff against S of T are the hockey-stick diagrams. For a long call the line sits at zero until S of T passes K, then rises one-for-one. For a long put the line falls as S of T rises, and it is zero once S of T is above K. The kink is at the strike.
Four positions exist, because each contract has two sides. The long side buys the choice and pays p. The short side sells the choice and receives p. A short payoff is the long payoff with the sign flipped. Whenever the long side’s payoff is positive, the short side’s payoff is negative by the same amount. On a gain chart, the long hockey stick is shifted down by p, and the short hockey stick is shifted up by p. The vertical gap is the premium. The short side accepts the later obligation only because that premium is paid.
How to use it
Write the payoff before you write a price. For a call, compute max(S of T minus K, 0). For a put, compute max(K minus S of T, 0). Then decide whether you are looking at payoff or gain. If gain, subtract p for a long position and remember the short position received p, which is the upward shift.
What usually goes wrong
Exercising a call when the share is below the strike throws away the “not a duty” clause and manufactures a loss the payoff function sets to zero. Forgetting to subtract p makes a winning payoff look like a winning trade. The buyer can be up 30 on the option and down overall if p was larger than 30.
Questions people ask
What is the call worth at expiry if the share is below the strike?
Zero. In the example, a share at 90 against a strike of 120 would lose money if exercised, so the holder lets it expire. Gain is then negative p, the premium already paid, not negative 30.
Why is a put’s payoff bounded?
The put payoff is the maximum of K minus the terminal price, and zero. The largest that gap can be, in this expression, is K, when the terminal price contributes nothing above zero.
What does the short side receive?
The premium p at the start. On the gain diagram that receipt shifts the short call and the short put up by p. In exchange, the short side faces the opposite of the holder’s later payoff.