Put-call parity

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. This is because if the price at expiry is above the strike price, the call will be exercised, while if it is below, the put will be exercised, and thus in either case one unit of the asset will be purchased for the strike price, exactly as in a forward contract.

The validity of this relationship requires that certain assumptions be satisfied; these are specified and the relationship derived below. In practice transaction costs and financing costs (leverage) mean this relationship will not exactly hold, but in liquid markets the relationship is close to exact.

Put–call parity is a static replication, and thus requires minimal assumptions, namely the existence of a forward contract. In the absence of traded forward contracts, the forward contract can be replaced (indeed, itself replicated) by the ability to buy the underlying asset and finance this by borrowing for fixed term (e.g., borrowing bonds), or conversely to borrow and sell (short) the underlying asset and loan the received money for term, in both cases yielding a self-financing portfolio.

These assumptions do not require any transactions between the initial date and expiry, and are thus significantly weaker than those of the Black–Scholes model, which requires dynamic replication and continual transaction in the underlying.

Replication assumes one can enter into derivative transactions, which requires leverage (and capital costs to back this), and buying and selling entails transaction costs, notably the bid-ask spread. The relationship thus only holds exactly in an ideal frictionless market with unlimited liquidity. However, real world markets may be sufficiently liquid that the relationship is close to exact, most significantly FX markets in major currencies or major stock indices, in the absence of market turbulence.

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