Portfolio optimisation under non-linear drawdown constraints in a semimartingale financial model

TL;DR

This paper addresses portfolio optimization with non-linear drawdown constraints in a general semimartingale market, transforming it into an unconstrained problem with a modified utility, and providing explicit solutions.

Contribution

It introduces a method to solve constrained portfolio optimization by converting it into an unconstrained problem with a modified utility function in a semimartingale setting.

Findings

Explicit formulas for the value function and optimal policy.

Equivalence between constrained and unconstrained problems.

General utility functions and drawdown constraints handled.

Abstract

A drawdown constraint forces the current wealth to remain above a given function of its maximum to date. We consider the portfolio optimisation problem of maximising the long-term growth rate of the expected utility of wealth subject to a drawdown constraint, as in the original setup of Grossman and Zhou (1993). We work in an abstract semimartingale financial market model with a general class of utility functions and drawdown constraints. We solve the problem by showing that it is in fact equivalent to an unconstrained problem with a suitably modified utility function. Both the value function and the optimal investment policy for the drawdown problem are given explicitly in terms of their counterparts in the unconstrained problem.