Deriving Put–Call Parity from Long Call and Short Put Payoffs

Summary

The document clarifies the payoff sign for a short put in the standard European put–call parity relationship. Selling a put gives the seller the negative of the long put payoff: the short position’s payoff is the negative of the option holder’s payoff. This resolves the question’s confusion between an obligation associated with exercise and the payoff received by the option writer.

Combining a long call and a short put with the same strike and expiry produces a payoff equal to the underlying price at expiry minus the strike. This is the payoff of a forward contract struck at that level, which explains the parity relationship. The explanation is a terminal payoff argument; it does not derive the full pricing equation or discuss financing, dividends, transaction costs, or early exercise. Its scope is the European option setup stated in the question.

Key ideas

• A short put has the negative of the long put’s payoff from the seller’s perspective.
• A long call and short put with matching strike and expiry combine to a linear payoff.
• The combined payoff equals the underlying’s terminal price minus the strike.
• This payoff replication explains the forward component of put–call parity.

Tags

# Put Call Parity confusion


# Put Call Parity confusion












My question concerns an ambiguity in the wikipedia article about Put Call Parity. In the first sentence: "In financial mathematics, put–call parity defines a relationship between the price of a European call option and European put option, both with the identical strike price and expiry, namely that a portfolio of a long call option and a short put option is equivalent to (and hence has the same value as) a single forward contract at this strike price and expiry." is about a "short" put option which means that we have the obligation to sell the put but in the derivation a long put is used, i.e. a put with payoff profile $(K - S_T, 0)^+$. But a short put would be actually $K-S_T \in \mathbb{R}$ if the option gets exercised. Edit: My question concerns the definition of "short put". In http://www.investopedia.com/terms/s/short-put.asp it is definied as the obligation to sell and then its not the payoff: $-(K-S_T)^+$?

## Answer by Antoine Conze (score 2)

https://quant.stackexchange.com/a/34478

Being short a put simply means that you have sold the put, hence its payoff is from your point of view $-(K - S_T)^+$. When your are long a call and short a put your total payoff is $(S_T - K)^+ -(K - S_T)^+ = S_T - K$, hence the call/put parity.

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