Global Closed-form Approximation of Free Boundary for Optimal Investment Stopping Problems

TL;DR

This paper develops a global closed-form approximation for the free boundary in optimal investment stopping problems, significantly reducing computational costs while maintaining accuracy for various utility functions.

Contribution

It introduces a novel closed-form approximation for the dual free boundary in a complex utility maximization problem involving optimal stopping.

Findings

The approximation is robust and accurate across different utility functions.

It significantly reduces computational time compared to traditional methods.

Numerical results confirm the effectiveness of the approximation.

Abstract

In this paper we study a utility maximization problem with both optimal control and optimal stopping in a finite time horizon. The value function can be characterized by a variational equation that involves a free boundary problem of a fully nonlinear partial differential equation. Using the dual control method, we derive the asymptotic properties of the dual value function and the associated dual free boundary for a class of utility functions, including power and non-HARA utilities. We construct a global closed-form approximation to the dual free boundary, which greatly reduces the computational cost. Using the duality relation, we find the approximate formulas for the optimal value function, trading strategy, and exercise boundary for the optimal investment stopping problem. Numerical examples show the approximation is robust, accurate and fast.