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
The payoff of a call option at expiry is:
- max(0, S − X) — correct
- max(0, X − S)Why not: That is the put payoff, which pays when the underlying is below the strike.
- S − X in all casesWhy not: Would make the payoff negative whenever the underlying finished below the strike. A holder simply declines to exercise instead.
The holder walks away when the underlying is below the exercise price, which is why the payoff has a floor at zero. max(0, X − S) is the put.
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.
The payoff must recover the premium: $50 + $3. At $50 the option is worthless and the buyer is down the full 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.
One party's gain is the other's loss on the same contract, which is what makes options a zero-sum instrument between the two counterparties.
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