Arbitrage and duality in nondominated discrete-time models

TL;DR

This paper develops a theoretical framework for discrete-time financial markets without a dominating probability measure, establishing fundamental duality results and superhedging strategies in this complex setting.

Contribution

It introduces a nondominated measure-theoretic approach to arbitrage and duality in discrete-time markets, extending classical results to more general models.

Findings

Equivalence between no arbitrage and existence of martingale measures.

Existence of optimal superhedging strategies in nondominated models.

Minimal superhedging price characterized by supremum over martingale measures.

Abstract

We consider a nondominated model of a discrete-time financial market where stocks are traded dynamically, and options are available for static hedging. In a general measure-theoretic setting, we show that absence of arbitrage in a quasi-sure sense is equivalent to the existence of a suitable family of martingale measures. In the arbitrage-free case, we show that optimal superhedging strategies exist for general contingent claims, and that the minimal superhedging price is given by the supremum over the martingale measures. Moreover, we obtain a nondominated version of the Optional Decomposition Theorem.