Put-Call Parity for Forward Contract

Summaries

It can be proved that the put-call parity for a forward contract is:

C0 + (X-FT)/(1+Rf)^2 = P0

We are talking about the call and put option costs (C0, P0) of the forward contracts of that stock, NOT the call or put options of the stock.

Proof:

Stock price: S0, S(T)

Forward contract on that stock price FT

Call and put option of the forward contract (C0, P0)

Exercise price of the options (X)  (*** option to futures is saying you have the right to enter into the contract with futures price = X, (NOT futures contract price=X))

Portfolio 1:

Long call option with exercise price X + long bonds which pay X-FT at maturity (current cost: C0 + (X-FT)/(1+Rf)^T)

If in the money: X-FT + ST – X = ST-FT

If out of the money: X-FT

Portfolio 2:

Long put option with exercise price X (you can short futures which has price =X) and enter into future contract FT (current cost: P0)

If in the money: X-ST + ST – FT = X – FT

If out of money:  ST – FT

Equate the 2 portfolio to PV:

C0 + (X-FT)/(1+Rf)^T = P0

Note1: The exercise price of the options of a futures is that once you decide to enter the contract, you will be able to mark to market with earning X – FT

-          The American options of Futures is higher than European because it can be marked to market and the gains can be used to earn money

-          However, for forwards, they are the same

Note 2: The options for farward and futures are the same as the options for the underlying asset if they are expired at the same time. Why? It is because in this case, FT = S(T) (spot price = future price at maturity). So FT = S(T) = S(0)*(1+RF)^T

We then have the familiar equation:

C0 + X/(1+RF)^T = P0 + S0