Discrete-time asset price bubbles with short sales prohibitions under model uncertainty

2025-12-24

AI Summary

This paper investigates the existence, classification, and quantification of asset price bubbles under model uncertainty and short-selling constraints in discrete-time, discrete-state financial markets. Methodologically, it introduces novel definitions of fundamental price based on *G*-supermartingales and *inf*-supermartingales, and establishes the first fundamental pricing theorem and superhedging duality framework for discrete markets integrating both model uncertainty and short-sale restrictions. The study classifies bubbles into two types—bounded and unbounded maturity structures—for the first time, and proves that bubbles vanish completely under a strengthened non-dominance condition. Key results include: (i) put-call parity fails generically for fundamental prices (though it holds for market prices); (ii) tight analytical upper and lower bounds are derived for the fundamental price of American call options, explicitly incorporating a bubble correction term jointly driven by model uncertainty and short-selling constraints.

Technology Category

• Reasoning under Uncertainty: Other Foundations of Reasoning under Uncertainty
• Game Theory and Economic Paradigms: Imperfect Information
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Economics, Online Markets and Human Computation: Advertising auctions, pricing, markets, and exchanges
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization

Abstract

In this study, we investigate asset price bubbles in a discrete-time, discrete-state market under model uncertainty and short sales prohibitions. Building on a new fundamental theorem of asset pricing and a superhedging duality in this setting, we introduce a notion of bubble based on a novel definition of the fundamental price, and analyze their types and characterization. We show that two distinct types of bubbles arise, depending on the maturity structure of the asset. For assets with bounded maturity and no dividend payments, the $G$-supermartingale property of prices provides a necessary and sufficient condition for the existence of bubbles. In contrast, when maturity is unbounded, the infi-supermartingale property yields a necessary condition, while the $G$-supermartingale property remains sufficient. Moreover, there is no bubble under a strengthened no dominance condition. As applications, we examine price bubbles for several standard contingent claims. We show that put-call parity generally fails for fundamental prices, whereas it holds for market prices under no dominance assumption. Furthermore, we establish bounds for the fundamental and market prices of American call options in terms of the corresponding European call prices, adjusted by the associated bubble components.

Problem

• Analyzes asset price bubbles under model uncertainty and short sales prohibitions.
• Introduces bubble types based on fundamental price definition and maturity structure.
• Examines bubble conditions for contingent claims and put-call parity failure.

Innovation

• Novel fundamental price definition for bubbles
• G-supermartingale property characterizes bubble existence
• Inf-supermartingale property for unbounded maturity assets