[Paper Review] An algorithm for calculating the set of superhedging portfolios and strategies in markets with transaction costs

Andreas Löhne, Birgit Rudloff|arXiv (Cornell University)|Jul 28, 2011

Stochastic processes and financial applications15 citations

TL;DR

This paper proposes a recursive algorithm based on linear vector optimization to compute the set of superhedging portfolios and strategies for multiple assets with proportional transaction costs in discrete-time models. By reformulating the problem backward in the event tree and solving it via vector optimization, the method enables numerical computation of multi-asset superhedging under transaction costs, with forward reconstruction of strategies and connections to scalar superhedging via duality.

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.

Motivation & Objective

• To develop a numerically tractable method for computing the full set of superhedging portfolios in multi-asset markets with proportional transaction costs.
• To reformulate the superhedging problem as a sequence of linear vector optimization problems for computational feasibility.
• To enable the reconstruction of superhedging strategies by traversing the event tree forward after backward computation.
• To establish a theoretical link between scalar superhedging prices and the proposed set-valued approach using recent duality theory in vector optimization.
• To provide a practical framework for pricing and hedging contingent claims in realistic, transaction cost-affected markets.

Proposed method

• The problem is solved recursively by working backward through the event tree, applying set operations at each node to determine the set of superhedging portfolios.
• Each stage of the backward recursion is formulated as a linear vector optimization problem, leveraging known algorithms for solution.
• The solution at each node involves computing the intersection and Minkowski sum of sets representing future portfolio constraints and asset positions.
• After backward computation, the superhedging strategy is reconstructed forward in the tree using the computed portfolio sets.
• The method uses duality theory for vector optimization to relate the set-valued solution to scalar superhedging prices.
• The approach is validated through examples involving correlated assets and basket options, demonstrating numerical feasibility.

Experimental results

Research questions

• RQ1 How can the set of superhedging portfolios be computed efficiently in a multi-asset market with proportional transaction costs?
• RQ2 What is the connection between set-valued superhedging and scalar superhedging prices in the presence of transaction costs?
• RQ3 Can linear vector optimization techniques be effectively adapted to solve multi-asset superhedging problems under transaction costs?
• RQ4 How can the corresponding superhedging strategy be reconstructed from the computed portfolio sets?
• RQ5 What is the role of duality in vector optimization in linking scalar and set-valued superhedging solutions?

Key findings

• The proposed algorithm enables the numerical computation of the full set of superhedging portfolios for multi-asset derivatives under proportional transaction costs.
• The problem is reformulated as a sequence of linear vector optimization problems, making it amenable to existing solution methods.
• The backward recursion with set operations allows for precise tracking of portfolio constraints across the event tree.
• The forward reconstruction of strategies ensures that the computed portfolios are implementable in practice.
• The connection to scalar superhedging is established through duality theory, validating the set-valued approach.
• Examples with basket options and correlated assets confirm the method’s feasibility and applicability in realistic market settings.