Examples and Counterexamples of Cost-efficiency in Incomplete Markets

ArXiv ID: 2407.08756 “View on arXiv”

Authors: Unknown

Abstract

We present a number of examples and counterexamples to illustrate the results on cost-efficiency in an incomplete market obtained in [“BS24”]. These examples and counterexamples do not only illustrate the results obtained in [“BS24”], but show the limitations of the results and the sharpness of the key assumptions. In particular, we make use of a simple 3-state model in which we are able to recover and illustrate all key results of the paper. This example also shows how our characterization of perfectly cost-efficient claims allows to solve an expected utility maximization problem in a simple incomplete market (trinomial model) and recover results from [“DS06, Chapter 3”], there obtained using duality.

Keywords: Incomplete markets, Cost-efficiency, Utility maximization, Trinomial model, Financial derivatives, Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 1.0/10
• Quadrant: Lab Rats
• Why: The paper presents advanced mathematical derivations using convex analysis and optimization theory in incomplete markets, but provides only theoretical examples without any backtesting, datasets, or implementation details.

  flowchart TD
    A["Research Goal:<br>Illustrate cost-efficiency<br>in incomplete markets"] --> B["Core Methodology:<br>3-state/Trinomial Model"]
    B --> C["Data/Input:<br>Key Assumptions<br>from [BS24"]]
    C --> D{"Computational Process:<br>Construct Examples<br>& Counterexamples"}
    D --> E{"Analyze Assumptions<br>for Sharpness & Limitations"}
    E --> F["Outcome 1:<br>Validated [BS24"] Results]
    E --> G["Outcome 2:<br>Solved EU Max Problem<br>via Cost-efficiency"]
    F & G --> H["Final Outcome:<br>Connected [BS24"] to<br>["DS06"] Duality Results]