Portfolio Optimization under Convex Incentive Schemes

TL;DR

This paper investigates a portfolio optimization problem where a manager's incentive scheme affects the utility function, using duality theory to establish existence and uniqueness of optimal portfolios in complex market models.

Contribution

It extends classical portfolio optimization by handling convex incentive schemes and proves conditions for the existence and uniqueness of solutions using duality, even in incomplete markets.

Findings

Existence and uniqueness of optimal portfolios are established under broad conditions.

Results are largely independent of the specific incentive scheme in many models.

Analysis includes complete, incomplete, and stochastic volatility market models.

Abstract

We consider the terminal wealth utility maximization problem from the point of view of a portfolio manager who is paid by an incentive scheme, which is given as a convex function $g$ of the terminal wealth. The manager's own utility function $U$ is assumed to be smooth and strictly concave, however the resulting utility function $U \circ g$ fails to be concave. As a consequence, the problem considered here does not fit into the classical portfolio optimization theory. Using duality theory, we prove wealth-independent existence and uniqueness of the optimal portfolio in general (incomplete) semimartingale markets as long as the unique optimizer of the dual problem has a continuous law. In many cases, this existence and uniqueness result is independent of the incentive scheme and depends only on the structure of the set of equivalent local martingale measures. As examples, we discuss (complete) one-dimensional models as well as (incomplete) lognormal mixture and popular stochastic volatility models. We also provide a detailed analysis of the case where the unique optimizer of the dual problem does not have a continuous law, leading to optimization problems whose solvability by duality methods depends on the initial wealth of the investor.