[mosta]

[ QUOTE ]
The theory seems to imply that a put in such a stock is worth just as much as a call. Makes no sense.

PairTheBoard

[/ QUOTE ]

yes, put-call parity is one of the first things you learn as an options trader. all that put-call parity amounts to is that buying a (at the money) call on 100 shares of stock and hedging it by selling 50 shares of stock short against it amounts to entirely the same thing (except for a minor wrinkle with basis exposure) as buying the (same strike) put on 100 shares and hedging it by buying 50 shares of stock. same risk exposure for stock price movement, time decay, and implied volatility change (different for interest rate and dividend change but those are usually pretty minor).

the way to understand BS is to understand the hedging/trading strategy that they use to derive the option value. suppose a stock is trading at $60 and I buy 1 contract of the 60 call for $1. one strategy would be to hold the calls and hope stock goes up over $61, where my profit starts. here's the BS strategy. (remember a "contract" is for a call on 100 shares--it's like 100 calls you might say.) when I buy the call at the same time I can sell 50 shares $60. (remember stock is currently trading at the strike price, 60.) now the game changes. now I don't need stock to go up, rather I can profit on an up move or a down move. consider the two scenarios:

a. stock goes up. what's my strategy here? I have the right to buy 100 shares at 60 a share (my call contract). I paid $1 for this. so to break even I need to sell 100 shares at an average price of $61. I already sold 50 shares at $60. so now if I sell 50 more shares at $62, I break even. or to be more cautious, what I might do is sell some at $60.50, $61, $61.50, $62, $63, etc a little at a time. once I'm short 100 shares, I could call it a day. hopefully I've worked my price average up over $61. I cover those short shares with my call contract and close everything out. my profit is the differency between my average sale price starting with teh first 50 shares I sold at $60, and $61 (60 for the exercise price of the call plus $1 for what the call cost me).

b. but what if stock goes down? well now the call is not looking so useful anymore. but I can still make money. I can buy my 50 short shares back. I break even if I can make a $1 buying the shares back. I sold 50 shares at $60, so if I can buy 50 shares at an average price of $58, I break even. hopefully I can buy it better than that. now my call is not something I'm thinking about using (I wouldn't exercise the right to buy at $60 when stock is under $60), but I made money on my short stock hedge.

now take another step. suppose you bought the calls at the money and sold the 50 shares, and then the stock kept running up and you kept selling a little at a time until you had sold 50 more shares. in a sense you're done. but actually you're not: now suppose stock drops back down suddenly to $60. what do you do? start the whole process over! sell 50 shares again. if stock runs up again, sell more and more until you've sold 50 more. if stock drops in price, buy more and more until you've bought back the 50. you can do this indefinitely. you can scalp stock against your options, because they stop you out from having any loss on an extreme price move. in fact yo make money on an extreme price move in either direction. one more refinement: you don't have to wait till stock as at strike and sell exactly 50 shares. whenever stock moves up you sell some and whenever it moves down you buy some. your stock position will vary between 0 and short 100 shares. if you can scalp on a continuous basis, then you can use calculus and differential equations to model this strategy.

and that is how we value the option. it doesn't matter whether stock goes up or down. I will scalp stock either way. I don't care which _way_ it moves, but I do care how much it moves. I need to make the price of the option in by trading stock against my options as it moves up or down. the more it moves the more I can trade stock and make little profits putting my hedge on and off (or holding my hedge and trying to catch a single big move-same thing either way, it scales). this is why the standard deviation of stock price movement is all we need to konw to value an option--because we have a money making strategy to trade against the option that is indifferent to which way stock moves. but it does need stock to move. time passing is lost opportunity for the stock to move and for me to trade against it. so as time passes options drop in value. you're biggest risk when you buy the calls and sell the 50 shares is that the stock price might not move at all.