Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On monotone completion of risk markets: Limit results for incomplete risk markets

Iman Khajepour, Geoffrey Pritchard, Danny Ralph, Golbon Zakeri

TL;DR

This work develops and analyzes an iterative framework for completing incomplete risk markets by refining bundles of traded instruments. By establishing an equivalence between risk-market equilibria and social welfare maximization, it proves that refining bundles monotonically improves welfare in finite-discrete settings and leads to convergence toward the complete-market outcome in countably infinite and general probability spaces. The authors leverage Zame's asymptotic-completeness results and duality theory for coherent risk measures to show that, under successive refinements, risk prices converge to a single, consistent pricing rule and that Arrow-Debreu securities can be uniformly approximated by finite bundles. The results are illustrated with examples using CVaR-based risk measures and Poisson-distributed futures, highlighting practical pathways for achieving welfare-enhancing market completion in diverse risk settings. Overall, the paper provides a rigorous, monotone-approach to market completion with convergence guarantees for risk pricing across finite, countable, and arbitrary probability spaces.

Abstract

We consider a competitive market with risk-averse participants. We assume that agents' risks are measured by coherent risk measures introduced by Artzner et al. (1999). Fundamental theorems of welfare economics have long established the equivalence of competitive equilibria and system welfare optimization (see, e.g., Samuelson (1947)). These have been extended to the case of risk-averse agents with complete risk markets in Ralph and Smeers (2015). In this paper, we consider risk trading in incomplete markets and introduce a mechanism to complete the market iteratively while monotonically enhancing welfare.

On monotone completion of risk markets: Limit results for incomplete risk markets

TL;DR

This work develops and analyzes an iterative framework for completing incomplete risk markets by refining bundles of traded instruments. By establishing an equivalence between risk-market equilibria and social welfare maximization, it proves that refining bundles monotonically improves welfare in finite-discrete settings and leads to convergence toward the complete-market outcome in countably infinite and general probability spaces. The authors leverage Zame's asymptotic-completeness results and duality theory for coherent risk measures to show that, under successive refinements, risk prices converge to a single, consistent pricing rule and that Arrow-Debreu securities can be uniformly approximated by finite bundles. The results are illustrated with examples using CVaR-based risk measures and Poisson-distributed futures, highlighting practical pathways for achieving welfare-enhancing market completion in diverse risk settings. Overall, the paper provides a rigorous, monotone-approach to market completion with convergence guarantees for risk pricing across finite, countable, and arbitrary probability spaces.

Abstract

We consider a competitive market with risk-averse participants. We assume that agents' risks are measured by coherent risk measures introduced by Artzner et al. (1999). Fundamental theorems of welfare economics have long established the equivalence of competitive equilibria and system welfare optimization (see, e.g., Samuelson (1947)). These have been extended to the case of risk-averse agents with complete risk markets in Ralph and Smeers (2015). In this paper, we consider risk trading in incomplete markets and introduce a mechanism to complete the market iteratively while monotonically enhancing welfare.

Paper Structure

This paper contains 9 sections, 7 theorems, 50 equations, 4 figures.

Table of Contents

  1. Introduction and background
  2. Incomplete Markets of Bundled Trade Instruments: finite discrete case
  3. Refining Bundles
  4. Incomplete Markets of Bundled Instruments: discrete distributions with countably infinite states
  5. Incomplete Markets of Bundled Trade Instruments: arbitrary probability space
  6. Preliminaries
  7. The social optimization problem
  8. Convergence of risk prices
  9. Proof of Theorem \ref{['th:prices_are_usc']}.

Key Result

Theorem 1

Consider the incomplete market set up ${\mathcal{M}_J}$ where $J$ is a partition of the scenario set $S$. Let $J_1 \cup J_2 =J$ be a refinement of the partition $J$, that is for each $S_j \in J$, $S_j = S_{j_1} \cup S_{j_2}$, where $S_{j_1} \cap S_{j_2} = \emptyset$. Then the incomplete market ${\ma

Figures (4)

  • Figure 1: Improvement of social welfare through bundle refinement. This figure illustrates the progression of incomplete markets towards the complete market social optimum and how refinement results in a non-decreasing trend.
  • Figure 2: Evolution of risk prices as bundles refine.
  • Figure 3: Progression of risk price with bundle refinement.
  • Figure 4: Price progression of the bundle containing scenario 64 across refinement iterations.

Theorems & Definitions (15)

  • Definition 1
  • Theorem 1
  • proof : proof
  • Proposition 1
  • proof
  • Definition 2
  • Theorem 2
  • Theorem 3
  • proof : Proof of Theorem \ref{['th:zy_thm']}.
  • Definition 3
  • ...and 5 more