An Information Theory of Asset Price Distributions

The Theil-Barbosa framework of "rational random behavior" in the theory of consumer demand involves a decision-maker who minimizes the sum of a loss functional u and the cost of information c(I). The definition of information I is that used in the information theory literature associated with Shannon, Wiener, Kullback, and Leibler. The present paper shows such a framework is appropriate for modelling the behavior of traders in speculative markets who announce bid-ask prices at which they are willing to trade. Traders will announce prices which will be drawn from a probability distribution, the selection of which is a function of the loss functional u and the cost of information. The lognormal distribution is derived as a special case. It is shown that the variance of such distributions depends on the marginal cost of information, but that the kurtosis depends on the shape of the loss functional. When the loss functional is less convex than a quadratic, distributions will be leptokurtic.