A unified Framework for Robust Modelling of Financial Markets in discrete time

TL;DR

This paper unifies pathwise and quasi-sure approaches to robust financial market modeling in discrete time, establishing fundamental theorems that connect different arbitrage notions and superhedging strategies.

Contribution

It provides a unified framework linking two major approaches in robust finance, proving key theorems and clarifying arbitrage concepts.

Findings

Established equivalence between pathwise and quasi-sure models.

Proved a Fundamental Theorem of Asset Pricing in the unified framework.

Demonstrated conditions for extending superhedging from measures to pathwise settings.

Abstract

We unify and establish equivalence between the pathwise and the quasi-sure approaches to robust modelling of financial markets in discrete time. In particular, we prove a Fundamental Theorem of Asset Pricing and a Superhedging Theorem, which encompass the formulations of [Bouchard, B., & Nutz, M. (2015). Arbitrage and duality in nondominated discrete-time models. The Annals of Applied Probability, 25(2), 823-859] and [Burzoni, M., Frittelli, M., Hou, Z., Maggis, M., & Obloj, J. (2019). Pointwise arbitrage pricing theory in discrete time. Mathematics of Operations Research]. In bringing the two streams of literature together, we also examine and relate their many different notions of arbitrage. We also clarify the relation between robust and classical $\mathbb{P}$-specific results. Furthermore, we prove when a superhedging property w.r.t. the set of martingale measures supported on a set of paths $\Omega$ may be extended to a pathwise superhedging on $\Omega$ without changing the superhedging price.