No-arbitrage concepts in topological vector lattices
Eckhard Platen, Stefan Tappe
TL;DR
The paper develops a unifying framework for no-arbitrage in topological vector lattices, replacing classical probability-measure arguments with convex-cone and Minkowski-functional techniques. It demonstrates that $NUPBR$, $NAA_1$, and $NA_1$ can diverge in general, yet relate to abstract FTAP results that hinge on separating functionals or measures in locally convex and Banach function spaces. By treating $L^0$ as a Fréchet lattice and extending to Banach function spaces and semimartingale markets without a numéraire, the authors derive broad FTAP-type equivalences and provide short, self-contained proofs leveraging topological vector lattice theory. The work thus deepens the theoretical understanding of no-arbitrage across diverse functional-analytic settings and clarifies when the classical FTAP with a separating measure holds in abstract spaces.
Abstract
We provide a general framework for no-arbitrage concepts in topological vector lattices, which covers many of the well-known no-arbitrage concepts as particular cases. The main structural condition we impose is that the outcomes of trading strategies with initial wealth zero and those with positive initial wealth have the structure of a convex cone. As one consequence of our approach, the concepts NUPBR, NAA$_1$ and NA$_1$ may fail to be equivalent in our general setting. Furthermore, we derive abstract versions of the fundamental theorem of asset pricing (FTAP), including an abstract FTAP on Banach function spaces, and investigate when the FTAP is warranted in its classical form with a separating measure. We also consider a financial market with semimartingales which does not need to have a numéraire, and derive results which show the links between the no-arbitrage concepts by only using the theory of topological vector lattices and well-known results from stochastic analysis in a sequence of short proofs.