A formalization of the asset pricing problem

Updated October 6

We assume that there is a given collection of assets. A linear combination of assets is a portfolio. Negative coefficients in a portfolio represent short-selling or borrowing of the corresponding asset. More general functions of assets are called derivative assets. An asset is positive if it is always possible to sell it for a positive price. The problem is to find a fair price P for all assets. The fair price must be linear (the price of a portfolio is the linear combination of the component asset prices) and positive (a positive asset must have a positive price) in order to avoid the possibility of arbitrage (getting something for nothing).

What I just described is the algebraic axiomatization of probability theory. This means that assets are random variables, and the price is an expectation. In other words, assets are characterized by the probability distributions of their values, whose expectation is their fair price. This is not to say that the expected value is calculated with respect to the probabilities of actual events in the future (or not necessarily). It could also be some sort of market expectation of the value of the asset. I used to think that the price was the risk expectation (i.e., with respect to uncertainty of future value), and that zero-price-variance assets are risk-free assets. That seems not to be the case necessarily, and I quickly got confused trying to follow its logical consequences. In particular, it seems that identifying the market expectation with the risk expectation leads to a contradiction with the Capital Asset Pricing Model.

I am thus going to adopt the interpretation that the price is some sort of market average of perceived value. Then, if an asset x existed such that P(x²)=P(x)², this would mean that there is only one possible price for the asset that anyone will pay. I'll call such an asset a benchmark. It is questionable whether such an asset exists. Back in the time of standard-backed currencies it might have made sense to say that gold was a benchmark, but in a world of freely floating fiat currencies I don't think there are any benchmarks left. Anyway, in a real market with imperfect information or bounded rationality there can't be any benchmarks either.