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December  2015, 7(4): 431-471. doi: 10.3934/jgm.2015.7.431

## Geometric arbitrage theory and market dynamics

 1 Core Dynamics GmbH, Scheuchzerstrasse 43, CH-8006, Zurich, Switzerland

Received  December 2011 Revised  August 2015 Published  October 2015

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:
$\bullet$ Write arbitrage as curvature of a principal fibre bundle.
$\bullet$ Parameterize arbitrage strategies by its holonomy.
$\bullet$ Give the Fundamental Theorem of Asset Pricing a differential homotopic characterization.
$\bullet$ Characterize Geometric Arbitrage Theory by five principles and show they are consistent with the classical theory of stochastic finance.
$\bullet$ 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.
$\bullet$ Prove that the no-free-lunch-with-vanishing-risk condition implies the zero curvature condition. The converse is in general not true and additionally requires the Novikov condition for the instantaneous Sharpe Ratio to be satisfied.
Citation: Simone Farinelli. Geometric arbitrage theory and market dynamics. Journal of Geometric Mechanics, 2015, 7 (4) : 431-471. doi: 10.3934/jgm.2015.7.431
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##### References:
 [1] Tomasz R. Bielecki, Igor Cialenco, Marek Rutkowski. Arbitrage-free pricing of derivatives in nonlinear market models. Probability, Uncertainty and Quantitative Risk, 2018, 3 (0) : 2-. doi: 10.1186/s41546-018-0027-x [2] Rod Cross, Victor Kozyakin. Double exponential instability of triangular arbitrage systems. Discrete & Continuous Dynamical Systems - B, 2013, 18 (2) : 349-376. doi: 10.3934/dcdsb.2013.18.349 [3] Alexander Schied, Iryna Voloshchenko. Pathwise no-arbitrage in a class of Delta hedging strategies. Probability, Uncertainty and Quantitative Risk, 2016, 1 (0) : 3-. doi: 10.1186/s41546-016-0003-2 [4] Ulrike Kant, Werner M. Seiler. Singularities in the geometric theory of differential equations. Conference Publications, 2011, 2011 (Special) : 784-793. doi: 10.3934/proc.2011.2011.784 [5] Lisa C Flatley, Robert S MacKay, Michael Waterson. Optimal strategies for operating energy storage in an arbitrage or smoothing market. Journal of Dynamics & Games, 2016, 3 (4) : 371-398. doi: 10.3934/jdg.2016020 [6] Editorial Office. Retraction: Xiao-Qian Jiang and Lun-Chuan Zhang, A pricing option approach based on backward stochastic differential equation theory. Discrete & Continuous Dynamical Systems - S, 2019, 12 (4-5) : 969-969. doi: 10.3934/dcdss.2019065 [7] María Teresa V. Martínez-Palacios, Adrián Hernández-Del-Valle, Ambrosio Ortiz-Ramírez. On the pricing of Asian options with geometric average of American type with stochastic interest rate: A stochastic optimal control approach. Journal of Dynamics & Games, 2019, 6 (1) : 53-64. doi: 10.3934/jdg.2019004 [8] Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397 [9] Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437 [10] Jia Yue, Nan-Jing Huang. Neutral and indifference pricing with stochastic correlation and volatility. Journal of Industrial & Management Optimization, 2018, 14 (1) : 199-229. doi: 10.3934/jimo.2017043 [11] Robert J. McCann. A glimpse into the differential topology and geometry of optimal transport. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1605-1621. doi: 10.3934/dcds.2014.34.1605 [12] Farid Tari. Geometric properties of the integral curves of an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2007, 17 (2) : 349-364. doi: 10.3934/dcds.2007.17.349 [13] Penka Georgieva, Aleksey Zinger. Real orientations, real Gromov-Witten theory, and real enumerative geometry. Electronic Research Announcements, 2017, 24: 87-99. doi: 10.3934/era.2017.24.010 [14] Elie Assémat, Marc Lapert, Dominique Sugny, Steffen J. Glaser. On the application of geometric optimal control theory to Nuclear Magnetic Resonance. Mathematical Control & Related Fields, 2013, 3 (4) : 375-396. doi: 10.3934/mcrf.2013.3.375 [15] Wen Chen, Song Wang. A finite difference method for pricing European and American options under a geometric Lévy process. Journal of Industrial & Management Optimization, 2015, 11 (1) : 241-264. doi: 10.3934/jimo.2015.11.241 [16] Seunghee Lee, Ganguk Hwang. A new analytical model for optimized cognitive radio networks based on stochastic geometry. Journal of Industrial & Management Optimization, 2017, 13 (4) : 1883-1899. doi: 10.3934/jimo.2017023 [17] Mariusz Michta. On solutions to stochastic differential inclusions. Conference Publications, 2003, 2003 (Special) : 618-622. doi: 10.3934/proc.2003.2003.618 [18] Christopher Cox, Renato Feres. Differential geometry of rigid bodies collisions and non-standard billiards. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6065-6099. doi: 10.3934/dcds.2016065 [19] Shouwen Fang, Peng Zhu. Differential Harnack estimates for backward heat equations with potentials under geometric flows. Communications on Pure & Applied Analysis, 2015, 14 (3) : 793-809. doi: 10.3934/cpaa.2015.14.793 [20] Víctor León, Bruno Scárdua. A geometric-analytic study of linear differential equations of order two. Electronic Research Archive, , () : -. doi: 10.3934/era.2020107

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