So I found myself after lunch today wandering over to Thackeray and poking my head into an introductory book on the math of such things. The notion of replication of financial instruments is kinda mind-blowing in its simultaneous usefulness and simplicity.
Imagine, for example, that I live in an idealized universe without interest rates. There is a stock market with one stock. The market price today is $10, and somehow I know tomorrow that it will either be $12 or $8. Someone offers me a "European call option" to be exercised tomorrow with strike price $10. This meaning that if I buy it, I will have the right (but not the obligation) to buy a share at $10, no matter what market price happens to be then.
How much money should I be willing to offer for this contract today?
My first reaction would definitely have been the one I'm told is common, to think that you'd have to know the probabilities that the stock will move up or down, and then simply do a little basic statistics. But it's not so! You can price the option with the information given.
Think about what the option will do for you: if the stock does move up, then you get a $2 profit buying the stock at $10 and selling it at $12. If it moves down, then you get no profit or loss, because you'd be dumb to buy at $10 when you could buy at the market price of $8. This tuple of possible outcomes ($2, $0) can be replicated by a portfolio of stock and cash. (in a scenario that takes into account interest rates, it would be a portfolio of stock and bonds) The portfolio is this: 1/2 a share of stock, and -$4. For tomorrow, 1/2 a share of stock and minus four dollars is worth exactly ($12/2-4=)$2 or ($8/2-4=)$0 according to whether the stock moves up or down.
So you just have to notice that a 1/2 share of stock can be gotten right now for $5, making the value of the option ($5-$4)=$1. This is true no matter what the probabilities involved are! (at least given all the usual "no-friction" assumptions about transaction costs being negligible that don't necessarily hold in the real world) If anybody was willing to buy or sell the option at any other price, an arbitrage opportunity would exist, because you could buy the replicating portfolio and sell the option, or vice versa, and make risk-free money.