Martingale pricing measures in incomplete markets via stochastic programming duality in the dual of L ∞

Abstract

We propose a new framework for analyzing pricing theory for incomplete markets and contingent claims, using conjugate duality and optimization theory. Various statements in the literature of the fundamental theorem of asset pricing give conditions under which an essentially arbitrage-free market is equivalent to the existence of an equivalent martingale measure, and a formula for the fair price of a contingent claim as an expectation with respect to such a measure. In the setting of incomplete markets, the fair price is not attainable as such a particular expectation, but rather as a supremum over an infinite set of equivalent martingale measures. Here, we consider the problem as a stochastic program and derive pricing results for quite general discrete time processes. It is shown that in its most general form, the martingale pricing measure is attainable if it is permitted to be finitely additive. This setup also gives rise to a natural way of analyzing models with risk preferences, spreads and margin constraints, and other problem variants. We consider a discrete time, multi-stage, infinite probability space setting and derive the basic results of arbitrage pricing in this framework.