Actuarial Outpost put-call parity and formulas for option payoffs
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# put-call parity and formulas for option payoffs

Posted 12-29-2008 at 12:03 AM by Marid Audran
Updated 01-15-2009 at 01:22 PM by Marid Audran (changing to a more commonly used notation)
Basic material here. I was reading McDonald's textbook and did not see things described in exactly this way.

Define the function
$x^+ = max(x,0)$.
Say K is the strike price of an option and X is the spot price of the underlying asset at expiration. Then here are the payoffs:
• Long call: $(X-K)^+$
• Short call: $-(X-K)^+$
• Long put: $(K-X)^+$
• Short put: $-(K-X)^+$
Put-call parity is related to the fact that
$(X-K)^+ - (K - X)^+ = X - K.$
That's the payoff from a long call plus a short put. But suppose you also enter into a forward contract (say the forward price is F) and lend the net present value of F-K. Then the payoff at expiration of this position is
$( X - F ) + ( F - K ) = X - K$.
So the price of long call plus short put must equal the price of a forward contract (zero) plus PV(F-K).
Posted in Exams, MFE-3F