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

Xinyu Chen, Zuo Quan Xu

Abstract

This paper studies an $α$-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 $α$-robust criterion interpolates between worst-case ($α=0$) and best-case ($α=1$) evaluations, generalizing both extremes through a continuous ambiguity attitude parameter. For weighted exponential utilities, we establish via rearrangement inequalities and comonotonicity theory that the $α$-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.

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

Abstract

This paper studies an -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 -robust criterion interpolates between worst-case () and best-case () evaluations, generalizing both extremes through a continuous ambiguity attitude parameter. For weighted exponential utilities, we establish via rearrangement inequalities and comonotonicity theory that the -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.

Paper Structure

This paper contains 21 sections, 11 theorems, 103 equations, 8 figures.

Table of Contents

  1. Introduction
  2. Notations and Conventions.
  3. Problem Formulation
  4. Financial Market Model
  5. Investment Problem
  6. Quantile Formulation
  7. A Static Optimization Problem
  8. Quantile Reformulation
  9. Optimal Solutions to \ref{['static4']} and \ref{['static2']}
  10. Numerical Study
  11. Numerical Methodology.
  12. Model Specification.
  13. Quantile of Pricing Kernel under Different $\theta$
  14. Relationship between Initial Endowment $x$ and Lagrange Multiplier $\lambda$
  15. Optimal Payoff under Different $\theta$
  16. ...and 6 more sections

Key Result

Lemma 3.1

For any $x>0$, the set $\mathscr{A}_x$ satisfies where $\rho$ is the pricing kernel (also called stochastic discount factor) defined by $\blacktriangleleft$$\blacktriangleleft$

Figures (8)

  • Figure 1: Quantile function $Q_\rho(p)$ of the pricing kernel $\rho$ under different market prices of risk $\theta$.
  • Figure 2: Relationship between initial endowment $x$ (horizontal axis) and Lagrange multiplier $\lambda$ (vertical axis).
  • Figure 3: Optimal payoff profiles $\rho \mapsto \overline{Q}(1-F_\rho(\rho))$ under different market prices of risk $\theta$.
  • Figure 4: Optimal payoff profiles $\rho \mapsto \overline{Q}(1-F_\rho(\rho))$ under different initial endowments $x$.
  • Figure 5: Optimal payoff profiles $\rho \mapsto \overline{Q}(1-F_\rho(\rho))$ under different utility function parameters $\gamma_1$, $\gamma_2$.
  • ...and 3 more figures

Theorems & Definitions (19)

  • Lemma 3.1
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4: Monotonicty and Concavity of $\mathcal{J}_\alpha$
  • proof
  • Proposition 3.5
  • proof
  • Lemma 3.6
  • proof
  • ...and 9 more