Financial Dynamics, Minority Game and Herding Model B. Zheng Zhejiang University.
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Transcript of Financial Dynamics, Minority Game and Herding Model B. Zheng Zhejiang University.
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Financial Dynamics,
Minority Game and Herding Model
B. Zheng
Zhejiang University
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Contents
I Introduction
II Financial dynamics
III Two-phase phenomenon
IV Minority Game
V Herding model
VI Conclusion
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I Introduction
Should physicists remain in traditional physics?
Two ways for penetrating to other subjects:
* fundamental chemistry, 地球物理 biophysics
* phenomenological econophysics social physics
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Scaling and universality exist widely in nature• chaos, turbulence• self-organized critical phenomena• earthquake, biology, medicine• financial dynamics, economics• society (traffic, internet, …)
Physical background strongly correlated self-similarity universality
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Methods• phenomenology of experimental data• models• Monte Carlo simulations• theoretical study
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II Financial dynamics
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Mantegna and Stanley, Nature 376 (1995)46
Large amount of data Universal scaling behavior
Financial index Y(t')
Variation Z(t) = Y(t' +t) – Y(t')
Probability distribution P(Z, t)
shorter t truncated Levy distribution longer t Gaussian
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Scaling form
Zero return
--- self-similarity in time direction usually robust or universal
)1,/(),( /1/1 tZPttZP
4.1),0( /1 ttP
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t
P(0,t)
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Let
Auto-correlation
exponentially decay
But
power-law decay!!
)'()1'()'( tYtYtY
2)'()'()'()( tYtYttYtA
2|)'(||)'(||)'(|)( tYtYttYtA
te
t
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t (min)
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t (min)
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Summary
* Y(t’)△ is short-range correlated* | Y(t’)|△ is long-range correlated
*
* for big Z, small t
* High-low asymmetry* Time reverse asymmetry ……
/1),0( ttP ZtZP ),(
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III Two-phase phenomenon
Index Y(t')
Variation Z(t) = Y(t' +t) – Y(t')
Conditional probability distribution
P(Z, r)
Here
r(t) = < | Y(t''+1)-Y(t'') - < Y(t''+1)-Y(t'')> | >
< … > is the average in [t', t'+t]
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Plerou, Gopikrishnan and Stanley, Nature 421 (2003) 130
Y(t') = Volume imbalance, t < 1 day
r small, P(Z, r) has a single peak
rc critical point
r big, P(Z, r) has double peaks
Our finding
Two-phase phenomenon exists also for
Y(t') = Financial index
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German DAX94-97 t = 10 rc = .15
Solid line: r < .1Dashed : .2 < r < .3Squares : .4 < r < .5Crosses : .6 < r < 1.0Triangles : 1.0 < r
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German DAX t = 20 rc = .30
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IV Minority Game
History : time steps, states
Strategies:
agents producers
s strategies 1 strategy and inactive
Scoring : minority wins
Price : Y(t') = buyers - sellers
m2mm22
aN pN
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This Minority game explains most of
stylized fact of financial markets
including long-range correlation, but
NOT the two-phase phenomenon
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Minority Game m = 2 s = 2 t = 10
Solid line: r < 30Dashed : 30 < r < 60Squares : 60 < r < 120Crosses : 120 < r
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Minority Game m = 2 s = 2 t = 50
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V Herding model
EZ model : Eguiluz and Zimmermann, Phys. Rev. Lett. 85 (2000)5659
N agents, at time t, pick agent i
1) with probability 1-a, connect to agent j, form a cluster;
2) with probability a , cluster i buy (sell), resolve the cluster i
Price variation : | Y(t')| = size of cluster △ i
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This herding model explains
the power-law decay (fat-tail) of P(Z, t), but
NOT the long-range correlation
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EZ model t = 10
Solid line: r < 20Dashed : 20 < r < 40Squares : 60 < r < 80Crosses : 120 < r
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EZ model t =100
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Interacting herding model
B. Zheng, F. Ren, S. Trimper and D.F. Zheng
1/a : rate of information transmission
Dynamic interaction
1/b is the highest rate
* take a small b * fix c to the ‘critical’ value : P(Z,t) obeys a power-law
scba /
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1
1
1
0
short-range anti-correlated
short-range correlated
long-range correlatedqualitatively explains the markets
unknown
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Interacting EZ model
t = 1001
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Interacting EZ model
t = 1001
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Interacting EZ model
t = 1001
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Interacting EZ model 20 < r <40
solid line: t = 50 dashed : t = 100 crosses : t = 200 diam. : DAX
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VI Conclusion
* There are two phases in financial markets
* There is no connection between long-range correlation and two-phase phenomenon
* The interacting dynamic herding model is rather successful including two-phase phenomenon, persistence probability ……
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谢谢
http://zimp.zju.edu.cn