$\alpha$-robust utility maximization with intractable claims: A quantile optimization approach

TL;DR

This paper introduces a novel quantile optimization approach for $lpha$-robust utility maximization with intractable claims, transforming a complex dynamic problem into a static, numerically solvable form.

Contribution

It develops a law-invariant, static quantile optimization framework for $lpha$-robust utility maximization, including optimality conditions and numerical methods.

Findings

Optimal payoffs depend on ambiguity attitude and market conditions.

The framework accommodates risk constraints like VaR and Expected Shortfall.

Numerical experiments illustrate the impact of ambiguity on optimal strategies.

Abstract

This paper studies an $\alpha$-robust utility maximization problem where an investor faces an intractable claim -- an exogenous contingent claim with known marginal distribution but unspecified dependence structure with financial market returns. The $\alpha$-robust criterion interpolates between worst-case ($\alpha=0$) and best-case ($\alpha=1$) evaluations, generalizing both extremes through a continuous ambiguity attitude parameter. For weighted exponential utilities, we establish via rearrangement inequalities and comonotonicity theory that the $\alpha$-robust risk measure is law-invariant, depending only on marginal distributions. This transforms the dynamic stochastic control problem into a concave static quantile optimization over a convex domain. We derive optimality conditions via calculus of variations and characterize the optimal quantile as the solution to a two-dimensional first-order ordinary differential equation system, which is a system of variational inequalities with mixed boundary conditions, enabling numerical solution. Our framework naturally accommodates additional risk constraints such as Value-at-Risk and Expected Shortfall. Numerical experiments reveal how ambiguity attitude, market conditions, and claim characteristics interact to shape optimal payoffs.