[mosta]

jason's probabilistic argument and my trading/hedging strategy are two different ways to see put-call parity. here's a third one: consider the combination of long call and short put on the same strike. what is that position? if stock is above strike at expiration you exercise the call. if stock is below strike at expiration you get assigned on your put. so regardless, the combo amounts to a contract to buy the stock at the strike price at expiration, whether stock is above or below. what's that? a forward contract. what's the value of a forward? parity plus basis. there is no time value/intrinsic value to a forward. thus all the time value/optionality of the long call and short put must cancel out--must be the same.

suppose stock is at strike. the call is trading $1 and the put is trading $2. (suppose the specialist is bearish and doesn't know how to price options and thinks the put should be worth more because he "knows" stock is going down.) what do you do? buy the call, sell the put, sell 100 shares, and never look at the position again. you pay $1, collect $2, sell stock at strike, and it back at strike at expiration either by exercising your call or being assigned on your put depending on whether stock has gone up or down. riskless $1 profit/arbitrage. (okay there are interest and dividend factors/risks, and that is why call and put are different in that regard.) you have a forward hedged/arb'd against the underlying. you have no risks, not for stock price or time (yes for interest and dividend...).

you can also do it where stock is not at strike. then there will be parity in one optoin. but the value above parity must be the same or you get an arb.