Classifying Financial Markets up to Isomorphism

TL;DR

This paper introduces a formal framework for classifying financial markets based on isomorphism, defining invariants like the absolute market price of risk, and explores market classification and mutual fund theorems.

Contribution

It defines a novel notion of market isomorphism in discrete and continuous time and classifies markets using a new invariant, advancing the understanding of market equivalence.

Findings

Classifies complete one-period markets.

Introduces the absolute market price of risk as an invariant.

Proves mutual fund theorems for markets with automorphisms.

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.