An algorithm for calculating the set of superhedging portfolios in markets with transaction costs

TL;DR

This paper introduces a new algorithm for explicitly calculating superhedging portfolios in markets with transaction costs by reformulating the problem as a sequence of linear vector optimization problems, enabling efficient numerical solutions.

Contribution

The paper's main contribution is establishing a connection between superhedging portfolio calculation and linear vector optimization, facilitating numerical solutions for multi-asset markets with transaction costs.

Findings

The algorithm effectively computes superhedging sets in multi-asset markets.

It relates existing scalar superhedging algorithms to set-valued approaches via duality.

Examples demonstrate the method's application to correlated assets and basket options.

Abstract

We study the explicit calculation of the set of superhedging portfolios of contingent claims in a discrete-time market model for d assets with proportional transaction costs. The set of superhedging portfolios can be obtained by a recursive construction involving set operations, going backward in the event tree. We reformulate the problem as a sequence of linear vector optimization problems and solve it by adapting known algorithms. The corresponding superhedging strategy can be obtained going forward in the tree. Examples are given involving multiple correlated assets and basket options. Furthermore, we relate existing algorithms for the calculation of the scalar superhedging price to the set-valued algorithm by a recent duality theory for vector optimization problems. The main contribution of the paper is to establish the connection to linear vector optimization, which allows to solve numerically multi-asset superhedging problems under transaction costs.