June 7th, 2008, 1:06 pm
There isn't such a thing as a physical probability distribution. Probability theory is epistemology, not ontology. Probability theory is like theology. If you believe in the God of the Christians, then you can use that to figure out useful things. When should you go on holiday? What should you eat? Other Christians will also agree, and so you can predict their behaviour, such as where you could find them on a Sunday. This doesn't mean there is any real reason for me to be a Christian, or that Christianity will tell me anything about the real world, because there are other equally compelling religions.The only time that you could use probability theory to calculate an observable quantity, is when there are enough constraints - symmetries of some kind or another - to fully determine the distribution. This would be the case in a casino for example. We can assume that a roulette wheel really does have an equal chance of hitting each number, because any other distribution implies that we have some degree of knowledge about which number will come up next. If you could find a way of obtaining that knowledge, then you would be banned from the casino. The simple fact that you are allowed in the casino means that you are a sucker, and suckers should use the discrete unform distribution.If you did choose some arbitrary probability distribution for a stock (power laws seem to be quite fashionable, apparently this God is the smightiest God because he brings forth credit crunches upon the withered blasphemers and their Gaussian whores) then you could calculate the price of an option. That price might even mean something to you, but it wouldn't mean much to me unless we agree about the distribution, and why would we do that?