On using shadow prices in portfolio optimization with transaction costs

TL;DR

This paper demonstrates how shadow prices can be effectively used in portfolio optimization with transaction costs, providing a dual approach for deriving and verifying solutions in Merton's problem.

Contribution

It shows that the dual approach with shadow prices can be used for both deriving and verifying solutions in portfolio optimization with transaction costs, including determining the shadow price process.

Findings

Shadow price process can be explicitly determined.

Dual approach applies to Merton's problem with transaction costs.

Verification of solutions using shadow prices is feasible.

Abstract

In frictionless markets, utility maximization problems are typically solved either by stochastic control or by martingale methods. Beginning with the seminal paper of Davis and Norman [Math. Oper. Res. 15 (1990) 676--713], stochastic control theory has also been used to solve various problems of this type in the presence of proportional transaction costs. Martingale methods, on the other hand, have so far only been used to derive general structural results. These apply the duality theory for frictionless markets typically to a fictitious shadow price process lying within the bid-ask bounds of the real price process. In this paper, we show that this dual approach can actually be used for both deriving a candidate solution and verification in Merton's problem with logarithmic utility and proportional transaction costs. In particular, we determine the shadow price process.