1 / 87

股市可以預測嗎 ? — 碎形觀點 Markets are unpredictable, but some exploitable

股市可以預測嗎 ? — 碎形觀點 Markets are unpredictable, but some exploitable. 廖思善 Sy-Sang Liaw Department of Physics National Chung-Hsing Univ, Taiwan October, 2009. Taiwan Stock Index (1971-2005). Dow Jones Industrial Average (DJIA index 1900-2007). An example of time series. Fourier Transform.

chadwilliams
Download Presentation

股市可以預測嗎 ? — 碎形觀點 Markets are unpredictable, but some exploitable

An Image/Link below is provided (as is) to download presentation Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author. Content is provided to you AS IS for your information and personal use only. Download presentation by click this link. While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server. During download, if you can't get a presentation, the file might be deleted by the publisher.

E N D

Presentation Transcript

  1. 股市可以預測嗎?—碎形觀點Markets are unpredictable, but some exploitable 廖思善 Sy-Sang Liaw Department of Physics National Chung-Hsing Univ, Taiwan October, 2009

  2. Taiwan Stock Index (1971-2005)

  3. Dow Jones Industrial Average (DJIA index 1900-2007)

  4. An example of time series

  5. Fourier Transform

  6. Fourier Transform on the flashing of fireflies Analysis method for regular sequences

  7. FFT on chaotic time series produces no useful information. External frequency

  8. A typical Random walk

  9. Gaussian distributions Central Limit Theorem

  10. Louis Bachelier (1870 – 1946) • PhD thesis: The Theory of Speculation, (published 1900). • Bachelier's work on random walks predated Einstein's celebrated study of Brownian motion by five years. • Black-Scholes model (1997 Nobel prize) assumes the price follows a Brownian motion.

  11. Fractals Benoit B. Mandelbrot (1975) http://www.fourmilab.ch/images/Romanesco/

  12. How long is the coast?

  13. Infinite structures

  14. Fractal time series D = 1.5 Random walk D = 1.3 D = 1.1 D = 1.7 D = 1.9

  15. Multifractals B.B. Mandelbrot

  16. Distribution of returns • Returns: Markets Normal Normal distribution This can not be explained by the Central limit theorem. R.N. Mantegna and H.E. Stanley, Nature 376, 46 (1995)

  17. Log-periodic oscillation N. Vandewalle, M. Ausloos, et al, Eur. Phys. J. B4, 139 (1998)

  18. Detrended Fluctuation Analysis(DFA) • (1) Time sequence of length N is divided into non- overlapping intervals of length L • (2) For each interval the linear trend is subtracted from the signal • (3) Calculate the rms fluctuationF(L) of the detrended signal and F(L) is averaged over all intervals • (4) The procedure is repeated for intervals of all length L<N • (5) One expects where H stands for the Hurst exponent C.K. Peng, et al, Phys. Rev. E49, 1685 (1994)

  19. Use of DFA on Polish stock index Crush at January 2008 Crush at March 1994 L. Czarnecki, D. Grech, and G. Pamula, Physica A387, 6801 (2008)

  20. T. Lux and M. Marchesi, Nature 397, 498 (1999) Stochastic multi-agent model

  21. Empirical Mode Decomposition N.E. Huang and Z. Wu, Review of Geophysics 46, RG2006 (2008)

  22. Fractal dimensions of Time series

  23. Examples of fractal functions White noise D = 2.0 Weierstrass function D=1.8 Fractal Brownian motions D = 1.4 Random walk D = 1.5 Riemann function D=1.226

  24. Calculations of the Fractal dimensions Hausdorff dimension Box-counting dimension (Shannon) Information dimension Correlation dimension None is geometrically intuitive. Fractal dimension = 2.315 Fractal dimension = 2.731

  25. Calculate fractal dimensions from turning angles Physica A388, 3100 (2009), Sy-Sang Liaw and Feng-Yuan Chiu

  26. Fractal dimension of DJIA index Red: points Blue: points Black: points Dow Jones 1900 - 2007

  27. Calculate fractal dimensions from Midpoint displacements S=2 S=4 S=6 mIRMD (modified inverse random midpoint displacement): scale Midpoint displacement

  28. Calculating fractal dimension using mIRMD D = 2 – slope for fractals Weierstrass function D=1.8 White noise Random walk sin(100t)

  29. Fractal dimension of Taiwan stock index Red: IRMD Blue: mIRMD log(s)

  30. Mono-fractalsWeierstrass function has single fractal dimensionat every scale everywhere

  31. Fractal dimension of S&P500 — at one minute intervals Bi-fractal! SP500—minutes (1987) D = 1.40 Crossover at D = 1.05 20 minutes

  32. Fractal dimension of S&P500--minutes Bi-fractal! SP500—minutes (1992) D = 1.38 D = 1.09 20 minutes

  33. Fractal dimension of S&P500—minutes (September 1987) mIRMD DFA

  34. Bi-fractals • A special kind of scale-dependent fractal has one fractal dimension for small scales and the other fractal dimension for scales larger than a certain value. We will call these fractals, bi-fractals. Mono-fractalsMono-fractals such as the Weierstrass function and the trajectory of a random walk have single fractal dimensionat every scale everywhere

  35. Bi-fractals have been observed in many real data, including • heart rate signals[1,2]; • fluctuations of fatigue crack growth[3]; • wind speed data[4]; • precipitation and river runoff records[5]; • stock indexes at one minute intervals[6] [1] T. Penzel, J.W. Kantelhardt, H.F. Becker, J.H. Peter, and A. Bunde, Comput. Cardiol., 30, 307 (2003). [2] S. Havlin, L.A.N. Amaral, Y. Ashkenazy, A.L. Goldberger, P.Ch. Ivanov, C.-K. Peng, and H.E. Stanley, and, Physica A274, 99 (1999). [3] N. Scafetta, A. Ray, and B.J. West, Physica A359, 1 (2006). [4] R.G. Kavasseri and R. Nagarajan, IEEE Trans. Circuits Syst., Part I: Fundamental Theory and Applications 51, 2255 (2004). [5] J.W. Kantelhardt, E. Koscielny-Bunde, D. Rybski, P. Braum, A. Bunde, and S. Havlin, J. Geophys. Res. 111, D01106 (2008). [6] Y. Liu, P. Cizeau, M. Meyer, C.-K. Peng, and H.E. Stanley, Physica A245, 437 (1997).

  36. Stock index at one minute intervals Taiwan stock index (2009 Jan-May) S&P500 (1987) log(<G(s)>) log(s) log(s)

  37. Stock indexes are intrinsically Bi-fractals

  38. Oct. 17 USA: S&P 500 (1987)

  39. Artificial S&P 500 (1987) Replace every return by 0, +1, or, -1 according to its sign in real data

  40. Bi-fractal property is preserved.

  41. Generate bi-fractalsdynamically Weakly persistentrandom walk model: • up – up -- probability p > 0.5 up • down – down – probability p > 0.5 down • up – down – probability q up • down – up – probability q down

  42. Trajectories generated using the weakly persistent random model (step length = 1) Log(<G(s)>) Black: p = 0.9, q = 0.5 Red: p = 0.8, q = 0.5 Blue: p = 0.7, q = 0.5 Brown: p = 0.6, q = 0.5 Log(S)

  43. Trajectories built using the weakly persistent random model (step length = random) p = 0.8, q = 0.5 p = 0.7, q = 0.5

  44. Taiwan

  45. US sp500

  46. US sp500

  47. US sp500

  48. US sp500

  49. US sp500

  50. US sp500

More Related