Bubbles in discrete-time models

Martin Herdegen () and Dörte Kreher ()
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Martin Herdegen: University of Warwick
Dörte Kreher: Humboldt-Universität zu Berlin

Finance and Stochastics, 2022, vol. 26, issue 4, No 6, 899-925

Abstract: Abstract We introduce a new definition of bubbles in discrete-time models based on the discounted stock price losing mass under an equivalent martingale measure at some finite drawdown. We provide equivalent probabilistic characterisations of this definition and give examples of discrete-time martingales that are bubbles and others that are not. In the Markovian case, we provide sufficient analytic conditions for the presence of bubbles. We also show that the existence of 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 bubbles in discrete time is consistent with the strict local martingale definition of bubbles in continuous time in the sense that a properly discretised strict local martingale in continuous time is a bubble in discrete time.

Keywords: Bubble; Strict local martingale; Markov martingale; Volterra integral equation; 60G42; 60J05; 91G99; 45D05 (search for similar items in EconPapers)
JEL-codes: C60 G10 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s00780-022-00487-6

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