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

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.