S-shaped Utility Maximization with VaR Constraint and Partial Information

Area of Science:

• Quantitative Finance
• Mathematical Economics
• Decision Theory

Background:

• Standard utility maximization models often assume known parameters and lack risk constraints.
• Incorporating Value at Risk (VaR) constraints and unobservable parameters presents significant analytical challenges.
• S-shaped utility functions capture realistic, non-monotonic risk preferences, but complicate optimization.

Purpose of the Study:

• To develop a framework for S-shaped utility maximization subject to a VaR constraint and an unobservable drift coefficient.
• To derive conditions for the existence and uniqueness of optimal solutions and Lagrange multipliers.
• To propose and evaluate computational methods for solving this complex financial optimization problem.

Main Methods:

• Bayesian filtering to handle the unobservable drift coefficient.
• Concavification principle to manage the non-concave S-shaped utility function.
• Change of measure techniques for risk-neutral pricing and duality.
• Semi-closed integral representation for the dual value function.

Main Results:

• A critical wealth level is identified, determining the feasibility and uniqueness of the constrained optimization problem.
• A semi-closed integral form for the dual value function is derived.
• The study establishes conditions for the existence of a unique optimal solution and Lagrange multiplier.

Conclusions:

• The proposed framework provides a robust method for analyzing optimal investment under complex utility preferences and risk constraints.
• The identified critical wealth level offers practical insights for portfolio management and risk assessment.
• The comparison of Lagrange, simulation, and deep neural network algorithms demonstrates the tractability of the problem with modern computational techniques.