Singular Perturbation Expansion for Utility Maximization with Order-$\epsilon$ Quadratic Transaction Costs

TL;DR

This paper develops an asymptotic expansion method for solving portfolio optimization problems with small quadratic transaction costs, providing explicit formulas and analyzing their behavior.

Contribution

It introduces a novel asymptotic expansion approach for the Hamilton-Jacobi-Bellman equation in the context of small quadratic transaction costs, deriving explicit first two terms.

Findings

Explicit formulas for the first two terms of the expansion

Analysis and simulation demonstrating the approximation's behavior

Abstract

We present an expansion for portfolio optimization in the presence of small, instantaneous, quadratic transaction costs. Specifically, the magnitude of transaction costs has a coefficient that is of the order $\epsilon$ small, which leads to the optimization problem having an asymptotically-singular Hamilton-Jacobi-Bellman equation whose solution can be expanded in powers of $\sqrt\epsilon$. In this paper we derive explicit formulae for the first two terms of this expansion. Analysis and simulation are provided to show the behavior of this approximating solution.