Examples and Counterexamples of Cost-efficiency in Incomplete Markets

TL;DR

This paper provides examples and counterexamples in a simple 3-state incomplete market to illustrate and test the limits of existing results on cost-efficiency, highlighting the assumptions' sharpness and applications to utility maximization.

Contribution

It introduces a simple 3-state model to demonstrate the limits of cost-efficiency results and connects these to utility maximization in incomplete markets.

Findings

Illustrates limitations of cost-efficiency results

Shows how to solve utility maximization using cost-efficiency characterization

Provides a simple example that recovers duality results from prior work

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.