I stumbled upon Peter Carr’s site and saw a very nice explanation on the arbitrage principles that govern option pricing. Usually explanations on option pricing are very cumbersome and not clear. Here is my take on it, but you can just go to his website and find it in the research papers under FAQ.
Suppose you have a stock that is priced at 1$ and it has a 50% on either going up to 2$ or going down to 0.5$. Thus, we assume there are only two states with equal probablity. Let us now assume there is a Call option written on that stock with strike price of 1$. Thus, in the up state it makes 1$ and in the down state it makes 0. Now comes the million dollar question.. how much is the option worth? If you take the expected value approach, you’d say it should be worth 0.5$ since that is the expected profit ()
There are a few problems with the expected value approach. We could also say that the stock should be worth 1.25$ in the future by the same expected value approach. This adds more confusion to the proper way to price the option.
Let us assume that we go with the first approach and we price it to be worth 0.5$. In that situation we could sell two call options and buy one stock. We have no initial cost. If the stock goes up, we make nothing, but if the stock goes down, we profit 0.5$ since the options expire worthless. That is in essence an arbitrage, since we have no risk at all and we stand to gain a profit (although in only one state of the world). This tells us that something is wrong with this approach. The question is, is their a proper way of doing this?
Let us assume that there is no interest in this world of ours for simplicity. This means that as time passes, we demand no compensation for our money if we take no risk. We know that the option pays 1$ in the upstate, and nothing in the downstate. We can construct a portfolio that replicates the option buy noticing that buying 2/3 of the stock and selling short 1/3 of a bond with face value of 1$ with exactly replicate the option price (do it yourself to make sure). The total cost would be 1/3$. What is that magical 1/3$ number we got? I claim that is should be the fair price of an option!
As a good exercise, you should try to figure out why this should be the fair price. But do not worry. In the next post, I’ll explain more about why this should be the fair price and touch upon what is known as state prices.
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