Superhedging in illiquid markets

TL;DR

This paper develops a dual framework for superhedging in discrete-time markets with convex trading costs and constraints, extending classical models to include nonlinear illiquidity effects and no cash account scenarios.

Contribution

It introduces dual characterizations of superhedging conditions in markets with convex costs and constraints, covering nonlinear illiquidity effects and markets without a cash account.

Findings

Derived dual representations valid under no arbitrage conditions.

Extended superhedging theory to markets with nonlinear costs.

Provided alternative conditions for market models with constraints.

Abstract

We study contingent claims in a discrete-time market model where trading costs are given by convex functions and portfolios are constrained by convex sets. In addition to classical frictionless markets and markets with transaction costs or bid-ask spreads, our framework covers markets with nonlinear illiquidity effects for large instantaneous trades. We derive dual characterizations of superhedging conditions for contingent claim processes in a market without a cash account. The characterizations are given in terms of stochastic discount factors that correspond to martingale densities in a market with a cash account. The dual representations are valid under a topological condition and a weak consistency condition reminiscent of the ``law of one price'', both of which are implied by the no arbitrage condition in the case of classical perfectly liquid market models. We give alternative sufficient conditions that apply to market models with nonlinear cost functions and portfolio constraints.