[jason1990]

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The reason BS was such a breakthrough result is that in computing this "fair" price for the option, which gives both buyer and seller equal EV, it was able to take the Stock "Trend" - assuming there is such a thing - out of the final calculation. The final calculation of the "fair" option price according to BS does not depend on the "Trend" except in how the Trend effects Volatility. This was an astonishing result by BS and had a huge freeing up effect on the options market. The BS mathematics has undergone tremendous scrutiny so it's almost certainly correct. However I still can't believe it.

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It's not about EV at all. It's about the fact that the option is not the only game in town. You can trade in the stock directly and take either side of the wager.

I'm sure you've seen it, but I'll repeat a canonical example. Today the stock is $100. Tomorrow it will be $200 with probability p and $50 with probability 1-p. I want to sell you an option to buy the stock tomorrow for $170. How much should you pay me?

How much will the option be worth tomorrow? It'll be worth $30 is the stock goes up and $0 is it goes down. Can you achieve that same result without going through me? Sure. Take $10 out of your pocket and borrow $10 from your uncle. Buy 1/5 of a share. If the stock goes up, sell it for $40, pay back your uncle, and you're left with $30. If it goes down, sell it for $10, pay back your uncle, and you're left with $0. So you can achieve the same result without my option and it cost you $10 out of your pocket. So you'd be a fool to pay me any more than $10 for this option. (In fact, this is exactly what I will be doing to make sure I have what I need to pay you tomorrow.)

But if I'm selling it for $9, then you've got a nice opportunity. Borrow 1/5 of a share from your uncle and sell it for $20. Buy my option. You've got $11 left. If the stock goes up, borrow $170 from your uncle, use the option, sell 4/5 share for $160, pay back your uncle his $170 plus his 1/5 share and you're left with $1. If the stock goes down, borrow $50 from your uncle, buy a share, sell 4/5 share for $40, pay back your uncle his $50 plus his 1/5 share and you're left with $1. Where does your free $1 come from? It comes from me. So I'd be a fool to sell the option for anything less than $10.

So the price is $10 and it has nothing to do with p. It therefore has nothing to do with EV. For example, if p=0.25, then your EV when you buy the option is $-2.50 (= $30*p - $10) and my EV when I sell it is $2.50. The option is not priced according to EV. It's priced according to the fact that it's nothing more than a shortcut. When I sell you the option, I'm just doing you a service. I'm making your life easier so that you don't have to go through the rigamarole of borrowing from your uncle, etc.

Now, it's an interesting fact that if you artificially set p=1/3, so that the stock has no "trend" and then price the option according to this fake EV, then you come to the right price. For no other value of p will you come to the correct price when you use the EV method. Now that may be something to be mystified by.

But the fact that the true EV doesn't matter is not mysterious. The option is a derivative. It's outcome is simply a shortcut for some other procedure. That other procedure is of the form, "if this happens, I'll do that; if that happens, I'll do this, etc." The procedure doesn't at all depend on the specific value of any of the probabilities of those events. So the startup cost for that procedure doesn't either. And that startup cost is exactly what you should pay for the option. It would be an entirely different story if the buyer of the option was unable to trade directly in the stock.