Arbitrage-free modeling under Knightian Uncertainty

TL;DR

This paper extends the Fundamental Theorem of Asset Pricing to markets under Knightian Uncertainty without relying on specific prior assumptions, using a functional analytic approach to establish arbitrage conditions.

Contribution

It introduces a general framework for modeling under Knightian Uncertainty, connecting arbitrage absence to approximate martingale measures and adapting classical notions like No Free Lunch.

Findings

Absence of arbitrage is equivalent to the existence of approximate martingale measures.

The approach does not require specific assumptions on priors or state space structure.

Specialization to discrete time yields true martingale measures.

Abstract

We study the Fundamental Theorem of Asset Pricing for a general financial market under Knightian Uncertainty. We adopt a functional analytic approach which require neither specific assumptions on the class of priors $\mathcal{P}$ nor on the structure of the state space. Several aspects of modeling under Knightian Uncertainty are considered and analyzed. We show the need for a suitable adaptation of the notion of No Free Lunch with Vanishing Risk and discuss its relation to the choice of an appropriate filtration. In an abstract setup, we show that absence of arbitrage is equivalent to the existence of \emph{approximate} martingale measures sharing the same polar set of $\mathcal{P}$. We then specialize the results to a discrete-time framework in order to obtain true martingale measures.