No need for snide remarks about whether or not I understood your last post. I clearly stated that option prices are risk neutral expectations in my prior post. This is equivalent to saying options are priced *as if* the underlying grows at the risk free rate. You initially said, "BS necessarily assumes that the underlying asset grows on average at the risk free rate.", which is incorrect. I'm interested to hear what other people have to say about this, but my understanding is that estimates for mu are far noisier than estimates for sigma. Nonetheless, your main point seems valid, I'll have to think about it a bit more.
Martinghoul, I don't see the issue with market completeness. Although there may be many possible risk neutral measures, the fair price for a derivative will still be discounted risk neutral expectation under one of these measures. Thus, as long as one chooses the same measure for the put and the call one can still get the put-call parity relation just by calculating Call - Put = E[exp(-rT)(S-K)^(+)] - E[exp(-rT)(K-S)^(+)] = E[exp(-rT)S] - E[exp(-rT)K] = S(0) -exp(-rT)K (using risk neutrality) with respect to whatever measure Q you choose to price with. I don't see why uniqueness of Q matters.
Initially I was, but I thought you were saying more generally that put call parity falls apart under incomplete markets. In any event, it seems natural that it does hold up considering real markets are incomplete.
Actually, it doesn't... The Collector (Espen Haug) has talked a lot about the violations of put-call parity that do occur in very incomplete mkts. In general, it's a very robust principle, indeed, but you can easily imagine it breaking down.
Well, you should come to me to do some wonderful trades, amico... I make you nice prices. In seriousness, mate, don't get me started on the subject. I can talk about why using LIBOR rates is wrong for a long long time.