Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces
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 bubbl...
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MDPI AG
2025-03-01
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| author | Mohamed Abdelghani Alexander Melnikov |
| author_facet | Mohamed Abdelghani Alexander Melnikov |
| author_sort | Mohamed Abdelghani |
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| description | 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. |
| format | Article |
| id | doaj-art-73550a1b9382488d92b50249a67f4bc8 |
| institution | DOAJ |
| issn | 2227-9091 |
| language | English |
| publishDate | 2025-03-01 |
| publisher | MDPI AG |
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| spelling | doaj-art-73550a1b9382488d92b50249a67f4bc82025-08-20T02:43:03ZengMDPI AGRisks2227-90912025-03-011335310.3390/risks13030053Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability SpacesMohamed Abdelghani0Alexander Melnikov1Wells Fargo, 150 E 42nd St, New York City, NY 10017, USADepartment of Mathematics and Statistics, University of Alberta, 11324 89 Ave NW, Edmonton, AB T6G 2J5, CanadaDeviation 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.https://www.mdpi.com/2227-9091/13/3/53exponential martingalediffusion process with jumpsGirsanov theoremregime shiftslocal optional martingalesdeflators |
| spellingShingle | Mohamed Abdelghani Alexander Melnikov Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces Risks exponential martingale diffusion process with jumps Girsanov theorem regime shifts local optional martingales deflators |
| title | Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces |
| title_full | Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces |
| title_fullStr | Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces |
| title_full_unstemmed | Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces |
| title_short | Modeling Financial Bubbles with Optional Semimartingales in Nonstandard Probability Spaces |
| title_sort | modeling financial bubbles with optional semimartingales in nonstandard probability spaces |
| topic | exponential martingale diffusion process with jumps Girsanov theorem regime shifts local optional martingales deflators |
| url | https://www.mdpi.com/2227-9091/13/3/53 |
| work_keys_str_mv | AT mohamedabdelghani modelingfinancialbubbleswithoptionalsemimartingalesinnonstandardprobabilityspaces AT alexandermelnikov modelingfinancialbubbleswithoptionalsemimartingalesinnonstandardprobabilityspaces |