Work out what a put option pays at expiry
The idea
At expiry a put option's payoff is max(0, X - S), where X is the exercise price and S the price of the underlying. The holder exercises only when S is below X, so the payoff is never negative — the option is simply abandoned otherwise. Profit subtracts the premium paid, so the buyer breaks even at X minus the premium, and the maximum gain is bounded because the underlying cannot fall below zero. The writer holds the mirror image: the premium received is the most that can be made, and the loss runs as far as the underlying can fall. A put is therefore the natural instrument for protecting a long position rather than for speculating on collapse.
Formulas
Payoff of a put at expiry
put payoff = max(0, X − Sₜ)
- X
- the exercise price
- Sₜ
- the price of the underlying at expiry
The mirror of the call. The most a put can pay is X, when the underlying goes to zero, which is why a put is bounded and a call is not.
A practice question
At expiry, the payoff of a put option is:
- max(0, X - S) — correct
- max(0, S - X)Why not: The call payoff, which pays when the underlying finishes above the strike.
- S - X in all casesWhy not: Both the wrong direction for a put and wrong in ignoring the holder's right to decline.
The holder gains only when the underlying has fallen below the exercise price, and abandons the option otherwise.
A harder one
A put with an exercise price of $40 expires when the underlying trades at $32. Its payoff is:
- $0Why not: The out-of-the-money answer, which would apply if the underlying finished at or above $40.
- $8 — correct
- $32Why not: The underlying price itself. The holder sells at $40 and buys at $32, so the gain is the difference.
Forty minus thirty-two is eight, and the payoff ignores the premium, which profit would then deduct.
The hardest tier
The maximum gain to the writer of a put is:
- the premium received — correct
- the exercise priceWhy not: That is closer to the writer's maximum loss, if the underlying falls to zero.
- unlimitedWhy not: The gain is capped at what was collected. No further money arrives however far the underlying rises.
Writing an option sells the right, so the best outcome is that it is never exercised and the premium is kept.
In the app: 18 practice questions on this outcome, a narrated walkthrough with diagrams, spaced review that brings it back before you forget it.
Take the free diagnostic →