No-arbitrage concepts in topological vector lattices

Eckhard Platen,Stefan Tappe

doi:10.1007/s11117-021-00848-z

Eckhard Platen, Stefan Tappe

Open Access

https://doi.org/10.1007/s11117-021-00848-z

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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.

Highlights

Let (, G, P) be a probability space, and let K0 ⊂ L0(, G, P) be a set of random variables, where we think of outcomes of trading strategies with initial wealth zero
The goal of this paper is to provide a general mathematical framework for no-arbitrage concepts which goes beyond the settings which have been considered in the literature so far
In this paper we present several abstract versions of the fundamental theorem of asset pricing (FTAP), including abstract FTAPs on locally convex spaces, an abstract FTAP on spaces of continuous functions on compact sets, and abstract FTAPs on Banach function spaces

Summary

Introduction

Let ( , G , P) be a probability space, and let K0 ⊂ L0( , G , P) be a set of random variables, where we think of outcomes of trading strategies with initial wealth zero. The appropriate concepts are NUPBR, NAA1 and NA1 It is well-known that for suitable semimartingale models in continuous time these three conditions are equivalent, and that they are satisfied if and only if there exists an equivalent local martingale deflator; see [41], and the earlier papers [6] and [23]. We will show that in a topological vector lattice the concepts NUPBR, NAA1 and NA1 are generally not equivalent. Topological vector lattices provide a suitable framework for abstract versions of the fundamental theorem of asset pricing (FTAP); in particular if the ideal U is a locally convex space or even a Banach function space. No-arbitrage concepts in topological vector lattices the abstract FTAP on locally convex spaces. 5 we review the no-arbitrage concepts in the particular situation where the topological vector lattice is the space L0 of random variables.

Topological vector lattices

The space of random variables

The fundamental theorem of asset pricing on Banach function spaces

Financial market with semimartingales