Consider Portfolio A: long 1 American put on a stock with strike price K and expiration at T; short 1 American call on the same stock with the same strike and expiration date; and long 1 share of stock.
(a) If you are long portfolio A and the call is exercised, show that you will have at least K dollars plus the value of the put at time T, no matter when the call is exercised.
(b) If you are long portfolio A, show that you are guaranteed a payoff of at least K at time T. Conclude that P − C + S0 ≥ Ke−rT .
Hint: Draw a timeline from 0 to T. Plot the different cash flows under different scenarios. When adding up, don’t forget TVM.