Utility Maximisation with Model-independent Constraints

ArXiv ID: 2512.24371 “View on arXiv”

Authors: Alexander M. G. Cox, Daniel Hernandez-Hernandez

Abstract

We consider an agent who has access to a financial market, including derivative contracts, who looks to maximise her utility. Whilst the agent looks to maximise utility over one probability measure, or class of probability measures, she must also ensure that the mark-to-market value of her portfolio remains above a given threshold. When the mark-to-market value is based on a more pessimistic valuation method, such as model-independent bounds, we recover a novel optimisation problem for the agent where the agents investment problem must satisfy a pathwise constraint. For complete markets, the expression of the optimal terminal wealth is given, using the max-plus decomposition for supermartingales. Moreover, for the Black-Scholes-Merton model the explicit form of the process involved in such decomposition is obtained, and we are able to investigate numerically optimal portfolios in the presence of options which are mispriced according to the agent’s beliefs.

Keywords: Utility Maximization, Derivatives Pricing, Supermartingales, Model-Independent Valuation, Optimal Portfolio, Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 3.0/10
• Quadrant: Lab Rats
• Why: The paper employs advanced mathematical techniques including utility theory, max-plus decomposition for supermartingales, and model-independent bounds, leading to high math complexity. However, it focuses on theoretical derivations and explicit solutions in the Black-Scholes model without presenting backtests, datasets, or implementation-heavy empirical results, resulting in low empirical rigor.

  flowchart TD
    A["Research Goal: Maximize utility subject to<br>model-independent mark-to-market constraints"] --> B["Methodology: Combine utility<br>maximization with supermartingale<br>decomposition for pathwise constraints"]
    B --> C["Data/Inputs: Agent's utility function,<br>market prices, model-independent<br>derivative bounds, pathwise wealth constraints"]
    C --> D["Computational Process 1: Derive optimal<br>terminal wealth for complete markets<br>via max-plus supermartingale decomposition"]
    C --> E["Computational Process 2: Apply explicit<br>decomposition in Black-Scholes-Merton model<br>to compute optimal portfolios"]
    D & E --> F["Key Findings: Explicit solution for<br>optimal wealth; numerical investigation<br>of optimal portfolios with mispriced options"]