Generalized Duality for Model-Free Superhedging given Marginals

TL;DR

This paper develops a generalized duality framework for model-free superhedging in discrete-time markets, accommodating upper semi-analytic claims and using measure sequences converging to martingales for pricing.

Contribution

It introduces a novel duality result that relaxes regularity assumptions on claims and extends risk-neutral pricing to a broader class of functions.

Findings

Established a portfolio-constrained duality for upper semi-analytic claims

Derived the generalized duality through probabilistic estimations and measure convergence

Extended superhedging duality beyond upper semicontinuous claims

Abstract

In a discrete-time financial market, a generalized duality is established for model-free superhedging, given marginal distributions of the underlying asset. Contrary to prior studies, we do not require contingent claims to be upper semicontinuous, allowing for upper semi-analytic ones. The generalized duality stipulates an extended version of risk-neutral pricing. To compute the model-free superhedging price, one needs to find the supremum of expected values of a contingent claim, evaluated not directly under martingale (risk-neutral) measures, but along sequences of measures that converge, in an appropriate sense, to martingale ones. To derive the main result, we first establish a portfolio-constrained duality for upper semi-analytic contingent claims, relying on Choquet's capacitability theorem. As we gradually fade out the portfolio constraint, the generalized duality emerges through delicate probabilistic estimations.