Double exponential instability of triangular arbitrage systems

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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

Abstract
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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 and 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.
[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.
[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.
[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.
[5]	BIS, "BIS Effective Exchange Rate Indices," Bank for International Settlements, Basel, Switzerland, 2012, http://www.bis.org/statistics/eer/index.htm.
[6]	G. Cassel, The present situation of the foreign exchanges. I, Economic Journal, 26 (1916), 62-65.
[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.
[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.
[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.
[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.
[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.
[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).
[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.
[14]	IMF, "SDR Valuation,'' International Monetary Fund, 2012, http://www.imf.org/external/np/fin/data/rms_sdrv.aspx.
[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.
[16]	J. M. Keynes, "A Tract on Monetary Reform,'' Macmilan, London UK, 1923.
[17]	M. Klau and S. S. Fung, The new BIS effective exchange rate indices, Bank for International Settlements Quarterly Review, 51-65, http://www.bis.org/publ/qtrpdf/r_qt0603e.pdf.
[18]	V. Kozyakin, B. O'Callaghan and A. Pokrovskii, Sequences of arbitrages, ArXiv.org e-Print archive, arXiv:1004.0561.
[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.
[20]	S. A. Ross, A simple approach to the valuation of risky streams, Journal of Business, 51 (1978), 453-475.
[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.
[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.

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.
[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.
[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.
[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.
[5]	BIS, "BIS Effective Exchange Rate Indices," Bank for International Settlements, Basel, Switzerland, 2012, http://www.bis.org/statistics/eer/index.htm.
[6]	G. Cassel, The present situation of the foreign exchanges. I, Economic Journal, 26 (1916), 62-65.
[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.
[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.
[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.
[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.
[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.
[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).
[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.
[14]	IMF, "SDR Valuation,'' International Monetary Fund, 2012, http://www.imf.org/external/np/fin/data/rms_sdrv.aspx.
[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.
[16]	J. M. Keynes, "A Tract on Monetary Reform,'' Macmilan, London UK, 1923.
[17]	M. Klau and S. S. Fung, The new BIS effective exchange rate indices, Bank for International Settlements Quarterly Review, 51-65, http://www.bis.org/publ/qtrpdf/r_qt0603e.pdf.
[18]	V. Kozyakin, B. O'Callaghan and A. Pokrovskii, Sequences of arbitrages, ArXiv.org e-Print archive, arXiv:1004.0561.
[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.
[20]	S. A. Ross, A simple approach to the valuation of risky streams, Journal of Business, 51 (1978), 453-475.
[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.
[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.

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