Detection of critical events by methods of multifractal analysis

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Detection of critical events by methods of
multifractal analysis

• Ilyas A. Agaev
• Department of Computational Physics
• Saint-Petersburg State University
• e-mail ilya-agaev_at_yandex.ru

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Contents

• Introduction to econophysics
• What is econophysics?
• Methodology of econophysics
• Fractals
• Iterated function systems
• Introduction to theory of fractals
• Multifractals
• Generalized fractal dimensions
• Local Holder exponents
• Function of multifractal spectrum
• Case study
• Multifractal analysis
• Detection of crisis on financial markets

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What is econophysics?
Computational physics
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Methodology of econophysics
Multifractal analysis (R/S-analysis, Hurst
exponent, Local Holder exponent, MMAR)
Chaos and nonlinear dynamics (Lyapunov exponents,
attractors, embedding dimensions)
Methodology of econophysics
Statistical physics (Fokker-Plank equation,
Kolmogorov equation, renormalization group
methods)
Artificial neural networks (Clusterisation,
forecasts)
Stochastic processes (Itos processes, stable
Levi distributions)
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Financial markets as complex systems
Financial markets
Complex systems

• Open systems
• Multi agent
• Adaptive and self-organizing
• Scale invariance

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Econophysics publications
Black-Scholes-Merton 1973
Modeling hypothesis Efficient market Absence of
arbitrage Gaussian dynamics of returns Brownian
motion
Black-Scholes pricing formula C SN(d1) -
Xe-r(T-t)N(d2)
Reference book Options, Futures and other
derivatives/J. Hull, 2001
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Econophysics publications
Mantegna-Stanley Physica A 239 (1997)

• Experimental data (logarithm of prices) fit to
• Gaussian distribution until 2 std.
• Levy distribution until 5 std.
• Then they appear truncate

Crush of linear paradigm
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Econophysics publications
Stanley et al. Physica A 299 (2001)
Log-log cumulative distribution for stocks power
law behavior on tails of distribution
Presence of scaling in investigated data
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Introduction to fractals
Fractal is a structure, composed of parts,
which in some sense similar to the whole
structure B. Mandelbrot
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Introduction to fractals
The basis of fractal geometry is the idea of
self-similarity S. Bozhokin
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Introduction to fractals
Nature shows us another level of complexity.
Amount of different scales of lengths in
natural structures is almost infinite B.
Mandelbrot
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Iterated Function Systems
Real fem
IFS fem
50x zoom of IFS fem
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Iterated Function Systems
Affine transformation
Values of coefficients and corresponding p
Resulting fem for 5000, 10000, 50000 iterations
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Iterated Function Systems
Without the first line in the table one obtains
the fern without stalk
The first two lines in the table are responsible
for the stalk growth
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Fractal dimension
Whats the length of Norway coastline?
Length changes as measurement tool does
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Fractal dimension
Whats the length of Norway coastline?
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Definitions
Box-counting method
If N(? ) ? 1/? d at ? ? 0
Fractal is a set with fractal (Hausdorf)
dimension greater than its topological dimension
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Fractal functions
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Scaling properties of Wierstrass function
From homogeneity C(bt)b2-DC(t)
Fractal Wierstrass function with b1.5, D1.8
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Scaling properties of Wierstrass function
Change of variables t ? b4t c(t) ?
b4(2-D)c(t)
Fractal Wierstrass function with b1.5, D1.8
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Multifractals
Important
Fractal dimension average all over the
fractal Local properties of fractal are, in
general, different
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Generalized dimensions
Definition
Reney dimensions
Artificial multifractal
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Generalized dimensions
Definition
Renée dimensions
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Special cases of generalized dimensions
Right-hand side of expression can be recognized
as definition of fractal dimension. Its rough
characteristic of fractal, doesnt provide any
information about its statistical properties.
D1 is called information dimension because it
makes use of p?ln(p) form associated with the
usual definition of information for a
probability distribution. A numerator accurate to
sign represent to entropy of fractal set.
Correlation sum defines the probability that two
randomly taken points are divided by distance
less than ? . D2 defines dependence of
correlation sum on ? ? 0. Thats why D2 is called
correlation dimension.
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Local Holder exponents
More convenient tool
Scaling relation
where ?I - scaling index or local Holder exponent
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Local Holder exponents
More convenient tool
Scaling relation
where ?I - scaling index or local Holder exponent
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Function of multifractal spectra
Distribution of scaling indexes
What is number of cells that have a scaling index
in the range between ? and ? d? ?
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Function of multifractal spectra
Distribution of scaling indexes
What is number of cells that have a scaling index
in the range between ? and ? d? ?
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Properties of multifractal spectra
Determining of the most important dimensions
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Properties of multifractal spectra
Determining of the most important dimensions
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Properties of multifractal spectra
Determining of the most important dimensions
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Multifractal analysis
Definitions
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MF spectral function
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Singularity at financial markets
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Description of major USA market crashes
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DJIA 1980-1988
Log-price
?
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DJIA 1995-2002
Log-price
?
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RUR/USD 1992-1994
RUR/USD
?
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RUR/USD 1994-1999
RUR/USD
?
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Detection of 1987 crash
Log-price
?
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Detection of 2001crash
Log-price
?
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Acknowledgements