Classifying financial markets up to isomorphism

Abstract

Two markets should be considered isomorphic if they are financially indistinguishable. We define a notion of isomorphism for financial markets in both discrete and continuous time. We then seek to identify the distinct isomorphism classes, that is to classify markets. We classify complete one-period markets. We define an invariant of continuous-time complete markets which we call the absolute market price of risk. This invariant plays a role analogous to the curvature in Riemannian geometry. We classify markets when the absolute market price of risk is deterministic. We show that, in general, markets with non-trivial automorphism groups admit mutual fund theorems. We prove a number of such theorems.