Arbitrage and deflators in illiquid markets

TL;DR

This paper develops a stochastic model for discrete-time trading in illiquid markets with convex costs and constraints, exploring arbitrage notions and their relation to deflators without assuming a cash account.

Contribution

It introduces a unified framework for markets with nonlinear illiquidity effects, generalizing classical arbitrage and linking it to state price deflators.

Findings

Two types of arbitrage are characterized: marginal and scalable.

The relation between arbitrage and deflators is established through auxiliary market models.

The framework encompasses markets with transaction costs, bid-ask spreads, and nonlinear illiquidity effects.

Abstract

This paper presents a stochastic model for discrete-time trading in financial markets where trading costs are given by convex cost functions and portfolios are constrained by convex sets. The model does not assume the existence of a cash account/numeraire. In addition to classical frictionless markets and markets with transaction costs or bid-ask spreads, our framework covers markets with nonlinear illiquidity effects for large instantaneous trades. In the presence of nonlinearities, the classical notion of arbitrage turns out to have two equally meaningful generalizations, a marginal and a scalable one. We study their relations to state price deflators by analyzing two auxiliary market models describing the local and global behavior of the cost functions and constraints.