Put-Call Parity

Definition: Put-call parity is the relationship between the price of a put option and a call option on a stock according to a standard model.

Define:

r as the risk-free interest rate, constant over time, in an environment with no liquidity constraints
S as a stock's price
t as the current date
T as the expiration date of a put option and a call option
K as the strike price of the put option and call option
C(S,t) as the price of the call option when the current stock price is S and the current date is t
P(S,t) as the price of the put option when the current stock price is S and the current date is t

Then the relationship is:

P(S,t) = C(S,t) - S + Ke^-r(T-t)

The relationship is derived from the fact that combinations of options can make portfolios that are equivalent to holding the stock through time T, and that they must return exactly the same amount or an arbitrage would be available to traders.

(Econterms)

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