Examples and Counterexamples of Cost-efficiency in Incomplete Markets

TL;DR

This paper investigates cost-efficiency of claims in incomplete markets using a tractable 3-state model to illuminate and test BS24 results, and to expose the sharpness of its assumptions. It recasts cost-efficiency as a distributional problem over the convex closure $\overline{\mathrm{conv}}(F)$ with cost $c(Z)=\sup_{\xi\in\Xi} \mathbb{E}[\xi Z]$, and analyzes the minimax, maximin, and their convexified variants, linking discrete trinomial insights to duality in continuous settings. A key contribution is the explicit characterization of perfectly cost-efficient distributions (notably when $z=3y-2x$ in a 3-state setup) and the demonstration that minimax and maximin can diverge unless perfect cost-efficiency holds; the results also show how expected utility maximization in incomplete markets can be solved in this framework and how convex order governs superhedging costs. Overall, the work clarifies limitations and sharpness of prior results (BS24), provides concrete, tractable examples for incomplete-market pricing and hedging, and connects distributional properties to pricing kernels and optimization under uncertainty.

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.

Figures (3)

• Figure 1: Illustration of the fact that distributional superhedging costs decrease with convex order: Superhedging costs of normal distributions with fixed means as functions of the variance in the stochastic volatility model of Example \ref{['coexa']}.
• Figure 2: Illustration of the solutions to the 3-states example in the cases $2x-3y+z>0$ (top, $x=1,\, y=2, \, z=5$) $2x-3y+z=0$ (middle, the cost-efficient case, $x=1,\, y=2, \, z=4$) and $2x-3y+z<0$ (bottom, $x=1,\, y=2, \, z=3$). Convexified Minimax (red), minimax (blue), convexified maximin (green) and maximin (black) problems.
• Figure 3: Illustration of the KKM argument in BS24 for the 3-state model with $x=1$, $y=2$ and $z=3$.

Theorems & Definitions (11)

• Remark 2.1
• Remark 2.2
• Example 2.3
• Proposition 4.1
• proof
• Example 4.2
• Example 4.3
• Corollary 5.1
• proof
• Remark 5.2