Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces

Mohamed Abdelghani () and Alexander Melnikov ()
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Mohamed Abdelghani: Wells Fargo, 150 E 42nd St, New York City, NY 10017, USA
Alexander Melnikov: Department of Mathematics and Statistics, University of Alberta, 11324 89 Ave NW, Edmonton, AB T6G 2J5, Canada

Risks, 2025, vol. 13, issue 3, 1-29

Abstract: Deviation of an asset price from its fundamental value, commonly referred to as a price bubble, is a well-known phenomenon in financial markets. Mathematically, a bubble arises when the deflated price process transitions from a martingale to a strict local martingale. This paper explores price bubbles using the framework of optional semimartingale calculus within nonstandard probability spaces, where the underlying filtration is not necessarily right-continuous or complete. We present two formulations for financial markets with bubbles: one in which asset prices are modeled as càdlàg semimartingales and another where they are modeled as làdlàg semimartingales. In both models, we demonstrate that the formation and re-emergence of price bubbles are intrinsically tied to the lack of right continuity in the underlying filtration. These theoretical findings are illustrated with practical examples, offering novel insights into bubble dynamics that hold significance for both academics and practitioners in the field of mathematical finance.

Keywords: exponential martingale; diffusion process with jumps; Girsanov theorem; regime shifts; local optional martingales; deflators; làdlàg semimartingales; unusual conditions (search for similar items in EconPapers)
JEL-codes: C G0 G1 G2 G3 K2 M2 M4 (search for similar items in EconPapers)
Date: 2025
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