Paper: arXiv 2605.00854

Authors: Naohiro Yoshida

Abstract

This paper proposes a simple and parsimonious discrete-time simulation model to describe the endogenous formation and periodic collapse of financial bubbles. While existing literature has extensively explored the statistical properties of locally explosive bubble dynamics, capturing the micro-level interplay of investor herd behavior and panic selling within a unified framework remains a challenge. Our model addresses this by introducing a cubic function of market momentum to determine the balance of trading directions. This mechanism drives both trend-following behavior during the bubble phase and sudden market crashes when the momentum exceeds a critical threshold. Furthermore, inspired by the self-exciting nature of the Hawkes process, the model endogenizes``market frenzy" by linking trading frequency directly to the accumulated momentum. Simulation results demonstrate that this minimal setup successfully replicates the complex, nonlinear dynamics of bubbles, including simultaneous surges in liquidity and price, followed by dramatic crashes.

Complexity vs Empirical Score

  • Math Complexity: 4.5/10
  • Empirical Rigor: 3.0/10
  • Quadrant: Philosophers — conceptual discussion, limited math and data

Why this score: The paper relies on a conceptual, stylized discrete-time simulation model rather than rigorous empirical validation against real-world market datasets. While it incorporates nonlinear dynamics (cubic functions) and stochastic processes, these serve as a conceptual framework for stylized facts rather than deep mathematical derivation or backtested trading strategies.

Research Flowchart

  flowchart TD
    A[<b>Goal</b>: Model bubble formation & periodic crashes] --> B[<b>Methodology</b>: Discrete-time simulation]
    B --> C[<b>Inputs</b>: Cubic momentum function & Hawkes-inspired frequency]
    C --> D[<b>Process</b>: Endogenize market frenzy & feedback loops]
    D --> E[<b>Outcomes</b>: Replicated non-linear bubble/crash dynamics]
    E --> F[<b>Findings</b>: Combined price surges & liquidity collapse]

Cited References (32)

  1. Nabla discrete fractional calculus and nabla inequalities (2010)
  2. Right nabla discrete fractional calculus (2011)
  3. Linear systems of fractional nabla difference equations (2011)
  4. Large deviations for fractional Poisson processes (2013)
  5. Fractional Poisson process: long-range dependence and applications in ruin theory (2014)
  6. Bubbles, rational expectations and financial markets (1982)
  7. A Langevin approach to stock market fluctuations and crashes (1998)
  8. An Introduction to the Theory of Point Processes (1988)
and 24 more
  1. A simple discrete-time approximation of continuous-time bubbles (1998)
  2. Fractional poisson process (2003)
  3. On a Poisson hyperbolic staircase (1999)
  4. On the long-range dependence of fractional Poisson and negative binomial processes (2016)
  5. The fractional Poisson process and the inverse stable subordinator (2011)
  6. Discrete Mittag-Leffler function and its applications (New (2003)
  7. Caputo type fractional difference operator and its application on discrete time scales (2015)
  8. Discrete Mittag-Leffler functions in linear fractional difference equations (2011)
  9. Probability with Martingales (1991)
  10. Bubble Modeling and Tagging: A Stochastic Nonlinear Autoregression Approach (2024) — link
  11. Explosive rational bubbles in stock prices? (1988)
  12. Explosive behavior in the 1990s Nasdaq: When did exuberance escalate asset values? (2011)
  13. Testing for multiple bubbles: Historical episodes of exuberance and collapse in the S&P 500 (2015)
  14. Spectra of some self-exciting and mutually exciting point processes (1971)
  15. Quadratic Hawkes processes for financial prices (2017)
  16. Non-parametric estimation of quadratic Hawkes processes for order book events (2022)
  17. Market fundamentals versus price-level dynamics (1980)
  18. Asset bubbles and overlapping generations (1985)
  19. Pitfalls in testing for explosive bubbles in asset prices (1991)
  20. Intrinsic bubbles: The case of stock prices (1991)
  21. A simple mechanism for financial bubbles: time-varying momentum horizon (2019)
  22. Statistical theory of reliability and life testing: probability models (1975)
  23. Records (1998)
  24. Extremal processes (1964)

0 of 32 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.