Model-independent Superhedging under Portfolio Constraints

TL;DR

This paper develops a model-independent superhedging framework in discrete-time markets with portfolio constraints, using optimal transport theory to establish duality and connect to convex risk measures, covering various Delta and Gamma constraints.

Contribution

It introduces a general superhedging duality under portfolio constraints using optimal transport, extending the fundamental theorem of asset pricing in a model-independent setting.

Findings

Established a superhedging duality using optimal transport theory.

Connected superhedging to convex risk measures.

Covered a broad class of Delta and Gamma constraints.

Abstract

In a discrete-time market, we study model-independent superhedging, while the semi-static superhedging portfolio consists of {\it three} parts: static positions in liquidly traded vanilla calls, static positions in other tradable, yet possibly less liquid, exotic options, and a dynamic trading strategy in risky assets under certain constraints. By considering the limit order book of each tradable exotic option and employing the Monge-Kantorovich theory of optimal transport, we establish a general superhedging duality, which admits a natural connection to convex risk measures. With the aid of this duality, we derive a model-independent version of the fundamental theorem of asset pricing. The notion "finite optimal arbitrage profit", weaker than no-arbitrage, is also introduced. It is worth noting that our method covers a large class of Delta constraints as well as Gamma constraint.