1. Detection of financial crisis by methods of multifractal analysis
I. Agaev
Department of Computational Physics
Saint-Petersburg State University
JAAS 2004

2. Contents Introduction to econophysics Fractals Multifractals
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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3. What is econophysics? Econophysics Complex systems theory Economic,
Numerical tools
Complex systems
theory
Economic,
finance
Econophysics
Methodology
Empirical data
Computational
physics
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4. 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
(Ito’s processes, stable Levi
distributions)
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5. Financial markets as complex systems
Open systems
Multi agent
Adaptive and self-organizing
Scale invariance
Quotes of GBP/USD in different scales
2 hours quotes
Weekly quotes
Monthly quotes
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6. 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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7. 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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8. 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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9. 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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10. Introduction to fractals
“The basis of fractal geometry is the idea of self-similarity”
S. Bozhokin
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11. 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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12. Iterated Function Systems
Real fem
IFS fem
50x zoom of IFS fem
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13. Iterated Function Systems
Affine transformation
Values of coefficients
and corresponding p
Resulting fem for
5000, 10000, 50000
iterations
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14. 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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15. Fractal dimension What’s the length of Norway coastline?
Length changes as
measurement tool
does
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16. Fractal dimension L( ) = a 1-D
What’s the length of Norway coastline?
L( ) = a 1-D
D – fractal (Hausdorf)
dimension
Reference book: “Fractals”
J. Feder, 1988
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17. 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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18. Fractal functions D=1.2 Wierstrass function is scale-invariant D=1.5
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19. Scaling properties of Wierstrass function
From homogeneity
C(bt)=b2-DC(t)
Fractal Wierstrass function with b=1.5, D=1.8
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20. Scaling properties of Wierstrass function
Change of variables
t  b4t
c(t)  b4(2-D)c(t)
Fractal Wierstrass function with b=1.5, D=1.8
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21. Multifractals Important
Fractal dimension – “average” all over the fractal
Local properties of fractal are, in general, different
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22. Generalized dimensions
Definition:
Reney
dimensions
Artificial multifractal
Artificial monofractal
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23. Generalized dimensions
Definition:
Renée
dimensions
S&P 500
British pound
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24. Special cases of generalized dimensions
Right-hand side of expression can be recognized as definition of fractal dimension. It’s rough characteristic of fractal, doesn’t provide any information about it’s 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. That’s why D2 is called correlation dimension.
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25. Local Holder exponents
More convenient
tool
Scaling relation:
where I - scaling index or local Holder exponent
Extreme cases:
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26. Local Holder exponents
More convenient
tool
Scaling relation:
where I - scaling index or local Holder exponent
Legendre
transform
The link between {q,(q)} and { ,f()}
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27. Function of multifractal spectra
Distribution of
scaling indexes
What is number of cells that have a scaling index in
the range between  and  + d ?
For monofractals:
For multifractals:
Non-homogeneous Cantor’s set
Homogeneous Cantor’s set
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28. Function of multifractal spectra
Distribution of
scaling indexes
What is number of cells that have a scaling index in
the range between  and  + d ?
For monofractals:
For multifractals:
S&P 500 index
British pound
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29. Properties of multifractal spectra
Determining of
the most important
dimensions
f() D0
min
max 0
Using function of multifractal spectra
to determine fractal dimension
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30. Properties of multifractal spectra
Determining of
the most important
dimensions D1
f() 
Using function of multifractal spectra
to determine information dimension
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31. Properties of multifractal spectra
Determining of
the most important
dimensions 
D2/2
f() 2
2-D2
Using function of multifractal spectra
to determine correlation dimension
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32. Multifractal analysis
Definitions
Let Y(t) is the asset price
X(t,t) = (ln Y(t+t) - ln Y(t))2
Divide [0,T] into N intervals
of length t and define
sample sum:
Define the scaling function:
The spectrum of fractal dimensions
of squared log-returns X(t,1) is defined as
If Dq  D0 for some q then X(t,1) is multifractal time series
For monofractal time series scaling function (q)
is linear: (q)=D0(q-1)
Remarks:
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33. MF spectral function Multifractal series can be characterized
by local Holder exponent (t):
as t  0
Remark: in classical asset pricing model
(geometrical brownian motion) (t)=1
The multifractal spectrum function f() describes the
distribution of local Holder exponent in multifractal process:
where N(t) is the number of intervals
of size t characterized by the fixed 
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34. Description of major USA market crashes
Oil embargo
Inflation (15-17%)
High oil prices
Declined debt pays
Summer 1982
Computer trading
Trade & budget deficits
Overvaluation
October 1987
Asian crisis
Internationality of
US corp.
Overvaluation
Autumn 1998
September 2001
Terror in New York
Overvaluation
Economic problems
High-tech crisis
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35. Singularity at financial markets
Remark:
as =1, f(x) becomes a differentiable function
as =0, f(x) has a nonremovable discontinuity
- local Holder exponents (t)
Local Holder exponents are convenient
measurement tool of singularity
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36. DJIA
Log-price 
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37. DJIA
Log-price 
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38. Detection of 1987 crash
Log-price 
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39. Detection of 2001crash
Log-price 
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40. Acknowledgements
Professor Yu. Kuperin, Saint-Petersburg State University
Professor S. Slavyanov, Saint-Petersburg State University
Professor C. Zenger, Technische Universität München
My family – dad, mom and sister
My friends – Oleg, Timothy, Alex and other
JAAS 2004