Stochastic dynamical model for stock-stock correlations

Physical Review E - Statistical, Nonlinear, and Soft Matter Physics

Volumn 70, Issue 2 2, 2004, Pages

  Ma, Wen Jong ^a   Hu, Chin Kun ^a   Amritkar, Ravindra E ^b  

^a INSTITUTE OF PHYSICS   (Taiwan)

^b PHYSICAL RESEARCH LABORATORY   (India)

Author keywords

[No Author keywords available]

Indexed keywords

COMPUTER SIMULATION; CORRELATION METHODS; EIGENVALUES AND EIGENFUNCTIONS; MATHEMATICAL MODELS; MATRIX ALGEBRA; PARAMETER ESTIMATION; SPECTRUM ANALYSIS; STATISTICAL METHODS;

COUPLING CONSTANTS; EIGENVALUE DISTRIBUTION; RANDOM NOISE; STOCK-STOCK CORRELATIONS;

RANDOM PROCESSES;

EID: 37649031583     PISSN: 15393755     EISSN: None     Source Type: Journal    
DOI: 10.1103/PhysRevE.70.026101     Document Type: Article

References (43)

1. M. L. Mehta, Random Matrices, 2nd ed. (Academic Press, New York, 1991).
2. See, for example, V. V. Flambaum et al., Phys. Rev. A 50, 267 (1994); A. D. Mirlin, Phys. Rep. 326, 259 (2000); W.-J. Ma et al., Physica A 321, 364 (2003); S. N. Taraskin and S. R. Elliott, Phys. Rev. B 65, 052201 (2002).
3. See, for example, V. V. Flambaum et al., Phys. Rev. A 50, 267 (1994); A. D. Mirlin, Phys. Rep. 326, 259 (2000); W.-J. Ma et al., Physica A 321, 364 (2003); S. N. Taraskin and S. R. Elliott, Phys. Rev. B 65, 052201 (2002).
4. See, for example, V. V. Flambaum et al., Phys. Rev. A 50, 267 (1994); A. D. Mirlin, Phys. Rep. 326, 259 (2000); W.-J. Ma et al., Physica A 321, 364 (2003); S. N. Taraskin and S. R. Elliott, Phys. Rev. B 65, 052201 (2002).
5. See, for example, V. V. Flambaum et al., Phys. Rev. A 50, 267 (1994); A. D. Mirlin, Phys. Rep. 326, 259 (2000); W.-J. Ma et al., Physica A 321, 364 (2003); S. N. Taraskin and S. R. Elliott, Phys. Rev. B 65, 052201 (2002).
6. L. Laloux, P. Cizeau, J.-P. Bouchaud, and M. Potters, Phys. Rev. Lett. 83, 1467 (1999).
7. V. Plerou et al., Phys. Rev. Lett. 83, 1471 (1999); V. Plerou et al., Phys. Rev. E 65, 66126 (2002).
8. V. Plerou et al., Phys. Rev. Lett. 83, 1471 (1999); V. Plerou et al., Phys. Rev. E 65, 66126 (2002).
9. M. Barthélemy, B. Gondran, and E. Guichard, Phys. Rev. E 66, 056110 (2002).
10. M. S. Santhanam and P. K. Patra, Phys. Rev. E 64, 016102 (2001).
11. J. Voit, The Statistical Mechanics of Financial Markets (Springer, New York, 2001); R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge University Press, Cambridge, 2000); J.-P. Bouchaud and M. Potters, Theory of Financial Risk: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000).
12. J. Voit, The Statistical Mechanics of Financial Markets (Springer, New York, 2001); R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge University Press, Cambridge, 2000); J.-P. Bouchaud and M. Potters, Theory of Financial Risk: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000).
13. J. Voit, The Statistical Mechanics of Financial Markets (Springer, New York, 2001); R. N. Mantegna and H. E. Stanley, An Introduction to Econophysics: Correlations and Complexity in Finance (Cambridge University Press, Cambridge, 2000); J.-P. Bouchaud and M. Potters, Theory of Financial Risk: From Statistical Physics to Risk Management (Cambridge University Press, Cambridge, 2000).
14. E. J. Elton and M. J. Gruber, Modern Portfolio Theory and Investment Analysis, 3rd ed. (Wiley, New York, 1995); J. Y. Campbell, A. W. Lo, and A. C. MacKinlay, The Econometrics of Financial Markets (Princeton University, Princeton, NJ, 1997).
15. E. J. Elton and M. J. Gruber, Modern Portfolio Theory and Investment Analysis, 3rd ed. (Wiley, New York, 1995); J. Y. Campbell, A. W. Lo, and A. C. MacKinlay, The Econometrics of Financial Markets (Princeton University, Princeton, NJ, 1997).
16. L. Bachelier, Ann. Sci. Ec. Normale Super. 17, 21 (1900); English translation of The Random Character of Stock Market Prices, edited by P. H. Cootner (MIT, Cambridge, MA, 1964).
17. L. Bachelier, Ann. Sci. Ec. Normale Super. 17, 21 (1900); English translation of The Random Character of Stock Market Prices, edited by P. H. Cootner (MIT, Cambridge, MA, 1964).
18. T. Lux and M. Marchest, Nature (London) 397, 498 (1999).
19. J. D. Noh, Phys. Rev. E 61, 5981 (2000).
20. S. Drozdz et al., Physica A 287, 440 (2000).
21. A. M. Sengupta, and P. P. Mitra, Phys. Rev. E 60, 3389 (1999).
22. N. Vandewalle, F. Brisbois, and X. Tordoir, Quant. Finance 1, 372 (2001) ; G. Bonanno, F. Lillo, and R. N. Mantegna, ibid. 1, 96 (2001); R. N. Mantegna, Eur. Phys. J. B 11, 193 (1999).
23. N. Vandewalle, F. Brisbois, and X. Tordoir, Quant. Finance 1, 372 (2001) ; G. Bonanno, F. Lillo, and R. N. Mantegna, ibid. 1, 96 (2001); R. N. Mantegna, Eur. Phys. J. B 11, 193 (1999).
24. N. Vandewalle, F. Brisbois, and X. Tordoir, Quant. Finance 1, 372 (2001) ; G. Bonanno, F. Lillo, and R. N. Mantegna, ibid. 1, 96 (2001); R. N. Mantegna, Eur. Phys. J. B 11, 193 (1999).
25. L. Giada and M. Marsili, Phys. Rev. E 63, 061101 (2001).
26. K. Kaneko, Physica D 124, 322 (1998); 34, 1 (1989); S. Jalan and R. E. Amritkar, Phys. Rev. Lett. 90, 014101 (2003); P. M. Gade and C.-K. Hu, Phys. Rev. E 60, 4966 (1999); 62, 6409 (2000).
27. K. Kaneko, Physica D 124, 322 (1998); 34, 1 (1989); S. Jalan and R. E. Amritkar, Phys. Rev. Lett. 90, 014101 (2003); P. M. Gade and C.-K. Hu, Phys. Rev. E 60, 4966 (1999); 62, 6409 (2000).
28. K. Kaneko, Physica D 124, 322 (1998); 34, 1 (1989); S. Jalan and R. E. Amritkar, Phys. Rev. Lett. 90, 014101 (2003); P. M. Gade and C.-K. Hu, Phys. Rev. E 60, 4966 (1999); 62, 6409 (2000).
29. K. Kaneko, Physica D 124, 322 (1998); 34, 1 (1989); S. Jalan and R. E. Amritkar, Phys. Rev. Lett. 90, 014101 (2003); P. M. Gade and C.-K. Hu, Phys. Rev. E 60, 4966 (1999); 62, 6409 (2000).
30. K. Kaneko, Physica D 124, 322 (1998); 34, 1 (1989); S. Jalan and R. E. Amritkar, Phys. Rev. Lett. 90, 014101 (2003); P. M. Gade and C.-K. Hu, Phys. Rev. E 60, 4966 (1999); 62, 6409 (2000).
31. s.
32. M.
33. G provides the statistical overlap in Eq. (11), leading to the blocked structure in the correlation matrix which accounts for the presence of those deviated eigenvalues from the bulk of the distribution.
34. M describes then the maximum possible mean square fluctuation over time, and the components of the vector f giving that maximum describe the corresponding portfolio.
35. 2 + 1(1 + o(ε̃ 1/ε̃N)), for 1/N ≪ ε̃ ≪ 1.
36. i normalized by the price, which justifies the use of the stochastically identical walks in our model.
37. max,min=1.7578..., 0.4545....
38. We survey the time sequences of trading prices from the Trades and Quotes (TAQ) database for one pool of N=345 US stocks selected from S&P 500, which have complete records over the four years 1996-1999.
39. We also use our model to fit the spectral data of Ref. [4] with N=1000 and 1=6448 (not shown) and find good agreement. For this fit the groups were not chosen from eigenvector analysis.
40. D. Korenblum and D. Shalloway, Phys. Rev. E 67, 056704 (2003), and references therein.
41. M. Marsili, Quant. Finance 2, 297 (2002).
42. 2〉's with the market data. (In practice, it is necessary to carry out certain coarse graining for this mean-field model by neglecting certain stock-by-stock differences in the real market data.) The estimation is then further finely adjusted to match the spectra. In the procedure, there is always some compromise between reproducing the grouped activities in the eigenvectors and matching the distribution spectra.
43. Normally, the inverse participation ratio [4,5] is used to estimate the fraction of components which significantly contribute to that eigenvector. However, it cannot tell us about the elements (walks) which are common to different groups, an information crucial to determine the different groups in our case. Hence, we use a different procedure to determine the groups as described in the text.