Dynamic trading under integer constraints

TL;DR

This paper explores the impact of integer trading constraints on no-arbitrage pricing and hedging, developing a new theory for non-rational price processes and characterizing the set of attainable prices.

Contribution

It introduces a novel theory of integer arbitrage free pricing and hedging for non-rational asset prices, including an FTAP and analysis of price sets.

Findings

Integer constraints have little effect on rational price models.

A new FTAP characterizes integer arbitrage free pricing.

Price sets are either empty or dense in an interval.

Abstract

In this paper we investigate discrete time trading under integer constraints, that is, we assume that the offered goods or shares are traded in integer quantities instead of the usual real quantity assumption. For finite probability spaces and rational asset prices this has little effect on the core of the theory of no-arbitrage pricing. For price processes not restricted to the rational numbers, a novel theory of integer arbitrage free pricing and hedging emerges. We establish an FTAP, involving a set of absolutely continuous martingale measures satisfying an additional property. The set of prices of a contingent claim is no longer an interval, but is either empty or dense in an interval. We also discuss superhedging with integral portfolios.