Coupled Harmonic Oscillators Model for Financial Time Series
Palavras-chave:
Econophysics, Harmonic oscillator network, Financial market model, Heavy-tailed distribution, Multifractal time series
Resumo
Neste artigo, apresentamos um modelo baseado em osciladores harmônicos dissipativos linearmente acoplados para caracterizar a dinâmica dos preços de ativos, cujos elementos fundamentais são forças restauradoras, inércia e amortecimento. As variáveis dinâmicas do sistema físico são substituídas por diferenças logarítmicas entre os preços dos ativos e preços de referência calculados por meio de médias móveis. O modelo é aplicado à ações que compõem o índice brasileiro Ibovespa. Os coeficientes de amortecimento e as constantes elásticas estimados sugerem a presença de forças atrativas e repulsivas no mercado, com tendência à regressão à média e a movimentos amortecidos conjuntos do mercado. As distribuições desses parâmetros indicam variações significativas no grau de interconexão entre as ações, oferecendo percepções valiosas para a seleção de ativos em estratégias de hedge. Quantidades análogas às energias cinética e potencial exibem distribuições com caudas em lei de potência e comportamento multifractal, e suas dinâmicas revelam indícios de eventos que impactaram tanto os mercados domésticos quanto os internacionais.
Referências
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[2] Caldeira, A.O., Leggett, A.J.: Quantum Tunnelling in a Dissipative System. Annals of Physics 149, 374–456 (1983)
[3] Bosco de Magalh˜aes, A.R., Mokarzel, S.G., Nemes, M.C., Terra Cunha, M.O.: Decay Rate and Decoherence Control in Coupled Dissipative Cavities. Physica A: Statistical Mechanics and its Applications 341, 234–250 (2004)
[4] Strogatz, S.H.: From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators. Physica D: Nonlinear Phenomena 143, 1–20 (2000)
[5] D¨orfler, F., Bullo, F.: Synchronization in Complex Networks of Phase Oscillators: A Survey. Automatica 50, 1539–1564 (2014)
[6] Agarwal, G.S.: In: Hohler, G. (ed.) Quantum Statistical Theories of Spontaneous Emission and their Relation to Other Approaches. Springer Tracts in Modern Physics, vol. 70. Springer, Berlin (1974)
[7] Lidar, D.A., Birgitta Whaley, K.: In: Benatti, F., Floreanini, R. (eds.) Decoherence-Free Subspaces and Subsystems. Lecture Notes in Physics, vol. 622, pp. 83–120. Springer, Berlin (2003)
[8] Bosco de Magalh˜aes, A.R.: Searching for Robust Quantum Memories in Many Coupled Oscillators. Physics Letters A 375, 4120–4129 (2011)
[9] Frisch, R.: Propagation Problems and Impulse Problems in Dynamic Economics. Economic Essays in Honor of Gustav Cassel, pp. 171–205. George Allen & Unwin, London (1933)
[10] Takayasu, M., Mizuno, T., Takayasu, H.: Potential Force Observed in Market Dynamics. Physica A: Statistical Mechanics and its Applications 370, 91–97 (2006)
[11] Ataullah, A., Tippett, M.: Equity Prices as a Simple Harmonic Oscillator with Noise. Physica A: Statistical Mechanics and its Applications 382, 557–564 (2007)
[12] Ye, C., Huang, J.P.: Non-Classical Oscillator Model for Persistent Fluctuations in Stock Markets. Physica A: Statistical Mechanics and its Applications 387, 1255–1263 (2008)
[13] Nastasiuk, V.A.: Fisher Information and Quantum Potential Well Model for Finance. Physics Letters A 379, 1998–2000 (2015)
[14] Kulesza, S., Belej, M.: Local Real Estate Markets in Poland as a Network of Damped Harmonic Oscillators. Acta Physica Polonica A 127(3-A), 99–102 (2015)
[15] Meng, X., Zhang, J., Xu, J., Guo, H.: Quantum Spatial-Periodic Harmonic Model for Daily Price-Limited Stock Markets. Physica A: Statistical Mechanics and its Applications 438, 154–160 (2015)
[16] Meng, X., Zhang, J., Guo, H.: Quantum Brownian Motion Model for the Stock Market. Physica A: Statistical Mechanics and its Applications 452, 281–288 (2016)
[17] Xavier, P.O.C., Atman, A.P.F., Bosco de Magalh˜aes, A.R.: Equation-Based Model for the Stock Market. Physical Review E 96, 032305 (2017)
[18] Ahn, K., Choi, M.Y., Dai, B., Sohn, S., Yang, B.: Modeling Stock Return Distributions with a Quantum Harmonic Oscillator. Europhysics Letters 120(3), 38003 (2017)
[19] Orrell, D.: A Quantum Model of Supply and Demand. Physica A: Statistical Mechanics and its Applications 539, 122928 (2020)
[20] Garcia, M.M., Machado Pereira, A.C., Acebal, J.L., Bosco de Magalh˜aes, A.R.: Forecast Model for Financial Time Series: An Approach Based on Harmonic Oscillators. Physica A: Statistical Mechanics and its Applications 549, 124365 (2020)
[21] Enders, W.: Applied Econometric Time Series, 4th edn. Wiley, ??? (2015)
[22] Resende, C.C., Pereira, A.C.M., Cardoso, R.T.N., Bosco de Magalh˜aes, A.R.: Investigating Market Efficiency Through a Forecasting Model Based on Differential Equations. Physica A: Statistical Mechanics and its Applications 474, 199–212 (2017)
[23] Tsay, R.S.: Analysis of Financial Time Series, 3rd edn. Wiley, ??? (2010)
[24] Granger, C.W.J., Newbold, P.: Spurious regressions in econometrics. Journal of Econometrics 2(2), 111–120 (1974)
[25] Campbell, J.Y., Lo, A.W., MacKinlay, A.C.: The Econometrics of Financial Markets. Princeton University Press, Princeton, New Jersey (1997)
[26] Hamilton, J.D.: Time Series Analysis. Princeton University Presss, Princeton (1994)
[27] Sanders, A.: The Subprime Crisis and its Role in the Financial Crisis. Journal of Housing Economics 17, 254–261 (2008)
[28] Cucinotta, D., Vanelli, M.: WHO Declares COVID-19 a Pandemic. Acta Biomed. 91, 157–160 (2020)
[29] Melo Modenesi, A., Luporini, V., Pimentel, D.: In: Arestis, P., Troncoso Baltar, C., Prates, D.M. (eds.) Asymmetric Exchange Rate Pass-Through: Evidence, Inflation Dynamics and Policy Implications for Brazil (1999–2016), pp. 69–99. Springer, Cham (2017)
[30] Avritzer, L.: The Rousseff Impeachment and the Crisis of Democracy in Brazil. Critical Policy Studies 11, 352–357 (2017)
[31] Nunes, F., Melo, C.R.: Impeachment, Political Crisis and Democracy in Brazil. Revista de Ciencia Pol´ıtica 37(2), 281–304 (2017)
[32] Silvestre, B.S., Monteiro, M.S., Viana, F.L.E., Sousa-Filho, J.M.: Challenges for Sustainable Supply Chain Management: When Stakeholder Collaboration Becomes Conducive to Corruption. Journal of Cleaner Production 194, 766–776 (2018)
[33] Mantegna, R.N., Stanley, H.E.: Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, Cambridge (1999)
[34] Lux, T., Marchesi, M.: Scaling and Criticality in a Stochastic Multi-Agent Model of a Financial Market. Nature (London) 397, 498–500 (1999)
[35] Gopikrishnan, P., Plerou, V., Nunes Amaral, L.A., Meyer, M., Stanley, H.E.: Scaling of the distribution of fluctuations of financial market indices. Phys. Rev. E 60, 5305–5316 (1999)
[36] Cajueiro, D.O., Tabak, B.M.: The hurst exponent over time: testing the assertion that emerging markets are becoming more efficient. Physica A: Statistical Mechanics and its Applications 336(3), 521–537 (2004)
[37] Di Matteo, T., Aste, T., Dacorogna, M.M.: Long-Term Memories of Developed and Emerging Markets: Using the Scaling Analysis to Characterize their Stage of Development. Journal of Banking & Finance 29, 827–851 (2005)
[38] Atman, B. A. P. F. Amin Gon¸calves: Influence of the investor’s behavior on the complexity of the stock market. Brazilian Journal of Physics (2012)
[39] Kwapie´n, J., Dro˙zd˙z, S.: Physical Approach to Complex Systems. Physics Reports 515, 115–226 (2012)
[40] Yang, G., Zhu, C.G., An, K.N., Huang, J.P.: Overall Fluctuations and Fat Tails in an Artificial Financial Market: The Two-Sided Impact of Leveraged Trading. Physics Letters A 379, 1857–1863 (2015)
[41] Botta, F., Moat, H.S., Stanley, H.E., Preis, T.: Quantifying Stock Return Distributions in Financial Markets. PLoS ONE 10(9), 0135600 (2015)
[42] Buonocore, R.J., Aste, T., Di Matteo, T.: Measuring Multiscaling in Financial Time-Series. Chaos, Solitons & Fractals 88, 38–47 (2016)
[43] Bosco de Magalhães, A.R.: Wealth Dynamics in a Market with Information Asymmetries. Physical Review E 107, 014305 (2023)
[44] Ducha, F.A., Atman, A.P.F., Bosco de Magalhães, A.R.: Information Flux in Complex Networks: Path to Stylized Facts. Physica A: Statistical Mechanics and its Applications 566, 125638 (2021)
[45] Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-Law Distributions in Empirical Data. SIAM Review 51(4), 661–703 (2009)
[46] Hurst, H.E.: Long-term storage capacity of reservoirs. Transactions of the American society of civil engineers 116(1), 770–799 (1951)
[47] Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Mosaic organization of dna nucleotides. Phys. Rev. E 49, 1685–1689 (1994)
[48] Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A., Stanley, H.E.: Multifractal detrended fluctuation analysis of nonstationary time series. Physica A: Statistical Mechanics and its Applications 316(1), 87–114 (2002)
[2] Caldeira, A.O., Leggett, A.J.: Quantum Tunnelling in a Dissipative System. Annals of Physics 149, 374–456 (1983)
[3] Bosco de Magalh˜aes, A.R., Mokarzel, S.G., Nemes, M.C., Terra Cunha, M.O.: Decay Rate and Decoherence Control in Coupled Dissipative Cavities. Physica A: Statistical Mechanics and its Applications 341, 234–250 (2004)
[4] Strogatz, S.H.: From Kuramoto to Crawford: Exploring the Onset of Synchronization in Populations of Coupled Oscillators. Physica D: Nonlinear Phenomena 143, 1–20 (2000)
[5] D¨orfler, F., Bullo, F.: Synchronization in Complex Networks of Phase Oscillators: A Survey. Automatica 50, 1539–1564 (2014)
[6] Agarwal, G.S.: In: Hohler, G. (ed.) Quantum Statistical Theories of Spontaneous Emission and their Relation to Other Approaches. Springer Tracts in Modern Physics, vol. 70. Springer, Berlin (1974)
[7] Lidar, D.A., Birgitta Whaley, K.: In: Benatti, F., Floreanini, R. (eds.) Decoherence-Free Subspaces and Subsystems. Lecture Notes in Physics, vol. 622, pp. 83–120. Springer, Berlin (2003)
[8] Bosco de Magalh˜aes, A.R.: Searching for Robust Quantum Memories in Many Coupled Oscillators. Physics Letters A 375, 4120–4129 (2011)
[9] Frisch, R.: Propagation Problems and Impulse Problems in Dynamic Economics. Economic Essays in Honor of Gustav Cassel, pp. 171–205. George Allen & Unwin, London (1933)
[10] Takayasu, M., Mizuno, T., Takayasu, H.: Potential Force Observed in Market Dynamics. Physica A: Statistical Mechanics and its Applications 370, 91–97 (2006)
[11] Ataullah, A., Tippett, M.: Equity Prices as a Simple Harmonic Oscillator with Noise. Physica A: Statistical Mechanics and its Applications 382, 557–564 (2007)
[12] Ye, C., Huang, J.P.: Non-Classical Oscillator Model for Persistent Fluctuations in Stock Markets. Physica A: Statistical Mechanics and its Applications 387, 1255–1263 (2008)
[13] Nastasiuk, V.A.: Fisher Information and Quantum Potential Well Model for Finance. Physics Letters A 379, 1998–2000 (2015)
[14] Kulesza, S., Belej, M.: Local Real Estate Markets in Poland as a Network of Damped Harmonic Oscillators. Acta Physica Polonica A 127(3-A), 99–102 (2015)
[15] Meng, X., Zhang, J., Xu, J., Guo, H.: Quantum Spatial-Periodic Harmonic Model for Daily Price-Limited Stock Markets. Physica A: Statistical Mechanics and its Applications 438, 154–160 (2015)
[16] Meng, X., Zhang, J., Guo, H.: Quantum Brownian Motion Model for the Stock Market. Physica A: Statistical Mechanics and its Applications 452, 281–288 (2016)
[17] Xavier, P.O.C., Atman, A.P.F., Bosco de Magalh˜aes, A.R.: Equation-Based Model for the Stock Market. Physical Review E 96, 032305 (2017)
[18] Ahn, K., Choi, M.Y., Dai, B., Sohn, S., Yang, B.: Modeling Stock Return Distributions with a Quantum Harmonic Oscillator. Europhysics Letters 120(3), 38003 (2017)
[19] Orrell, D.: A Quantum Model of Supply and Demand. Physica A: Statistical Mechanics and its Applications 539, 122928 (2020)
[20] Garcia, M.M., Machado Pereira, A.C., Acebal, J.L., Bosco de Magalh˜aes, A.R.: Forecast Model for Financial Time Series: An Approach Based on Harmonic Oscillators. Physica A: Statistical Mechanics and its Applications 549, 124365 (2020)
[21] Enders, W.: Applied Econometric Time Series, 4th edn. Wiley, ??? (2015)
[22] Resende, C.C., Pereira, A.C.M., Cardoso, R.T.N., Bosco de Magalh˜aes, A.R.: Investigating Market Efficiency Through a Forecasting Model Based on Differential Equations. Physica A: Statistical Mechanics and its Applications 474, 199–212 (2017)
[23] Tsay, R.S.: Analysis of Financial Time Series, 3rd edn. Wiley, ??? (2010)
[24] Granger, C.W.J., Newbold, P.: Spurious regressions in econometrics. Journal of Econometrics 2(2), 111–120 (1974)
[25] Campbell, J.Y., Lo, A.W., MacKinlay, A.C.: The Econometrics of Financial Markets. Princeton University Press, Princeton, New Jersey (1997)
[26] Hamilton, J.D.: Time Series Analysis. Princeton University Presss, Princeton (1994)
[27] Sanders, A.: The Subprime Crisis and its Role in the Financial Crisis. Journal of Housing Economics 17, 254–261 (2008)
[28] Cucinotta, D., Vanelli, M.: WHO Declares COVID-19 a Pandemic. Acta Biomed. 91, 157–160 (2020)
[29] Melo Modenesi, A., Luporini, V., Pimentel, D.: In: Arestis, P., Troncoso Baltar, C., Prates, D.M. (eds.) Asymmetric Exchange Rate Pass-Through: Evidence, Inflation Dynamics and Policy Implications for Brazil (1999–2016), pp. 69–99. Springer, Cham (2017)
[30] Avritzer, L.: The Rousseff Impeachment and the Crisis of Democracy in Brazil. Critical Policy Studies 11, 352–357 (2017)
[31] Nunes, F., Melo, C.R.: Impeachment, Political Crisis and Democracy in Brazil. Revista de Ciencia Pol´ıtica 37(2), 281–304 (2017)
[32] Silvestre, B.S., Monteiro, M.S., Viana, F.L.E., Sousa-Filho, J.M.: Challenges for Sustainable Supply Chain Management: When Stakeholder Collaboration Becomes Conducive to Corruption. Journal of Cleaner Production 194, 766–776 (2018)
[33] Mantegna, R.N., Stanley, H.E.: Introduction to Econophysics: Correlations and Complexity in Finance. Cambridge University Press, Cambridge (1999)
[34] Lux, T., Marchesi, M.: Scaling and Criticality in a Stochastic Multi-Agent Model of a Financial Market. Nature (London) 397, 498–500 (1999)
[35] Gopikrishnan, P., Plerou, V., Nunes Amaral, L.A., Meyer, M., Stanley, H.E.: Scaling of the distribution of fluctuations of financial market indices. Phys. Rev. E 60, 5305–5316 (1999)
[36] Cajueiro, D.O., Tabak, B.M.: The hurst exponent over time: testing the assertion that emerging markets are becoming more efficient. Physica A: Statistical Mechanics and its Applications 336(3), 521–537 (2004)
[37] Di Matteo, T., Aste, T., Dacorogna, M.M.: Long-Term Memories of Developed and Emerging Markets: Using the Scaling Analysis to Characterize their Stage of Development. Journal of Banking & Finance 29, 827–851 (2005)
[38] Atman, B. A. P. F. Amin Gon¸calves: Influence of the investor’s behavior on the complexity of the stock market. Brazilian Journal of Physics (2012)
[39] Kwapie´n, J., Dro˙zd˙z, S.: Physical Approach to Complex Systems. Physics Reports 515, 115–226 (2012)
[40] Yang, G., Zhu, C.G., An, K.N., Huang, J.P.: Overall Fluctuations and Fat Tails in an Artificial Financial Market: The Two-Sided Impact of Leveraged Trading. Physics Letters A 379, 1857–1863 (2015)
[41] Botta, F., Moat, H.S., Stanley, H.E., Preis, T.: Quantifying Stock Return Distributions in Financial Markets. PLoS ONE 10(9), 0135600 (2015)
[42] Buonocore, R.J., Aste, T., Di Matteo, T.: Measuring Multiscaling in Financial Time-Series. Chaos, Solitons & Fractals 88, 38–47 (2016)
[43] Bosco de Magalhães, A.R.: Wealth Dynamics in a Market with Information Asymmetries. Physical Review E 107, 014305 (2023)
[44] Ducha, F.A., Atman, A.P.F., Bosco de Magalhães, A.R.: Information Flux in Complex Networks: Path to Stylized Facts. Physica A: Statistical Mechanics and its Applications 566, 125638 (2021)
[45] Clauset, A., Shalizi, C.R., Newman, M.E.J.: Power-Law Distributions in Empirical Data. SIAM Review 51(4), 661–703 (2009)
[46] Hurst, H.E.: Long-term storage capacity of reservoirs. Transactions of the American society of civil engineers 116(1), 770–799 (1951)
[47] Peng, C.-K., Buldyrev, S.V., Havlin, S., Simons, M., Stanley, H.E., Goldberger, A.L.: Mosaic organization of dna nucleotides. Phys. Rev. E 49, 1685–1689 (1994)
[48] Kantelhardt, J.W., Zschiegner, S.A., Koscielny-Bunde, E., Havlin, S., Bunde, A., Stanley, H.E.: Multifractal detrended fluctuation analysis of nonstationary time series. Physica A: Statistical Mechanics and its Applications 316(1), 87–114 (2002)
Publicado
2026-06-01
Como Citar
de Oliveira, M. R. A., Raad, L. V. V., & de Magalhães, A. R. B. (2026). Coupled Harmonic Oscillators Model for Financial Time Series. Revista De Matemática Da UFOP, 1(1). https://doi.org/10.63801/rmat.v1i1.8528
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