# Put-Call Parity for Options on Forwards

• An option can be created for a forward contract.

• Recall that at contract initiation a forward will have a value of zero.

• European options on forwards will have the same price as European options on futures contracts; but American options on forwards will have a different price from American options on futures.

• The payoff values for options on forwards are:

Call payoff = Max[0,ST-X]

Put payoff = Max [0,X-ST]

ST = Underlying's price at expiration

X = Strike price of option

• Put-Call Parity for Options on Forwards:

p0 = c0 + ((X - F(0,T))/(1+rF)T)

• p0 = Today's price for a European put on a futures contract
• c0 = Today's price for a European call on a futures contract
• X = The strike price
• F(0,T) = Price at time "0" for a forward expiring at time "T"
• X - F(0,T) = The face value in long position of a zero coupon bond

Excluding storage costs, F(0,T) can be substituted with the formula for pricing a forward: S0(1+rF)T

With the put call parity formula and forward pricing formula, investors can apply the put call parity relationship to exploit mispricings.

## Early Exercise of American Options on Forwards and Futures

• Deep in the money American call options on futures behave similarly to the underlying futures contract, so it may make sense for the option holder to exercise the option and establish the futures position, in order to obtain interest from the margin account.

• The same logic can be applied to American put options on futures contracts.

• Recall that forward contracts will not pay any cash until expiration so this logic will not apply to options on forward contracts.

• Therefore, an American call on a forward is the same as a European call on a forward.

• However, because of the potential incentive to exercise early (i.e., to earn margin account interest) an American call on a futures contract is different from a European call on a futures contract and is thus more expensive.