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Exact solution to two-body financial dealer model: revisited from the viewpoint of kinetic theory

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  • Kiyoshi Kanazawa
  • Hideki Takayasu
  • Misako Takayasu

Abstract

The two-body stochastic dealer model is revisited to provide an exact solution to the average order-book profile using the kinetic approach. The dealer model is a microscopic financial model where individual traders make decisions on limit-order prices stochastically and then reach agreements on transactions. In the literature, this model was solved for several cases: an exact solution for two-body traders $N=2$ and a mean-field solution for many traders $N\gg 1$. Remarkably, while kinetic theory plays a significant role in the mean-field analysis for $N\gg 1$, its role is still elusive for the case of $N=2$. In this paper, we revisit the two-body dealer model $N=2$ to clarify the utility of the kinetic theory. We first derive the exact master-Liouville equations for the two-body dealer model by several methods. We next illustrate the physical picture of the master-Liouville equation from the viewpoint of the probability currents. The master-Liouville equations are then solved exactly to derive the order-book profile and the average transaction interval. Furthermore, we introduce a generalised two-body dealer model by incorporating interaction between traders via the market midprice and exactly solve the model within the kinetic framework. We finally confirm our exact solution by numerical simulations. This work provides a systematic mathematical basis for the econophysics model by developing better mathematical intuition.

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  • Kiyoshi Kanazawa & Hideki Takayasu & Misako Takayasu, 2022. "Exact solution to two-body financial dealer model: revisited from the viewpoint of kinetic theory," Papers 2205.15558, arXiv.org.
    • Handle: RePEc:arx:papers:2205.15558
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    References listed on IDEAS

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    1. Kiyoshi Kanazawa & Takumi Sueshige & Hideki Takayasu & Misako Takayasu, 2017. "Derivation of the Boltzmann Equation for Financial Brownian Motion: Direct Observation of the Collective Motion of High-Frequency Traders," Papers 1703.06739, arXiv.org, revised Mar 2018.
    2. Takayasu, Hideki & Miura, Hitoshi & Hirabayashi, Tadashi & Hamada, Koichi, 1992. "Statistical properties of deterministic threshold elements — the case of market price," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 184(1), pages 127-134.
    3. Sato, Aki-Hiro & Takayasu, Hideki, 1998. "Dynamic numerical models of stock market price: from microscopic determinism to macroscopic randomness," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 250(1), pages 231-252.
    4. Mantegna,Rosario N. & Stanley,H. Eugene, 2007. "Introduction to Econophysics," Cambridge Books, Cambridge University Press, number 9780521039871.
    5. Kiyoshi Kanazawa & Didier Sornette, 2021. "Ubiquitous power law scaling in nonlinear self-excited Hawkes processes," Papers 2102.00242, arXiv.org, revised Oct 2021.
    6. Kiyoshi Kanazawa & Takumi Sueshige & Hideki Takayasu & Misako Takayasu, 2018. "Kinetic Theory for Finance Brownian Motion from Microscopic Dynamics," Papers 1802.05993, arXiv.org.
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