Transcript Detection of financial crisis by methods of multifractal

Detection of financial crisis by
methods of multifractal analysis
I. Agaev
Department of Computational Physics
Saint-Petersburg State University
e-mail: [email protected]
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 f ( )
• Case study
• Multifractal analysis
• Detection of crisis on financial markets
What is econophysics?
Complex systems
theory
Computational
physics
Economic,
finance
Methodology
Numerical tools
Empirical data
Econophysics
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)
Financial markets as complex systems
Financial markets
Complex systems
1. Open systems
2. Multi agent
3. Adaptive and
self-organizing
4. Scale invariance
Quotes of GBP/USD in different scales
2 hours quotes
Weekly quotes
Monthly quotes
Econophysics publications
Black-Scholes-Merton
1973
Modeling hypothesis:
Efficient market
Absence of arbitrage
Gaussian dynamics of returns
Brownian motion
…
Black-Scholes pricing formula:
-r(T-t)
C = SN(d1) - Xe
N(d2)
Reference book: “Options, Futures
and other derivatives”/J. Hull, 2001
Econophysics publications
Mantegna-Stanley
Physica A 239 (1997)
Experimental data (logarithm of prices) fit to
1. Gaussian distribution until 2 std.
2. Levy distribution until 5 std.
3. Then they appear truncate
Crush of
linear
paradigm
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
Introduction to fractals
“Fractal is a structure, composed of parts, which in some
sense similar to the whole structure”
B. Mandelbrot
Introduction to fractals
“The basis of fractal geometry is the idea of self-similarity”
S. Bozhokin
Introduction to fractals
“Nature shows us […] another level of complexity. Amount of
different scales of lengths in [natural] structures is almost infinite”
B. Mandelbrot
Iterated Function Systems
Real fem
IFS fem
50x zoom of IFS fem
Iterated Function Systems
Affine transformation
Values of coefficients
and corresponding p
Resulting fem for
5000, 10000, 50000
iterations
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
Fractal dimension
What’s the length of Norway coastline?
Length changes as
measurement tool
does
Fractal dimension
What’s the length of Norway coastline?
L( ) = a
1-D
D – fractal (Hausdorf)
dimension
Reference book: “Fractals”
J. Feder, 1988
Definitions
Box-counting method
If N( )  1/ d at   0
ln N ( )
D   lim
 0
ln 
Fractal – is a set with fractal (Hausdorf) dimension greater
than its topological dimension
Fractal functions
D=1.2
Wierstrass function is scale-invariant
(1  cos bnt )
C(t )  ReW (t )  
(2D )n
b
n 

D=1.5
D=1.8
Scaling properties of Wierstrass function
From homogeneity
C(bt)=b2-DC(t)
Fractal Wierstrass function with b=1.5, D=1.8
Scaling properties of Wierstrass function
Change of variables
t

c(t) 
b4 t
b4(2-D)c(t)
Fractal Wierstrass function with b=1.5, D=1.8
Multifractals
Fractal dimension – “average” all over the fractal
Local properties of fractal are, in general, different
Distribution of income
Number of families
Important
30
25
20
15
10
5
0
$
Figville
Tree City
Generalized dimensions
Definition:
Reney
dimensions
N( )
ln  piq
1
i 1
Dq  lim
 0 q  1
ln
Artificial multifractal

 (q)
1 q
Artificial monofractal
Generalized dimensions
Definition:
Renée
dimensions
N( )
ln  piq
1
i 1
Dq  lim
 0 q  1
ln
S&P 500

 (q)
1 q
British pound
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.
ln N( )
 0 ln 
D0   lim
N ( )
D1  lim
 0
 p ln p
i 1
i
i
ln 
N ( )
D2  lim
 0
p
i 1
ln 
2
i
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.
Local Holder exponents
More convenient
tool
Scaling relation:
pi ( i )  k  
i
i
where I - scaling index or local Holder exponent
Extreme cases:
d
dq
d
dq
 (q  )  qD
 D   min
q 
min    max
 D   max
q 
Local Holder exponents
More convenient
tool
Scaling relation:
pi ( i )  k  
i
i
where I - scaling index or local Holder exponent
The link between {q,(q)} and { ,f()}

Legendre
transform
d (q )
dq
d (q)
f ( )  q
  (q)
dq
Function of multifractal spectra
Distribution of
scaling indexes
What is number of cells that have a scaling index in
the range between  and  + d ? n( )d
For monofractals:
N ( )    D
For multifractals:
n( )    f ( )
Non-homogeneous
Cantor’s set
Homogeneous
Cantor’s set
Function of multifractal spectra
Distribution of
scaling indexes
What is number of cells that have a scaling index in
the range between  and  + d ? n( )d
For monofractals:
For multifractals:
S&P 500 index
N ( )    D
n( )  
 f ( )
British pound
Properties of multifractal spectra
Using function of multifractal spectra
to determine fractal dimension
D0
f()
Determining of
the most important
dimensions
min
0
max
Properties of multifractal spectra
Determining of
the most important
dimensions
Using function of multifractal spectra
to determine information dimension
f()
D1

D1
Properties of multifractal spectra
Determining of
the most important
dimensions
Using function of multifractal spectra
to determine correlation dimension
f()
2-D2
D2/2
2

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:
 (q)  lim
The spectrum of fractal dimensions
of squared log-returns X(t,1) is defined as
Remarks:
t 0
ln Z q (T , t )
ln t
Dq 
 (q)
q 1
•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)
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 
Description of major USA market crashes
Summer 1982
October 1987
•Oil embargo
•Inflation (15-17%)
•High oil prices
•Declined debt pays
•Computer trading
•Trade & budget deficits
•Overvaluation
Autumn 1998
September 2001
•Asian crisis
•Internationality of
US corp.
•Overvaluation
•Terror in New York
•Overvaluation
•Economic problems
•High-tech crisis
Singularity at financial markets

f (t  h)  f (t )  const  h
Remark:
as =1, f(x) becomes a differentiable function
as =0, f(x) has a nonremovable discontinuity
 (t )
f (t  h)  f (t )  const  h
- local Holder exponents (t)
Local Holder exponents are convenient
measurement tool of singularity
DJIA 1980-1988
Log-price

DJIA 1995-2002
Log-price

Detection of 1987 crash
Log-price

Detection of 2001crash
Log-price

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