Optimal Investment with an Unbounded Random Endowment and Utility-Based Pricing

TL;DR

This paper addresses optimal investment strategies for agents with unbounded endowments and utility functions supporting both gains and losses, providing existence results, arbitrage conditions, and new insights into utility-based pricing methods.

Contribution

It establishes the existence of optimal strategies under broad conditions and characterizes utility-based prices, including marginal utility-based and indifference prices, in complex unbounded settings.

Findings

Existence of optimal trading strategies under supermartingale constraints

Characterization of utility-based price processes as local martingales

New results on continuity and asymptotics of utility indifference prices

Abstract

This paper studies the problem of maximizing the expected utility of terminal wealth for a financial agent with an unbounded random endowment, and with a utility function which supports both positive and negative wealth. We prove the existence of an optimal trading strategy within a class of permissible strategies -- those strategies whose wealth process is a supermartingale under all pricing measures with finite relative entropy. We give necessary and sufficient conditions for the absence of utility-based arbitrage, and for the existence of a solution to the primal problem. We consider two utility-based methods which can be used to price contingent claims. Firstly we investigate marginal utility-based price processes (MUBPP's). We show that such processes can be characterized as local martingales under the normalized optimal dual measure for the utility maximizing investor. Finally, we present some new results on utility indifference prices, including continuity properties and volume asymptotics for the case of a general utility function, unbounded endowment and unbounded contingent claims.