Put-Call Parity

The "Put-Call Parity" equation is an instance of the Law of One Price. Its mathematical form is

     C - P = S - X * exp(-rT),

where S is the price of a stock at time 0, C is the price at time 0 of a European call option on S with strike price X and expiration time T, P is the price at time 0 of a European put option on S with strike price X and expiration time T, and r is the risk-free interest rate (assumed to be constant).

The term X * exp(-rT)in the equation is constant, and it represents the value at time 0 of the amount X paid with certainty at time T. Figuring out the value of a call and a put with the given strike for any given price of the stock on the expiration date will show that being long such a call and short such a put will always have the save payoff at time T as the stock itself less a payment of X at the same instant, which invokes the Law of One Price.

Put-Call Parity is model independent, meaning that it will hold regardless of under what model prices are determined. Considered differently, any model that violates Put-Call Parity cannot be realistic.