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Geometric Arbitrage Theory and Market Dynamics Reloaded

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  • Simone Farinelli

Abstract

We have embedded the classical theory of stochastic finance into a differential geometric framework called Geometric Arbitrage Theory and show that it is possible to: --Write arbitrage as curvature of a principal fibre bundle. --Parameterize arbitrage strategies by its holonomy. --Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization. --Characterize Geometric Arbitrage Theory by five principles and show they they are consistent with the classical theory of stochastic finance. --Derive for a closed market the equilibrium solution for market portfolio and dynamics in the cases where: -->Arbitrage is allowed but minimized. -->Arbitrage is not allowed. --Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition.

Suggested Citation

  • Simone Farinelli, 2009. "Geometric Arbitrage Theory and Market Dynamics Reloaded," Papers 0910.1671, arXiv.org, revised Jul 2021.
    • Handle: RePEc:arx:papers:0910.1671
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    Cited by:

    1. Jean-Pierre Magnot, 2019. "On Mathematical Structures On Pairwise Comparisons Matrices With Coefficients In A Group Arising From Quantum Gravity," Post-Print hal-01835958, HAL.
    2. Jean Pierre Magnot, 2018. "On Multidisciplinary Potential Applications of Gauge Theories," Biostatistics and Biometrics Open Access Journal, Juniper Publishers Inc., vol. 7(1), pages 1-2, May.
    3. Simone Farinelli & Hideyuki Takada, 2019. "The Black-Scholes Equation in Presence of Arbitrage," Papers 1904.11565, arXiv.org, revised Oct 2021.
    4. Wanxiao Tang & Jun Zhao & Peibiao Zhao, 2019. "Geometric No-Arbitrage Analysis in the Dynamic Financial Market with Transaction Costs," JRFM, MDPI, vol. 12(1), pages 1-17, February.
    5. Jean-Pierre Magnot, 2018. "On Mathematical Structures On Pairwise Comparisons Matrices With Coefficients In A Group Arising From Quantum Gravity," Working Papers hal-01835958, HAL.
    6. Kanjamapornkul, Kabin & Pinčák, Richard & Bartoš, Erik, 2020. "Cohomology theory for financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 546(C).
    7. P. Liebrich, 2019. "A Relation between Short-Term and Long-Term Arbitrage," Papers 1909.00570, arXiv.org.
    8. Simone Farinelli & Hideyuki Takada, 2022. "The Black–Scholes equation in the presence of arbitrage," Quantitative Finance, Taylor & Francis Journals, vol. 22(12), pages 2155-2170, December.
    9. Simone Farinelli & Hideyuki Takada, 2019. "When Risks and Uncertainties Collide: Mathematical Finance for Arbitrage Markets in a Quantum Mechanical View," Papers 1906.07164, arXiv.org, revised Jan 2021.

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