Bubbles in discrete time models

TL;DR

This paper introduces a new discrete-time definition of speculative bubbles based on discounted stock prices losing mass, providing probabilistic characterizations, examples, and conditions for their existence, linking to continuous-time models.

Contribution

It offers a novel discrete-time framework for defining and analyzing speculative bubbles, connecting them to martingale properties and integral equations.

Findings

Discrete-time bubbles are characterized by discounted prices losing mass.

Sufficient conditions for bubbles are derived in the Markovian case.

The discrete-time definition aligns with continuous-time strict local martingale bubbles.

Abstract

We introduce a new definition of speculative bubbles in discrete-time models based on the discounted stock price losing mass at some finite drop-down under an equivalent martingale measure. We provide equivalent probabilistic characterisations of this definition and give examples of discrete-time martingales that are speculative bubbles and those that are not. In the Markovian case, we provide sufficient analytic conditions for the presence of speculative bubbles. We also show that the existence of speculative bubbles is directly linked to the existence of a non-trivial solution to a linear Volterra integral equation of the second kind involving the Markov kernel. Finally, we show that our definition of speculative bubbles in discrete time is consistent with the strict local martingale definition of speculative bubbles in continuous time in the sense that a properly discretised strict local martingale in continuous time is a speculative bubble in discrete time.