Multistage Portfolio Optimization: A Duality Result in Conic Market Models

TL;DR

This paper establishes a duality framework for multi-stage portfolio optimization in markets with transaction costs, using set optimization to handle vector-valued portfolios without requiring liquidation into a numeraire.

Contribution

It introduces a duality result for multi-stage portfolio optimization in conic market models, extending the theory to vector-valued portfolios under partial ordering.

Findings

Proves a strong duality relationship in conic market models.

Embeds portfolio optimization into set-optimization framework.

Applicable to markets with proportional transaction costs.

Abstract

We prove a general duality result for multi-stage portfolio optimization problems in markets with proportional transaction costs. The financial market is described by Kabanov's model of foreign exchange markets over a finite probability space and finite-horizon discrete time steps. This framework allows us to compare vector-valued portfolios under a partial ordering, so that our model does not require liquidation into some numeraire at terminal time. We embed the vector-valued portfolio problem into the set-optimization framework, and generate a problem dual to portfolio optimization. Using recent results in the development of set optimization, we then show that a strong duality relationship holds between the problems.