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
A put holder will exercise when the underlying price is:
- below the exercise price - correct
- above the exercise priceWhy not: Selling at the strike when the market pays more would destroy value. That is when the put is abandoned.
- equal to the exercise priceWhy not: Exercise gains nothing at the strike; the payoff is zero either way.
A put gives the right to sell at the exercise price. That right is only worth using when the market would pay you less than that price, because then selling through the put gets you a better deal. Exercise when S < X, where S is the market price and X the exercise price. If the market pays more than X, you would sell in the market instead and let the put lapse. At S = X, exercising gains nothing.
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.
A put lets you sell at the exercise price, here $40. The market price is only $32, so you can sell for $8 more than the market pays. Payoff = max(0, X - S) = max(0, 40 - 32) = $8. This is the payoff before the premium. Profit would then take away whatever premium was paid for the option.
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.
The writer sells the put and is paid the premium straight away. The best result is that the put is never exercised, because the market price stays above the exercise price. Then the writer keeps the premium and owes nothing. Maximum gain = Premium received. However well things go, no extra money arrives, so the gain is capped at the premium. The exercise price is nearer to the writer's worst-case loss.
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