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

2025-06-25

AI Summary

This paper studies the optimal investment problem for a constant relative risk aversion (CRRA) investor under the Black–Scholes framework when facing stochastic endowments. Traditional approaches struggle with the resulting non-Markovian structure induced by endowment uncertainty. To overcome this, we develop a novel dual-methodology framework that yields an explicit decomposition of the optimal portfolio strategy into two components: (i) the benchmark Merton solution—corresponding to the no-endowment case—and (ii) a linear–exponential adjustment term driven solely by the endowment-to-wealth ratio. This decomposition transparently uncovers the time-varying mechanism through which stochastic endowments influence dynamic asset allocation. The resulting closed-form solution is both analytically tractable and economically interpretable, providing a rigorous yet practical theoretical foundation for investment decisions in realistic settings involving exogenous income, labor endowments, or insurance payouts.

Technology Category

• Reasoning under Uncertainty: Stochastic Optimization
• Search and Optimization: Mixed Discrete/Continuous Search
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Economics, Online Markets and Human Computation: The sharing economy
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Browser and end-point security

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.

Problem

• Optimal investment with random endowment in Black-Scholes market
• Explicit solution for trading strategy with risk aversion
• Duality-based decomposition into baseline strategy and additive shift

Innovation

• Duality arguments derive optimal trading strategy
• Strategy decomposes into base and additive shift
• Shift depends linearly on endowment-to-wealth ratio