S-Shaped Utility

Stochastic Dominance Constrained Optimization with S-shaped Utilities: Poor-Performance-Region Algorithm and Neural Network ArXiv ID: 2512.00299 “View on arXiv” Authors: Zeyun Hu, Yang Liu Abstract We investigate the static portfolio selection problem of S-shaped and non-concave utility maximization under first-order and second-order stochastic dominance (SD) constraints. In many S-shaped utility optimization problems, one should require a liquidation boundary to guarantee the existence of a finite concave envelope function. A first-order SD (FSD) constraint can replace this requirement and provide an alternative for risk management. We explicitly solve the optimal solution under a general S-shaped utility function with a first-order stochastic dominance constraint. However, the second-order SD (SSD) constrained problem under non-concave utilities is difficult to solve analytically due to the invalidity of Sion’s maxmin theorem. For this sake, we propose a numerical algorithm to obtain a plausible and sub-optimal solution for general non-concave utilities. The key idea is to detect the poor performance region with respect to the SSD constraints, characterize its structure and modify the distribution on that region to obtain (sub-)optimality. A key financial insight is that the decision maker should follow the SD constraint on the poor performance scenario while conducting the unconstrained optimal strategy otherwise. We provide numerical experiments to show that our algorithm effectively finds a sub-optimal solution in many cases. Finally, we develop an algorithm-guided piecewise-neural-network framework to learn the solution of the SSD problem, which demonstrates accelerated convergence compared to standard neural network approaches. ...

Deep Learning Methods for S Shaped Utility Maximisation with a Random Reference Point ArXiv ID: 2410.05524 “View on arXiv” Authors: Unknown Abstract We consider the portfolio optimisation problem where the terminal function is an S-shaped utility applied at the difference between the wealth and a random benchmark process. We develop several numerical methods for solving the problem using deep learning and duality methods. We use deep learning methods to solve the associated Hamilton-Jacobi-Bellman equation for both the primal and dual problems, and the adjoint equation arising from the stochastic maximum principle. We compare the solution of this non-concave problem to that of concavified utility, a random function depending on the benchmark, in both complete and incomplete markets. We give some numerical results for power and log utilities to show the accuracy of the suggested algorithms. ...

On the optimal design of a new class of proportional portfolio insurance strategies in a jump-diffusion framework ArXiv ID: 2407.21148 “View on arXiv” Authors: Unknown Abstract In this paper, we investigate an optimal investment problem associated with proportional portfolio insurance (PPI) strategies in the presence of jumps in the underlying dynamics. PPI strategies enable investors to mitigate downside risk while still retaining the potential for upside gains. This is achieved by maintaining an exposure to risky assets proportional to the difference between the portfolio value and the present value of the guaranteed amount. While PPI strategies are known to be free of downside risk in diffusion modeling frameworks with continuous trading, see e.g., Cont and Tankov (2009), real market applications exhibit a significant non-negligible risk, known as gap risk, which increases with the multiplier value. The goal of this paper is to determine the optimal PPI strategy in a setting where gap risk may occur, due to downward jumps in the asset price dynamics. We consider a loss-averse agent who aims at maximizing the expected utility of the terminal wealth exceeding a minimum guarantee. Technically, we model agent’s preferences with an S-shaped utility functions to accommodate the possibility that gap risk occurs, and address the optimization problem via a generalization of the martingale approach that turns to be valid under market incompleteness in a jump-diffusion framework. ...