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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
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:
  • Short call:
  • Long put:
  • Short put:
Put-call parity is related to the fact that
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
So the price of long call plus short put must equal the price of a forward contract (zero) plus PV(F-K).
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