Why a Put’s Expiration Payoff Is Strike Minus Asset Value

Summary

The document clarifies why a put option’s expiration payoff is the positive part of the strike price minus the asset’s market price, rather than the full strike price whenever the option is exercised. Exercising a put lets its holder sell the asset for the strike, but the holder must give up an asset that still has market value. The payoff therefore measures the net economic gain from that exchange.

The answer also distinguishes physical delivery from cash settlement. With physical settlement, the holder delivers the asset and receives the strike price; with cash settlement, the writer pays the difference between strike and spot when that difference is positive. Both settlement methods express the same intrinsic payoff at maturity. This explanation concerns a European put at expiration and does not cover the option’s premium, early exercise, or the full valuation of an option before maturity.

Key ideas

• A put gives its holder the right to sell an asset at the strike price.
• The holder gives up an asset worth the spot price when exercising a physically settled put.
• The expiration payoff is the positive part of strike minus spot.
• Cash settlement can pay the same payoff without transferring the underlying asset.
• Expiration payoff does not include the option premium or describe pre-expiration value.

Tags

# Confusion about payoff for an option


# Confusion about payoff for an option












My teacher said that the payoff of a put is $\mathrm{max}(K-S_T, 0)$, where $K$ is the strike price and $S_T$ is the spot price at maturity. Why isn't it $K$ if $K-S_T > 0$ and $0$ otherwise (i.e. $K*\mathbf{I}_{K-S_T>0})$? If you exercise the option when $K-S_T>0$, then you make $K$ by selling it and otherwise make $0$; doesn't the extra term of $S_T$ assume that you will rebuy the asset after selling it? Or put another way, I'm confused in that $\mathrm{max}(K-S_T, 0)$ seems to capture the "value" of the put, rather than its payoff.

## Answer by Rylan (score 2, accepted)

https://quant.stackexchange.com/a/76796

A European put, which you're describing, gives the holder the right to sell the asset $S_t$ at time $T$ for price $K$. From the putholder's perspective, they receive $K$, but they have to part with an asset worth $S_T$.

It might also be worth noting that the "transaction" of "selling the stock $S_t$ for $K$ at time $T$" doesn't always literally happen when the putholder exercises their option. Sometimes, options are financially settled (aka cash settled), meaning that the putwriter just sends the putholder $K-S_T$ at time $T$, and never takes the stock $S_T$ or any money from the putholder.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.