Arbitrage and duality in nondominated discrete-time models

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 strategiesexist 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.