Option traders use put-call parity as a simple test for European style option price model. If the parity doesn’t hold, then there is a mispricing somewhere, and investors can profit from it without taking on any risk (arbitrage opportunity).
So how does put-call parity work? Let us take in consideration a portfolio composed of a call option and a bond, which has matures at expiration date of the option, and a face value equal to value the strike price. This portfolio should be equal to the value of a portfolio composed of a put option and a long position on the underlying stock. Because it is a European option, no early exercise is possible. If the expiration value is the same, then the present value of two portfolios must be the same. There are many sophisticated way to analyse put-call parity. But one of the simplest expressions of the parity is the following one where we take in consideration: c = current value of European Call; X = option strike price; = present value of a bond with face value of X, discounted at appropriate risk-free rate; p = current put value; s = current underlying stock price. Such equivalence ensures the non-arbitrage prices of a put option and a call option with same underlying security and same expiration date.
Also, from the put-call parity, we can infer that a protective put will deliver the exact same payoff as a fiduciary call. In a less technical term, if we hold a stock and long a put option on the stock will have the same payoff as we long a call option and invest in some risk-free bond with present value of the strike price (asset + put = cash equivalent + call).
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