Arbitrage concepts under trading restrictions in discrete-time financial markets

TL;DR

This paper investigates arbitrage and market viability in discrete-time financial markets with trading restrictions, establishing new links between portfolio optimization and fundamental asset pricing theorems.

Contribution

It introduces a weaker no-arbitrage condition linked to portfolio optimization, simplifying existing theories by avoiding semimartingale assumptions in discrete-time models.

Findings

Equivalence between portfolio optimization solvability and absence of arbitrage of the first kind.

Simplified proofs of fundamental theorems of asset pricing for discrete-time markets.

Illustrative examples including one-period models with borrowing constraints.

Abstract

In a discrete-time setting, we study arbitrage concepts in the presence of convex trading constraints. We show that solvability of portfolio optimization problems is equivalent to absence of arbitrage of the first kind, a condition weaker than classical absence of arbitrage opportunities. We center our analysis on this characterization of market viability and derive versions of the fundamental theorems of asset pricing based on portfolio optimization arguments. By considering specifically a discrete-time setup, we simplify existing results and proofs that rely on semimartingale theory, thus allowing for a clear understanding of the foundational economic concepts involved. We exemplify these concepts, as well as some unexpected situations, in the context of one-period factor models with arbitrage opportunities under borrowing constraints.