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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
##### References:
 [1] V. I. Arnold, Mathematical Methods of Classical Mechanics,, Graduate Texts in Mathematics, (1989). doi: 10.1007/978-1-4757-2063-1. [2] F. Bellini and M. Frittelli, On the existence of minimax martingale measures,, Mathematical Finance, 12 (2002), 1. doi: 10.1111/1467-9965.00001. [3] T. Björk, Arbitrage Theory in Continuous Time,, Oxford Finance, (2004). [4] T. Björk and H. Hult, A note on Wick products and the fractional Black-Scholes model,, Finance & Stochastics, 9 (2005), 197. doi: 10.1007/s00780-004-0144-5. [5] D. Bleecker, Gauge Theory and Variational Principles,, Addison-Wesley Publishing, (1981). [6] J. Cresson and S. Darses, Stochastic embedding of dynamical systems,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2736519. [7] F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage,, Springer-Verlag, (2006). [8] B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry-Methods and Applications: Part II. The Geometry and Topology of Manifolds,, Graduate Texts in Mathematics, (1985). doi: 10.1007/978-1-4612-1100-6. [9] B. Dupoyet, H. R. Fiebig and D. P. Musgrov, Gauge invariant lattice quantum field theory: Implications for statistical properties of high frequency financial markets,, Physica A, 389 (2010), 107. doi: 10.1016/j.physa.2009.09.002. [10] C. Dellachérie and P. A. Meyer, Probabilité et Potentiel II - Théorie des Martingales - Chapitres 5 à 8,, Hermann, (1980). [11] K. D. Elworthy, Stochastic Differential Equations on Manifolds,, London Mathematical Society Lecture Notes Series, (1982). [12] M. Eméry, Stochastic Calculus on Manifolds-With an Appendix by P. A. Meyer,, Springer, (1989). doi: 10.1007/978-3-642-75051-9. [13] S. Farinelli and S. Vazquez, Gauge invariance, geometry and arbitrage,, The Journal of Investment Strategies, 1 (2012), 23. [14] M. Fei-Te and M. Jin-Long, Solitary wave solutions of nonlinear financial markets: Data-modeling-concept-practicing,, Front. Phys. China, 2 (2007), 368. [15] B. Flesaker and L. Hughston, Positive Interest,, Risk, 9 (1996), 36. [16] H. Föllmer and A. Schied, Stochastic Finance: An Introduction In Discrete Time,, Second Edition, (2004). doi: 10.1515/9783110212075. [17] Y. E. Gliklikh, Global and Stochastic Analysis with Applications to Mathematical Physics,, Theoretical and Mathemtical Physics, (2011). doi: 10.1007/978-0-85729-163-9. [18] W. Hackenbroch and A. Thalmaier, Stochastische Analysis. Eine Einführung in die Theorie der stetigen Semimartingale,, Teubner Verlag, (1994). doi: 10.1007/978-3-663-11527-4. [19] L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis,, Springer, (2003). [20] E. P. Hsu, Stochastic Analysis on Manifolds,, Graduate Studies in Mathematics, 38 (2002). doi: 10.1090/gsm/038. [21] P. J. Hunt and J. E. Kennedy, Financial Derivatives in Theory and Practice,, Wiley Series in Probability and Statistics, (2004). doi: 10.1002/0470863617. [22] K. Ilinski, Gauge geometry of financial markets,, J. Phys. A: Math. Gen., 33 (2000). doi: 10.1088/0305-4470/33/1/102. [23] K. Ilinski, Physics of Finance: Gauge Modelling in Non-Equilibrium Pricing,, Wiley, (2001). [24] J. D. Jackson, Classical Electrodynamics,, Third Edition, (1998). [25] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I,, Wiley, (1996). [26] P. N. Malaney, The Index Number Problem: A Differential Geometric Approach,, PhD Thesis, (1997). [27] Y. Morisawa, Toward a geometric formulation of triangular arbitrage: An introduction to gauge theory of arbitrage,, Progress of Theoretical Physics Supplement, 179 (2009), 209. doi: 10.1143/PTPS.179.209. [28] E. Nelson, Dynamical Theories of Brownian Motion,, Princeton University Press, (1967). [29] Ph. E. Protter, Stochastic Integration and Differential Equations: Version 2.1,, Stochastic Modelling and Applied Probability, (2005). [30] L. C. G. Rogers, Equivalent martingale measures and no-arbitrage,, Stochastics, 51 (1994), 41. doi: 10.1080/17442509408833943. [31] W. Schachermayer, Optimal investment in incomplete markets when wealth may become negative,, Annals of Applied Probability, 11 (2001), 694. doi: 10.1214/aoap/1015345346. [32] L. Schwartz, Semi-martingales Sur des Variétés et Martingales Conformes sur des Variétés Analytiques Complexes,, Springer Lecture Notes in Mathematics, (1980). [33] S. E. Shreve, Stochastic Calculus for Finance,, Springer-Verlag, (2004). [34] M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media,, Texts and Monographs in Physics. Springer-Verlag, (1997). [35] A. Smith and C. Speed, Gauge Transforms in Stochastic Investment,, Proceedings of the 1998 AFIR Colloquim, (1998). [36] S. Sternberg, Lectures On Differential Geometry,, Second Edition, (1983). [37] D. W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold,, Mathematical Surveys and Monographs, 74 (2000). [38] E. Weinstein, Gauge theory and inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application,, Talk given at Perimeter Institute, (2006). [39] K. Yasue, Stochastic calculus of variations,, Journal of Functional Analysis, 41 (1981), 327. doi: 10.1016/0022-1236(81)90079-3. [40] K. Young, Foreign exchange market as a lattice gauge theory,, Am. J. Phys., 67 (1999). doi: 10.1119/1.19139.

show all references

##### References:
 [1] V. I. Arnold, Mathematical Methods of Classical Mechanics,, Graduate Texts in Mathematics, (1989). doi: 10.1007/978-1-4757-2063-1. [2] F. Bellini and M. Frittelli, On the existence of minimax martingale measures,, Mathematical Finance, 12 (2002), 1. doi: 10.1111/1467-9965.00001. [3] T. Björk, Arbitrage Theory in Continuous Time,, Oxford Finance, (2004). [4] T. Björk and H. Hult, A note on Wick products and the fractional Black-Scholes model,, Finance & Stochastics, 9 (2005), 197. doi: 10.1007/s00780-004-0144-5. [5] D. Bleecker, Gauge Theory and Variational Principles,, Addison-Wesley Publishing, (1981). [6] J. Cresson and S. Darses, Stochastic embedding of dynamical systems,, J. Math. Phys., 48 (2007). doi: 10.1063/1.2736519. [7] F. Delbaen and W. Schachermayer, The Mathematics of Arbitrage,, Springer-Verlag, (2006). [8] B. A. Dubrovin, A. T. Fomenko and S. P. Novikov, Modern Geometry-Methods and Applications: Part II. The Geometry and Topology of Manifolds,, Graduate Texts in Mathematics, (1985). doi: 10.1007/978-1-4612-1100-6. [9] B. Dupoyet, H. R. Fiebig and D. P. Musgrov, Gauge invariant lattice quantum field theory: Implications for statistical properties of high frequency financial markets,, Physica A, 389 (2010), 107. doi: 10.1016/j.physa.2009.09.002. [10] C. Dellachérie and P. A. Meyer, Probabilité et Potentiel II - Théorie des Martingales - Chapitres 5 à 8,, Hermann, (1980). [11] K. D. Elworthy, Stochastic Differential Equations on Manifolds,, London Mathematical Society Lecture Notes Series, (1982). [12] M. Eméry, Stochastic Calculus on Manifolds-With an Appendix by P. A. Meyer,, Springer, (1989). doi: 10.1007/978-3-642-75051-9. [13] S. Farinelli and S. Vazquez, Gauge invariance, geometry and arbitrage,, The Journal of Investment Strategies, 1 (2012), 23. [14] M. Fei-Te and M. Jin-Long, Solitary wave solutions of nonlinear financial markets: Data-modeling-concept-practicing,, Front. Phys. China, 2 (2007), 368. [15] B. Flesaker and L. Hughston, Positive Interest,, Risk, 9 (1996), 36. [16] H. Föllmer and A. Schied, Stochastic Finance: An Introduction In Discrete Time,, Second Edition, (2004). doi: 10.1515/9783110212075. [17] Y. E. Gliklikh, Global and Stochastic Analysis with Applications to Mathematical Physics,, Theoretical and Mathemtical Physics, (2011). doi: 10.1007/978-0-85729-163-9. [18] W. Hackenbroch and A. Thalmaier, Stochastische Analysis. Eine Einführung in die Theorie der stetigen Semimartingale,, Teubner Verlag, (1994). doi: 10.1007/978-3-663-11527-4. [19] L. Hörmander, The Analysis of Linear Partial Differential Operators I: Distribution Theory and Fourier Analysis,, Springer, (2003). [20] E. P. Hsu, Stochastic Analysis on Manifolds,, Graduate Studies in Mathematics, 38 (2002). doi: 10.1090/gsm/038. [21] P. J. Hunt and J. E. Kennedy, Financial Derivatives in Theory and Practice,, Wiley Series in Probability and Statistics, (2004). doi: 10.1002/0470863617. [22] K. Ilinski, Gauge geometry of financial markets,, J. Phys. A: Math. Gen., 33 (2000). doi: 10.1088/0305-4470/33/1/102. [23] K. Ilinski, Physics of Finance: Gauge Modelling in Non-Equilibrium Pricing,, Wiley, (2001). [24] J. D. Jackson, Classical Electrodynamics,, Third Edition, (1998). [25] S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Volume I,, Wiley, (1996). [26] P. N. Malaney, The Index Number Problem: A Differential Geometric Approach,, PhD Thesis, (1997). [27] Y. Morisawa, Toward a geometric formulation of triangular arbitrage: An introduction to gauge theory of arbitrage,, Progress of Theoretical Physics Supplement, 179 (2009), 209. doi: 10.1143/PTPS.179.209. [28] E. Nelson, Dynamical Theories of Brownian Motion,, Princeton University Press, (1967). [29] Ph. E. Protter, Stochastic Integration and Differential Equations: Version 2.1,, Stochastic Modelling and Applied Probability, (2005). [30] L. C. G. Rogers, Equivalent martingale measures and no-arbitrage,, Stochastics, 51 (1994), 41. doi: 10.1080/17442509408833943. [31] W. Schachermayer, Optimal investment in incomplete markets when wealth may become negative,, Annals of Applied Probability, 11 (2001), 694. doi: 10.1214/aoap/1015345346. [32] L. Schwartz, Semi-martingales Sur des Variétés et Martingales Conformes sur des Variétés Analytiques Complexes,, Springer Lecture Notes in Mathematics, (1980). [33] S. E. Shreve, Stochastic Calculus for Finance,, Springer-Verlag, (2004). [34] M. Šilhavý, The Mechanics and Thermodynamics of Continuous Media,, Texts and Monographs in Physics. Springer-Verlag, (1997). [35] A. Smith and C. Speed, Gauge Transforms in Stochastic Investment,, Proceedings of the 1998 AFIR Colloquim, (1998). [36] S. Sternberg, Lectures On Differential Geometry,, Second Edition, (1983). [37] D. W. Stroock, An Introduction to the Analysis of Paths on a Riemannian Manifold,, Mathematical Surveys and Monographs, 74 (2000). [38] E. Weinstein, Gauge theory and inflation: Enlarging the Wu-Yang Dictionary to a unifying Rosetta Stone for Geometry in Application,, Talk given at Perimeter Institute, (2006). [39] K. Yasue, Stochastic calculus of variations,, Journal of Functional Analysis, 41 (1981), 327. doi: 10.1016/0022-1236(81)90079-3. [40] K. Young, Foreign exchange market as a lattice gauge theory,, Am. J. Phys., 67 (1999). doi: 10.1119/1.19139.
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