Put-Call parity does not mean that both Options have an identical price, it refers to relationship with the Underlying asset or commodity and its cash value (as derived from the Strike Price). In other words:
Call Option + Cash = Put Option + Underlying
For example, if the option is for 100 shares of a given stock and the strike is $50 then the relationship is as follows:
Call Strike Price 50 + 5000 = Put Strike 50 + 100 Stock
Buying both a Call and Put at the same Strike Price with the same Expiration is the same as a forward contract with the same Expiration and and price as the Strike Price.
Put-Call Parity conventionally only applies to European Style Options. If adjustments are made for dividends and interest rates, however, Put-Call Parity is also applicable to American Style Options contracts. Dividends have the effect of increasing put values (and cause Call prices to decrease). Interest rate changes have the inverse effect on values. An increase in interest rates cause call values to rise and put prices to fall.
It is theoretically possible to find Arbitrage opportunities that may result from temporary divergences from Put-Call Parity. In practice, this is almost impossible to accomplish as high frequency algorithms operated by hedge funds and proprietary trading firms usually identify them (and remove the pricing inefficiency) before human traders are either aware of them or have an opportunity to profit. The box spread strategy is an example of an options trading strategy that seeks to take advantage of these scenarios.
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European Options: Put-Call Parity
Matt Brigida explains Put-Call parity in this video introducing the concept.
External LinksPut–call parity
Wikipedia article describing the mathematical concepts behind put-call parity.Understanding Put-Call Parity
Options Guide article with more detail on put-call parity, with examples and some formulas.