Financial market models in discrete time beyond the concave case

TL;DR

This paper develops a general framework for discrete-time financial market models without assuming concavity, establishing representation, no-arbitrage, super-hedging, and utility maximization results, including vector-valued models like currency markets.

Contribution

It introduces a non-concave, axiomatic approach to market models, extending classical results to more general, possibly vector-valued, settings.

Findings

Market models can be represented as normal integrands under upper-semicontinuity.

The no-arbitrage condition is extended and its implications are analyzed.

Results on super-hedging and utility maximization are generalized to this framework.

Abstract

In this article we propose a study of market models starting from a set of axioms, as one does in the case of risk measures. We define a market model simply as a mapping from the set of adapted strategies to the set of random variables describing the outcome of trading. We do not make any concavity assumptions. The first result is that under sequential upper-semicontinuity the market model can be represented as a normal integrand. We then extend the concept of no-arbitrage to this setup and study its consequences as the super-hedging theorem and utility maximization. Finally, we show how to extend the concepts and results to the case of vector-valued market models, an example of which is the Kabanov model of currency markets.