Model-Free Finance and Non-Lattice Integration

Abstract

Starting solely with a set of possible prices for a traded asset $S$ (in infinite discrete time) expressed in units of a numeraire, we explain how to construct a Daniell type of integral representing prices of integrable functions depending on the asset. Such functions include the values of simple dynamic portfolios obtained by trading with $S$ and the numeraire. The space of elementary integrable functions, i.e. the said portfolio values, is not a vector lattice. It then follows that the integral is not classical, i.e. it is not associated to a measure. The essential ingredient in constructing the integral is a weak version of the no-arbitrage condition but here expressed in terms of properties of the trajectory space. We also discuss the continuity conditions imposed by Leinert (Archiv der Mathematik, 1982) and K\"onig (Mathematische Annalen, 1982) in the abstract theory of non-lattice integration from a financial point of view and establish some connections between these continuity conditions and the existence of martingale measures