Work out what a call option pays at expiry
The idea
At expiry a call option's payoff is max(0, S - X), where S is the price of the underlying and X the exercise price. The holder exercises only when S exceeds X, so the payoff is never negative - the option is abandoned otherwise. Profit subtracts the premium paid, so the buyer breaks even at X plus the premium. The writer's position is the mirror image: premium received is the maximum gain, and the loss on a short call is theoretically unbounded as the underlying rises.
Formulas
Payoff of a call at expiry
call payoff = max(0, Sₜ − X)
- Sₜ
- the price of the underlying at expiry
- X
- the exercise price
The holder exercises only when it pays to, which is what the zero is doing. The payoff is never negative; the loss to a buyer comes from the premium, which the payoff does not include.
A practice question
A call option with an exercise price of $40 expires with the underlying at $55. Its payoff is:
- $55Why not: The underlying price itself. The holder pays $40 to acquire it, so the gain is the difference.
- $15 - correct
- $0Why not: The out-of-the-money answer. At $55 against a $40 strike the option is well in the money.
A call lets you buy at the exercise price. Payoff = max(0, S - X), where S is the market price and X the exercise price. Here S = 55 and X = 40, so payoff = max(0, 55 - 40) = $15. In plain words, you pay $40 for something worth $55, so you gain $15. Profit would be lower, because you would also subtract the premium (the price paid to buy the option).
A harder one
A call with an exercise price of $50 costs a $3 premium. The buyer breaks even at an underlying price of:
- $47Why not: Subtracts the premium instead of adding it. That is the put's breakeven direction.
- $53 - correct
- $50Why not: The strike, where the payoff is zero and the buyer is down the whole premium.
Break-even is the price at which you end up with no gain and no loss. For a call buyer, the option only starts to pay once the asset is above the exercise price, and the first part of that pays back the premium. Break-even = Exercise price + Premium = 50 + 3 = $53. At $50 the option pays nothing, so you are still down the full $3 premium.
The hardest tier
The payoff profiles of a call buyer and a call writer are best described as:
- mirror images summing to zero before fees - correct
- identical in shape and signWhy not: Same shape, opposite sign. One party's gain is precisely the other's loss.
- unrelated, since they are separate contractsWhy not: They are two sides of one contract, which is what makes the transfer zero-sum.
Every option has two sides: the buyer and the writer (seller). Whatever the buyer gains, the writer loses, and the other way round, so their payoff lines are mirror images and add up to zero (before any fees). Buyer's payoff + Writer's payoff = 0. This is why an option is called a zero-sum contract between the two parties: one person's gain is exactly the other person's loss.
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