Portfolio Optimization under Shortfall Risk Constraint

TL;DR

This paper develops a method for portfolio optimization that maximizes utility while respecting shortfall risk constraints, applicable in various market settings, and provides explicit solutions in specific models.

Contribution

It introduces a duality-based approach for utility maximization under shortfall risk constraints, extending to complex cases including consumption and explicit solutions in Black-Scholes markets.

Findings

Optimal wealth process characterized under constraints

Duality relation between value functions established

Explicit solutions provided for specific market models

Abstract

This paper solves a utility maximization problem under utility-based shortfall risk constraint, by proposing an approach using Lagrange multiplier and convex duality. Under mild conditions on the asymptotic elasticity of the utility function and the loss function, we find an optimal wealth process for the constrained problem and characterize the bi-dual relation between the respective value functions of the constrained problem and its dual. This approach applies to both complete and incomplete markets. Moreover, the extension to more complicated cases is illustrated by solving the problem with a consumption process added. Finally, we give an example of utility and loss functions in the Black-Scholes market where the solutions have explicit forms.