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

TL;DR

This paper analyzes asset price bubbles in a discrete-time, discrete-state market with model uncertainty and short sale restrictions, introducing new theoretical tools and characterizing bubble types based on maturity structure.

Contribution

It develops a new fundamental theorem of asset pricing and superhedging duality in this setting, defining bubbles via a novel fundamental price and analyzing their properties.

Findings

Two types of bubbles depend on asset maturity structure.

G-supermartingale property characterizes bubbles with bounded maturity.

Infi-supermartingale property relates to unbounded maturity assets.

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.