An Explicit Solution for the Problem of Optimal Investment with Random Endowment

TL;DR

This paper derives an explicit formula for the optimal investment strategy in a Black-Scholes market considering random endowment, providing a clear decomposition into baseline strategy and adjustment term.

Contribution

It introduces a novel explicit solution for the optimal investment problem with random endowment using duality, enhancing understanding of investment adjustments.

Findings

Explicit optimal strategy formula derived

Decomposition into baseline and adjustment components

Linear and exponential dependence on endowment ratio and time

Abstract

We consider the problem of optimal investment with random endowment in a Black--Scholes market for an agent with constant relative risk aversion. Using duality arguments, we derive an explicit expression for the optimal trading strategy, which can be decomposed into the optimal strategy in the absence of a random endowment and an additive shift term whose magnitude depends linearly on the endowment-to-wealth ratio and exponentially on time to maturity.