American Institute of Mathematical Sciences

Advanced
March  2013, 18(2): 349-376. doi: 10.3934/dcdsb.2013.18.349

Double exponential instability of triangular arbitrage systems

Rod Cross 1, and  Victor Kozyakin 2,

1. 

Department of Economics, University of Strathclyde, Sir William Duncan Building, 130 Rottenrow, Glasgow G4 0GE, United Kingdom
2. 

Institute for Information Transmission Problems, Russian Academy of Sciences, Bolshoj Karetny lane 19, Moscow 127994 GSP-4

Received  June 2012 Revised  July 2012 Published  November 2012

If financial markets displayed the informational efficiency postulated in the efficient markets hypothesis (EMH), arbitrage operations would be self-extinguishing. The present paper considers arbitrage sequences in foreign exchange (FX) markets, in which trading platforms and information are fragmented. In [18,9] it was shown that sequences of triangular arbitrage operations in FX markets containing $4$ currencies and trader-arbitrageurs tend to display periodicity or grow exponentially rather than being self-extinguishing. This paper extends the analysis to $5$ or higher-order currency worlds. The key findings are that in a $5$-currency world arbitrage sequences may also follow an exponential law as well as display periodicity, but that in higher-order currency worlds a double exponential law may additionally apply. There is an ``inheritance of instability'' in the higher-order currency worlds. Profitable arbitrage operations are thus endemic rather that displaying the self-extinguishing properties implied by the EMH.
Keywords: recurrent sequences, Limits to arbitrage, asynchronous systems., matrix products.
Mathematics Subject Classification: Primary: 91B26, 91B54, 91B64; Secondary: 15A6.
Citation: 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
References:
[1]

Q. F. Akram, D. Rime and L. Sarno, Arbitrage in the foreign exchange market: Turning on the microscope, Journal of International Economics, 76 (2008), 237-253, http://www.sciencedirect.com/science/article/pii/S0022199608000706. Google Scholar

[2]

E. A. Asarin, V. S. Kozyakin, M. A. Krasnosel$'$skiĭ et al., "Analiz Ustoichivosti Rassinkhronizovannykh Diskretnykh Sistem," Nauka, Moscow, 1992, http://eqworld.ipmnet.ru/ru/library/books/AsarinKozyakinKrasnoselskijKuznecov1992ru.pdf, in Russian.  Google Scholar

[3]

BIS, "Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity in 2010 - Final results," Bank for International Settlements, Basel, Switzerland, 2010, http://www.bis.org/publ/rpfxf10t.pdf. Google Scholar

[4]

BIS, "High-Frequency Trading in the Foreign Exchange Market: New Report from the Markets Committee," Bank for International Settlements, Basel, Switzerland, 2011, http://www.bis.org/publ/mktc05.pdf. Google Scholar

[5]

BIS, "BIS Effective Exchange Rate Indices," Bank for International Settlements, Basel, Switzerland, 2012, http://www.bis.org/statistics/eer/index.htm. Google Scholar

[6]

G. Cassel, The present situation of the foreign exchanges. I, Economic Journal, 26 (1916), 62-65. Google Scholar

[7]

A. A. Cournot, "Researches into the Mathematical Principles of the Theory of Wealth," Macmillan, New York, 1929, translated in 1897 by N. T. Bacon, first published in 1838. Google Scholar

[8]

V. Covrig and M. Melvin, Asymmetric information and price discovery in the FX market: Does Tokyo know more about the yen?, Journal of Empirical Finance, 9 (2002), 271-285, http://ideas.repec.org/a/eee/empfin/v9y2002i3p271-285.html. Google Scholar

[9]

R. Cross, V. Kozyakin, B. O'Callaghan et al., Periodic sequences of arbitrage: A tale of four currencies, Metroeconomica, 63 (2012), 250-294, arXiv:1112.5850. doi: 10.1111/j.1467-999X.2011.04140.x.  Google Scholar

[10]

C. D'Souza, Price discovery across geographic locations in the foreign exchange market, Bank of Canada Review, Spring 2008 (2008), 19-27, http://ideas.repec.org/a/bca/bcarev/v2008y2008ispring08p19-27.html. Google Scholar

[11]

P. H. Dybvig and S. A. Ross, Arbitrage, in "The New Palgrave Dictionary of Economics" (eds. S. N. Durlauf and L. E. Blume), Palgrave Macmillan, Basingstoke, 2008. Google Scholar

[12]

E. F. Fama, Efficient capital markets: A review of theory and empirical work, Journal of Finance, 25 (1970), 383-417, http://www.jstor.org/discover/10.2307/2325486?uid=2129&uid=2&uid=70&uid=4&sid=47699016684017, Papers and Proceedings of the Twenty-Eighth Annual Meeting of the American Finance Association New York, N. Y. December, 28-30, (1969) (May, 1970). Google Scholar

[13]

S. J. Grossman and J. E. Stiglitz, Information and competitive price systems, American Economic Review, 66 (1976), 246-253, http://e-m-h.org/GrSt76.pdf. Google Scholar

[14]

IMF, "SDR Valuation,'' International Monetary Fund, 2012, http://www.imf.org/external/np/fin/data/rms_sdrv.aspx. Google Scholar

[15]

R. Jungers, "The Joint Spectral Radius," vol. 385 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 2009, theory and applications. doi: 10.1007/978-3-540-95980-9.  Google Scholar

[16]

J. M. Keynes, "A Tract on Monetary Reform,'' Macmilan, London UK, 1923. Google Scholar

[17]

M. Klau and S. S. Fung, The new BIS effective exchange rate indices,, Bank for International Settlements Quarterly Review, (): 51.   Google Scholar

[18]

V. Kozyakin, B. O'Callaghan and A. Pokrovskii, Sequences of arbitrages,, ArXiv.org e-Print archive, ().   Google Scholar

[19]

B. R. Marshall, S. Treepongkaruna and M. Young, Exploitable arbitrage opportunities exist in the foreign exchange market, Discussion Paper, 10 September, Massey University, Palmerston North, New Zealand, 2007, http://wwwdocs.fce.unsw.edu.au/banking/seminar/2007/exploitablearbitrage_Marshall_Sept13.pdf. Google Scholar

[20]

S. A. Ross, A simple approach to the valuation of risky streams, Journal of Business, 51 (1978), 453-475. Google Scholar

[21]

P. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review, 6 (1965), 41-49, http://stevereads.com/papers_to_read/proof_that_properly_anticipated_prices_fluctuate_randomly.pdf. Google Scholar

[22]

Z. Xiaochuan, Reform the international monetary system, Bank for International Settlements Quarterly Review, 4 (2009), 1-3, http://www.bis.org/review/r090402c.pdf. Google Scholar

show all references

References:
[1]

Q. F. Akram, D. Rime and L. Sarno, Arbitrage in the foreign exchange market: Turning on the microscope, Journal of International Economics, 76 (2008), 237-253, http://www.sciencedirect.com/science/article/pii/S0022199608000706. Google Scholar

[2]

E. A. Asarin, V. S. Kozyakin, M. A. Krasnosel$'$skiĭ et al., "Analiz Ustoichivosti Rassinkhronizovannykh Diskretnykh Sistem," Nauka, Moscow, 1992, http://eqworld.ipmnet.ru/ru/library/books/AsarinKozyakinKrasnoselskijKuznecov1992ru.pdf, in Russian.  Google Scholar

[3]

BIS, "Triennial Central Bank Survey of Foreign Exchange and Derivatives Market Activity in 2010 - Final results," Bank for International Settlements, Basel, Switzerland, 2010, http://www.bis.org/publ/rpfxf10t.pdf. Google Scholar

[4]

BIS, "High-Frequency Trading in the Foreign Exchange Market: New Report from the Markets Committee," Bank for International Settlements, Basel, Switzerland, 2011, http://www.bis.org/publ/mktc05.pdf. Google Scholar

[5]

BIS, "BIS Effective Exchange Rate Indices," Bank for International Settlements, Basel, Switzerland, 2012, http://www.bis.org/statistics/eer/index.htm. Google Scholar

[6]

G. Cassel, The present situation of the foreign exchanges. I, Economic Journal, 26 (1916), 62-65. Google Scholar

[7]

A. A. Cournot, "Researches into the Mathematical Principles of the Theory of Wealth," Macmillan, New York, 1929, translated in 1897 by N. T. Bacon, first published in 1838. Google Scholar

[8]

V. Covrig and M. Melvin, Asymmetric information and price discovery in the FX market: Does Tokyo know more about the yen?, Journal of Empirical Finance, 9 (2002), 271-285, http://ideas.repec.org/a/eee/empfin/v9y2002i3p271-285.html. Google Scholar

[9]

R. Cross, V. Kozyakin, B. O'Callaghan et al., Periodic sequences of arbitrage: A tale of four currencies, Metroeconomica, 63 (2012), 250-294, arXiv:1112.5850. doi: 10.1111/j.1467-999X.2011.04140.x.  Google Scholar

[10]

C. D'Souza, Price discovery across geographic locations in the foreign exchange market, Bank of Canada Review, Spring 2008 (2008), 19-27, http://ideas.repec.org/a/bca/bcarev/v2008y2008ispring08p19-27.html. Google Scholar

[11]

P. H. Dybvig and S. A. Ross, Arbitrage, in "The New Palgrave Dictionary of Economics" (eds. S. N. Durlauf and L. E. Blume), Palgrave Macmillan, Basingstoke, 2008. Google Scholar

[12]

E. F. Fama, Efficient capital markets: A review of theory and empirical work, Journal of Finance, 25 (1970), 383-417, http://www.jstor.org/discover/10.2307/2325486?uid=2129&uid=2&uid=70&uid=4&sid=47699016684017, Papers and Proceedings of the Twenty-Eighth Annual Meeting of the American Finance Association New York, N. Y. December, 28-30, (1969) (May, 1970). Google Scholar

[13]

S. J. Grossman and J. E. Stiglitz, Information and competitive price systems, American Economic Review, 66 (1976), 246-253, http://e-m-h.org/GrSt76.pdf. Google Scholar

[14]

IMF, "SDR Valuation,'' International Monetary Fund, 2012, http://www.imf.org/external/np/fin/data/rms_sdrv.aspx. Google Scholar

[15]

R. Jungers, "The Joint Spectral Radius," vol. 385 of Lecture Notes in Control and Information Sciences, Springer-Verlag, Berlin, 2009, theory and applications. doi: 10.1007/978-3-540-95980-9.  Google Scholar

[16]

J. M. Keynes, "A Tract on Monetary Reform,'' Macmilan, London UK, 1923. Google Scholar

[17]

M. Klau and S. S. Fung, The new BIS effective exchange rate indices,, Bank for International Settlements Quarterly Review, (): 51.   Google Scholar

[18]

V. Kozyakin, B. O'Callaghan and A. Pokrovskii, Sequences of arbitrages,, ArXiv.org e-Print archive, ().   Google Scholar

[19]

B. R. Marshall, S. Treepongkaruna and M. Young, Exploitable arbitrage opportunities exist in the foreign exchange market, Discussion Paper, 10 September, Massey University, Palmerston North, New Zealand, 2007, http://wwwdocs.fce.unsw.edu.au/banking/seminar/2007/exploitablearbitrage_Marshall_Sept13.pdf. Google Scholar

[20]

S. A. Ross, A simple approach to the valuation of risky streams, Journal of Business, 51 (1978), 453-475. Google Scholar

[21]

P. Samuelson, Proof that properly anticipated prices fluctuate randomly, Industrial Management Review, 6 (1965), 41-49, http://stevereads.com/papers_to_read/proof_that_properly_anticipated_prices_fluctuate_randomly.pdf. Google Scholar

[22]

Z. Xiaochuan, Reform the international monetary system, Bank for International Settlements Quarterly Review, 4 (2009), 1-3, http://www.bis.org/review/r090402c.pdf. Google Scholar

[1]

Adel Alahmadi, Hamed Alsulami, S.K. Jain, Efim Zelmanov. On matrix wreath products of algebras. Electronic Research Announcements, 2017, 24: 78-86. doi: 10.3934/era.2017.24.009
[2]

Tobias Wichtrey. Harmonic limits of dynamical systems. Conference Publications, 2011, 2011 (Special) : 1432-1439. doi: 10.3934/proc.2011.2011.1432
[3]

Stefan Siegmund, Petr Stehlík. Time scale-induced asynchronous discrete dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (2) : 1011-1029. doi: 10.3934/dcdsb.2020151
[4]

P. Adda, J. L. Dimi, A. Iggidir, J. C. Kamgang, G. Sallet, J. J. Tewa. General models of host-parasite systems. Global analysis. Discrete & Continuous Dynamical Systems - B, 2007, 8 (1) : 1-17. doi: 10.3934/dcdsb.2007.8.1
[5]

Artur Avila, Thomas Roblin. Uniform exponential growth for some SL(2, R) matrix products. Journal of Modern Dynamics, 2009, 3 (4) : 549-554. doi: 10.3934/jmd.2009.3.549
[6]

Matteo Petrera, Yuri B. Suris. Geometry of the Kahan discretizations of planar quadratic Hamiltonian systems. Ⅱ. Systems with a linear Poisson tensor. Journal of Computational Dynamics, 2019, 6 (2) : 401-408. doi: 10.3934/jcd.2019020
[7]

Stefano Bianchini. A note on singular limits to hyperbolic systems of conservation laws. Communications on Pure & Applied Analysis, 2003, 2 (1) : 51-64. doi: 10.3934/cpaa.2003.2.51
[8]

Jacques Demongeot, Dan Istrate, Hajer Khlaifi, Lucile Mégret, Carla Taramasco, René Thomas. From conservative to dissipative non-linear differential systems. An application to the cardio-respiratory regulation. Discrete & Continuous Dynamical Systems - S, 2020, 13 (8) : 2121-2134. doi: 10.3934/dcdss.2020181
[9]

Denis de Carvalho Braga, Luis Fernando Mello, Carmen Rocşoreanu, Mihaela Sterpu. Lyapunov coefficients for non-symmetrically coupled identical dynamical systems. Application to coupled advertising models. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 785-803. doi: 10.3934/dcdsb.2009.11.785
[10]

Susanna Terracini, Juncheng Wei. DCDS-A Special Volume Qualitative properties of solutions of nonlinear elliptic equations and systems. Preface. Discrete & Continuous Dynamical Systems, 2014, 34 (6) : i-ii. doi: 10.3934/dcds.2014.34.6i
[11]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅰ: Dirichlet and Neumann boundary conditions. Communications on Pure & Applied Analysis, 2017, 16 (6) : 2357-2376. doi: 10.3934/cpaa.2017116
[12]

Anna Kostianko, Sergey Zelik. Inertial manifolds for 1D reaction-diffusion-advection systems. Part Ⅱ: periodic boundary conditions. Communications on Pure & Applied Analysis, 2018, 17 (1) : 285-317. doi: 10.3934/cpaa.2018017
[13]

Peizhao Yu, Guoshan Zhang, Yi Zhang. Decoupling of cubic polynomial matrix systems. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 13-26. doi: 10.3934/naco.2020012
[14]

Rajeshwari Majumdar, Phanuel Mariano, Hugo Panzo, Lowen Peng, Anthony Sisti. Lyapunov exponent and variance in the CLT for products of random matrices related to random Fibonacci sequences. Discrete & Continuous Dynamical Systems - B, 2020, 25 (12) : 4779-4799. doi: 10.3934/dcdsb.2020126
[15]

Ian H. Dinwoodie. Computational methods for asynchronous basins. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3391-3405. doi: 10.3934/dcdsb.2016103
[16]

Angelo B. Mingarelli. Nonlinear functionals in oscillation theory of matrix differential systems. Communications on Pure & Applied Analysis, 2004, 3 (1) : 75-84. doi: 10.3934/cpaa.2004.3.75
[17]

Chunlei Xie, Yujuan Sun. Construction and assignment of orthogonal sequences and zero correlation zone sequences for applications in CDMA systems. Advances in Mathematics of Communications, 2020, 14 (1) : 1-9. doi: 10.3934/amc.2020001
[18]

Huseyin Coskun. Nonlinear decomposition principle and fundamental matrix solutions for dynamic compartmental systems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (12) : 6553-6605. doi: 10.3934/dcdsb.2019155
[19]

Sonia Martínez, Jorge Cortés, Francesco Bullo. A catalog of inverse-kinematics planners for underactuated systems on matrix groups. Journal of Geometric Mechanics, 2009, 1 (4) : 445-460. doi: 10.3934/jgm.2009.1.445
[20]

Daniel Alpay, Eduard Tsekanovskiĭ. Subclasses of Herglotz-Nevanlinna matrix-valued functtons and linear systems. Conference Publications, 2001, 2001 (Special) : 1-13. doi: 10.3934/proc.2001.2001.1

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (69)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences