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 PhysicsNational 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

)7sin(9)4sin(24)3sin(6)sin(3.2)( xxxxxf

Fourier Transform on the flashing of fireflies

Analysis method for regular sequences

FFT on chaotic time series

)cos()()(

2)()( 3

2

2

tFtxdt

tdxtx

dt

txd

42.0,1,25.0,1,1 F

External frequency

produces no useful information.

A typical Random walk

Gaussian distributions

22 2/

4

1)(

xexf

2

)(0

dxxfx

Central Limit Theorem

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.

Fractals

http://www.fourmilab.ch/images/Romanesco/

Benoit B. Mandelbrot (1975)

How long is the coast?

Infinite structures

D = 1.1

Fractal time series D = 1.3 D = 1.5 Random walk

D = 1.7 D = 1.9

MultifractalsB.B. Mandelbrot

Distribution of returns

• Returns:

)()( tfttfr

R.N. Mantegna and H.E. Stanley, Nature 376, 46 (1995)

This can not be explained by the Central limit theorem.

Normal distribution

Normal

Markets

Log-periodic oscillation

)])log(cos(1)[log()( ttCttBAty cc

N. Vandewalle, M. Ausloos, et al, Eur. Phys. J. B4, 139 (1998)

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 fluctuation F(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

HLLF ~)(

C.K. Peng, et al, Phys. Rev. E49, 1685 (1994)

Use of DFA on Polish stock index

L. Czarnecki, D. Grech, and G. Pamula, Physica A387, 6801 (2008)

Crush at March 1994 Crush at January 2008

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

Empirical Mode Decomposition

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

Fractal dimensions of Time series

Examples of fractal functionsWhite noise D = 2.0

Riemann function D=1.226

Fractal Brownian motions D = 1.4

Weierstrass function D=1.8

Random walk D = 1.5

Calculations of the Fractal dimensions

• Hausdorff dimension

• Box-counting dimension

• (Shannon) Information dimension

• Correlation dimension

•

• Fractal dimension

= 2.315

Fractal dimension

= 2.731

None is geometrically intuitive.

Calculate fractal dimensions from turning angles

1

1

2

2

0,)()(2

1

)()()()(

2

1

k

k

oddjkjRjLk

oddj k

jkj

k

kjjkk

tftf

tftftftf

)1(1 2)(2 DkHkk

kk

Physica A388, 3100 (2009),

Sy-Sang Liaw and Feng-Yuan Chiu

Fractal dimension of DJIA index

Dow Jones 1900 - 2007

Red: points

Blue: points

Black: points

1213

1211 1212

)2(G )4(G

)log()2()(log( sDsG

2/1

2/

|2

)2/()2/()(|

1)(

sN

sx

sxfsxfxf

sNsG

mIRMD (modified inverse random midpoint displacement):

S=2

S=4

S=6

Midpoint displacement scale

Calculate fractal dimensions from Midpoint displacements

Calculating fractal dimension using mIRMD

))(log( sG

Weierstrass function D=1.8 White noise

sin(100t)

Random walk

D = 2 – slope for fractals

Fractal dimension of Taiwan stock index

Red: IRMD

Blue: mIRMD))(log( sG

log(s)

45.1D

Mono-fractalsWeierstrass function has single fractal dimension

at every scale everywhere

Fractal dimension of S&P500 — at one minute intervals

SP500—minutes (1987)

D = 1.05

D = 1.40

Bi-fractal!

20 minutes

Crossover at

Fractal dimension of S&P500--minutes

SP500—minutes (1992)

D = 1.38

D = 1.09

Bi-fractal!

20 minutes

Fractal dimension of S&P500—minutes (September 1987)

mIRMD

DFA

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 dimension at every scale everywhere

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).

log(<G(s)>)

log(s)

Stock index at one minute intervals

S&P500

(1987)

Taiwan stock index

(2009 Jan-May)

log(s)

Stock indexes are intrinsically Bi-fractals

Oct. 17

USA: S&P 500 (1987)

Artificial S&P 500 (1987)

Replace every return by 0, +1, or, -1 according to its sign in real data

Bi-fractal property is preserved.

Generate bi-fractals dynamically

Weakly persistent random 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

Trajectories generated using the weakly persistent random model (step length = 1)

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)

Log(<G(s)>)

Trajectories built using the weakly persistent random model (step length = random)

p = 0.8, q = 0.5 p = 0.7, q = 0.5

Taiwan

US sp500

US sp500

US sp500

US sp500

US sp500

US sp500

US sp500

Financial market is a weakly persistent

random walk !

As a consequence, the

financial market is

intrinsically more

unpredictable than

random walks.

The distribution of the returns, and accordingly, the general trend of the market, are mainly determined by external effects.

On the other hand …

• Short-range prediction is possible

Because the stock market is weakly persistent, for many moments, one knows when the market will be up or down with more than 50% chance (probability p > 0.5), so that one can always profit in the stock market (if transaction cost is neglected.)

Profits gained based on the weakly persistent random walk model

S&P500 1987

Net gain at selling

Net gain at buying

0

Average time interval between transactions is 10 minutes.

S&P500 1987-1992

1986 1987 1988 1989 19921990 1991

Net gain at selling Net gain

at buying

Profits gained based on the weakly persistent random walk model

Taiwan stock index 2009 Jan-May

Net gain at selling

Net gain at buying

0

Profits gained based on the weakly persistent random walk model

Random walk

Net gain at selling Net gain

at buying

0

Is the bi-fractal property universal for all stock indexes around the world?

Australia_AORD 2008,9—2009,6

Brazil_BVSP 2008,9—2009,6

China_SSEC 2008,9—2009,6

France_CAC40 2008,9—2009,6

Germany_XDAX 2008,9—2009,6

India_SENSEX 2008,9—2009,6

Japan_NIKKEI225 2008,9—2009,6

Portugal_PSI20 2008,9—2009,6

Taiwan_TAIEX 2008,9—2009,6

US_S&P500 1986,1—1992,12

China

Brazil

India

Japan

Australia

France

Germany

Portugal

Is the bi-fractal property found in every single stock?

Combination of a few stocks

Conclusions

• Markets are intrinsically unpredictable, but some are exploitable.

• WPRW does not work for single stocks.

• For a portfolio consisting of a few selected stocks, the WPRW is still a good model.

Thank you for your attention!