Shadow price in the power utility case

TL;DR

This paper extends the concept of shadow prices to the power utility maximization problem in a Black-Scholes market with transaction costs, providing a frictionless equivalent that simplifies analysis.

Contribution

It derives a shadow price process for power utility maximization with proportional transaction costs, generalizing previous work on utility maximization in such markets.

Findings

Derivation of a shadow price process within the bid-ask spread.

Equivalence of the shadow price approach to the original problem.

Application to infinite horizon power utility optimization.

Abstract

We consider the problem of maximizing expected power utility from consumption over an infinite horizon in the Black-Scholes model with proportional transaction costs, as studied in Shreve and Soner [Ann. Appl. Probab. 4 (1994) 609-692]. Similar to Kallsen and Muhle-Karbe [Ann. Appl. Probab. 20 (2010) 1341-1358], we derive a shadow price, that is, a frictionless price process with values in the bid-ask spread which leads to the same optimal policy.