Collective completeness and pricing-hedging duality II

2026-06-17

AI Summary

This study addresses collective pricing and hedging when risk exchanges form a finitely generated convex cone, extending the classical Fundamental Theorem of Asset Pricing. By introducing the notion of collective strong replicability and integrating tools from functional analysis, convex analysis, and equivalent martingale measure theory, the work establishes a novel framework accommodating infinite-dimensional trading opportunities and finite-dimensional convex-cone exchanges. Key contributions include proving that collective no-arbitrage implies closedness of the aggregate feasibility cone, characterizing the set of collective prices as a relatively open convex set, providing equivalent conditions for collective and strong completeness, and demonstrating that when the exchange structure is merely a convex cone rather than a vector space, the pricing–hedging duality undergoes a fundamental transformation—thereby yielding an enhanced version of the Collective Second Fundamental Theorem of Asset Pricing.

Abstract

This paper complements and extends Doldi, Frittelli and Maggis, Collective completeness and pricing-hedging duality, Math. Finan. Econ. 19, 757-784 (2025), by studying collective pricing and hedging when admissible risk exchanges form a finitely generated convex cone. The collective First Fundamental Theorem of Asset Pricing and the collective pricing-hedging duality are extended to this setting. A key contribution is a closedness result showing that no collective arbitrage implies the closedness of the aggregate feasibility cone combining infinite-dimensional trading opportunities with finite-dimensional exchanges. The paper also proves that no-collective-arbitrage prices for vectors of contingent claims form a relatively open convex set. Finally, strong collective replicability is introduced and shown to be equivalent to price uniqueness. This leads to an enhanced collective Second Fundamental Theorem of Asset Pricing, providing equivalent characterizations of collective completeness and strong collective completeness in terms of the uniqueness of the collective equivalent martingale measure. We highlight that several core aspects of the theory are substantially altered when exchanges belong to a convex cone rather than a vector space.

Problem

• collective completeness
• pricing-hedging duality
• convex cone
• collective arbitrage
• equivalent martingale measure

Innovation

• collective completeness
• pricing-hedging duality
• convex cone
• no-collective-arbitrage
• equivalent martingale measure