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Bubbles in discrete-time models
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Herdegen, Martin and Kreher, Dörte (2022) Bubbles in discrete-time models. Finance and Stochastics, 26 . pp. 899-925. doi:10.1007/s00780-022-00487-6 ISSN 0949-2984.
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Official URL: https://doi.org/10.1007/s00780-022-00487-6
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.
| Item Type: | Journal Article | ||||||||
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| Subjects: | Q Science > QA Mathematics | ||||||||
| Divisions: | Faculty of Science, Engineering and Medicine > Science > Mathematics | ||||||||
| Library of Congress Subject Headings (LCSH): | Martingales (Mathematics), Discrete-time systems, Markov processes, Volterra equations | ||||||||
| Journal or Publication Title: | Finance and Stochastics | ||||||||
| Publisher: | Springer | ||||||||
| ISSN: | 0949-2984 | ||||||||
| Official Date: | October 2022 | ||||||||
| Dates: |
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| Volume: | 26 | ||||||||
| Page Range: | pp. 899-925 | ||||||||
| DOI: | 10.1007/s00780-022-00487-6 | ||||||||
| Status: | Peer Reviewed | ||||||||
| Publication Status: | Published | ||||||||
| Access rights to Published version: | Open Access (Creative Commons) | ||||||||
| Date of first compliant deposit: | 20 July 2022 | ||||||||
| Date of first compliant Open Access: | 10 October 2022 | ||||||||
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