Question

Derive carefully the put-call parity formula using the law of one price.

Answer 1

In the put-call parity equation, S0 is the current price of the underlying asset, P0 is the price of the put on the asset, C0 is the price of the call on the asset and Ke^(-r x T) is the present value of the options' strike price (K). Both options have the same underlying asset, maturity and strike price.

Consider two portfolios, one with only the stock (under consideration) in it and a second one which is composed of a long call option, short put option and a zero coupon bond with a face value equal to the option's strike price.

Pay off of the First Portfolio = S $

Pay off of the Second Portfolio:

If Stock Price S > K, then

Call Position Pay off = (S-K) and Put Position Pay off = 0 and Bond Pay off = K

Net Pay Off = S-K + K = $ S

If Stock Price S < K, then

Call Position Pay Off = 0 and Put Position Pay Off = - (K - S)

Bond Pay Off = K

Net Pay Off = - K + S + K = $ S

As is observable, the net future payoffs for both portfolios is $ S irrespective of the asset's future price, option's strike price, maturity, and premiums. By the law of one price if two portfolios have the same payoff irrespective of future occurences, then both portfolios should have the same price.

Hence, Price of Portfolio 1 (= Price of the stock in the portfolio) = C - P + K / e^(r x T) (bond price should be equal to the expected future face value redemption discounted at the risk-free rate with bond tenure being equal to the option's maturity)

Therefore. S0 = C0 - P0 + Ke^(-r x T) or S0 + P0 = C0 + Ke^(-r x T)