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Detection of financial crisis by Detection of financial crisis by methods of multifractal analysismethods of multifractal analysis
I. AgaevDepartment of Computational Physics
Saint-Petersburg State University
e-mail: [email protected]
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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
)(f
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What is econophysics?
ComputationalComputationalphysicsphysics
Numerical toolsNumerical tools
Complex systemsComplex systemstheorytheory
Economic,Economic,financefinance
EconophysicsEconophysics
MethodologyMethodology Empirical dataEmpirical data
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Methodology of econophysics
Statistical physics (Fokker-Plank equation, Kolmogorov equation,
renormalization group methods)
Chaos and nonlinear dynamics(Lyapunov exponents, attractors,
embedding dimensions)
Artificial neural networks(Clusterisation, forecasts)
Multifractal analysis(R/S-analysis, Hurst exponent,Local Holder exponent, MMAR)
Methodology ofeconophysics
Stochastic processes(Ito’s processes, stable Levi
distributions)
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Financial markets as complex systems
Financial markets Complex systemsComplex systems
1. Open systems2. Multi agent3. Adaptive and
self-organizing4. Scale invariance
Quotes of GBP/USD in different scales
2 hours quotes Weekly quotes Monthly quotes
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Econophysics publicationsBlack-Scholes-MertonBlack-Scholes-Merton
19731973 Modeling hypothesis:Modeling hypothesis:
Efficient marketEfficient marketAbsence of arbitrageAbsence of arbitrage
Gaussian dynamics of returnsGaussian dynamics of returnsBrownian motionBrownian motion
……
Black-Scholes pricing formula:Black-Scholes pricing formula:
C = SN(dC = SN(d11) - Xe) - Xe-r(T-t)-r(T-t)
N(dN(d22))
Reference book: “Options, Reference book: “Options, FuturesFuturesand other derivatives”/J. Hull, and other derivatives”/J. Hull, 20012001
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Econophysics publicationsMantegna-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 ofCrush oflinearlinear
paradigmparadigm
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Econophysics publicationsStanley et al.
Physica A 299 (2001)Log-log cumulativeLog-log cumulative
distribution for stocks:distribution for stocks:power law behaviorpower law behavior
on tails of distributionon tails of distribution
Presence of scalingPresence of scalingin investigated datain investigated data
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Introduction to fractals““Fractal is a structure, composed of parts, which in Fractal is a structure, composed of parts, which in
somesomesense similar to the whole structure”sense similar to the whole structure”
B. MandelbrotB. Mandelbrot
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Introduction to fractals““The basis of fractal geometry is the idea of self-The basis of fractal geometry is the idea of self-
similarity”similarity”S. BozhokinS. Bozhokin
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Introduction to fractals““Nature shows us […] another level of complexity. Amount ofNature shows us […] another level of complexity. Amount of
different scales of lengths in [natural] structures is almost different scales of lengths in [natural] structures is almost infinite”infinite”
B. MandelbrotB. Mandelbrot
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Iterated Function Systems
IFS femIFS fem
Real femReal fem
50x zoom of IFS 50x zoom of IFS femfem
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Iterated Function Systems
AAffine transformationffine transformation
Values of coefficientsValues of coefficientsand corresponding and corresponding pp
Resulting fem forResulting fem for5000, 10000, 50000 5000, 10000, 50000 iterationsiterations
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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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Length changes asLength changes asmeasurement toolmeasurement tool
doesdoes
Fractal dimension
What’s the length of Norway coastline?What’s the length of Norway coastline?
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Fractal dimension
What’s the length of Norway coastline?What’s the length of Norway coastline?
L( ) = a 1-D
D – fractal (Hausdorf)dimension
Reference book: “Fractals”Reference book: “Fractals”J. Feder, 1988 J. Feder, 1988
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Definitions
FractalFractal – is a set with fractal (Hausdorf) dimension greater – is a set with fractal (Hausdorf) dimension greater than its topological dimensionthan its topological dimension
Box-counting methodBox-counting method
If If N(N( ) ) 1/ 1/ dd at at 0 0
0
ln ( )lim
ln
ND
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Fractal functions
(2 )
(1 cos )( ) Re ( )
n
D nn
b tC t W t
b
Wierstrass function is scale-invariantWierstrass function is scale-invariant
DD=1.=1.22
DD=1.=1.55
DD=1.=1.88
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Scaling properties of Wierstrass function
From homogeneityFrom homogeneityC(bt)C(bt)==bb22--DDC(t)C(t)
Fractal Wierstrass function with Fractal Wierstrass function with bb=1.5, =1.5, DD=1.8=1.8
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Scaling properties of Wierstrass function
Change of variablesChange of variablest t b b44ttc(t) c(t) b b4(2-D)4(2-D)c(t) c(t)
Fractal Wierstrass function with Fractal Wierstrass function with bb=1.5, =1.5, DD=1.8=1.8
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Multifractals
0
5
10
15
20
25
30N
umber of f
am
ilies
$
Distribution of income
Figville Tree City
Fractal dimension – “average” all over the fractalFractal dimension – “average” all over the fractalLocal properties of fractal are, in general, different Local properties of fractal are, in general, different
Important
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Generalized dimensions
( )
1
0
ln1 ( )
lim1 ln 1
Nqi
iq
pq
Dq q
Definition:Definition:
Artificial multifractal Artificial multifractal
Reney dimensions
Artificial monofractal Artificial monofractal
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British poundBritish pound
Generalized dimensions
( )
1
0
ln1 ( )
lim1 ln 1
Nqi
iq
pq
Dq q
Definition:Definition:Renée
dimensions
S&P 500 S&P 500
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Special cases of generalized dimensions
0 0
ln ( )lim
ln
ND
( )
11 0
lnlim
ln
N
i ii
p pD
Right-hand side of expression can be recognized as Right-hand side of expression can be recognized as definition of definition of fractal dimension.fractal dimension. It’s rough characteristic of It’s rough characteristic of fractal, doesn’t provide any information about it’s statistical fractal, doesn’t provide any information about it’s statistical properties.properties.
DD11 is called is called information dimensioninformation dimension because it makes use because it makes use
of of ppln(p)ln(p) form associated with the usual definition of form associated with the usual definition of “information” for a probability distribution. A numerator “information” for a probability distribution. A numerator accurate to sign represent to entropy of fractal set.accurate to sign represent to entropy of fractal set.
Correlation sum defines the probability that two randomly Correlation sum defines the probability that two randomly taken points are divided by distance less than taken points are divided by distance less than . D . D2 2 defines defines
dependence of correlation sum on dependence of correlation sum on 00.. That’s why DThat’s why D22 is is
called called correlation dimensioncorrelation dimension..
( )
2
12 0
limln
N
ii
pD
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Local Holder exponents
( ) ii i ip k
More convenienttool Scaling relation:Scaling relation:
where where II - - scaling indexscaling index or or local Holder exponentlocal Holder exponent
EExtreme casextreme cases:s: ( ) q qD
min
q
dD
dq
max
q
dD
dq
min max
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Local Holder exponents
( ) ii i ip k
More convenienttool Scaling relation:Scaling relation:
where where II - - scaling indexscaling index or or local Holder exponentlocal Holder exponent
( ) d q
dq
( )( ) ( )
d q
f q qdq
LegendreLegendretransformtransform
The link between The link between {q,{q,(q)}(q)} and and {{ ,f( ,f()})}
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Function of multifractal spectra
( ) n dDistribution ofscaling indexes
What is number of cells that have a scaling index inWhat is number of cells that have a scaling index inthe range between the range between and and + d + d ??
( ) DNFor monofractals:For monofractals:
For multifractals:For multifractals:( )( ) fn
Non-homogeneous Non-homogeneous Cantor’s setCantor’s set
Homogeneous Homogeneous Cantor’s setCantor’s set
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Function of multifractal spectra
( ) n dDistribution ofscaling indexes
What is number of cells that have a scaling index inWhat is number of cells that have a scaling index inthe range between the range between and and + d + d ??
( ) DNFor monofractals:For monofractals:
For multifractals:For multifractals:( )( ) fn
S&P 500 S&P 500 indexindex
British British poundpound
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f(f( ))
DD00
minmin maxmax00
Using function of multifractal spectraUsing function of multifractal spectrato determine to determine fractal dimensionfractal dimension
Properties of multifractal spectra
Determining of the most important
dimensions
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Properties of multifractal spectra
Determining of the most important
dimensions
DD11
f(f())
DD11
Using function of multifractal spectraUsing function of multifractal spectrato determine to determine information dimensioninformation dimension
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Properties of multifractal spectra
Determining of the most important
dimensions
DD22/2/2
f(f())
22
22-D-D22
Using function of multifractal spectraUsing function of multifractal spectrato determine to determine correlation dimensioncorrelation dimension
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Multifractal analysisDefinitions Let Let Y(t)Y(t) is the asset price is the asset price
X(t,X(t,t) = (ln Y(t+t) = (ln Y(t+t) - ln t) - ln
Y(t))Y(t))22
Divide [Divide [0,T0,T] into ] into NN intervals intervalsof length of length t t and define and define sample sum:sample sum:
Define the Define the scaling function:scaling function:0
ln ( , )( ) lim
ln
q
t
Z T tq
t
•If If DDqq D D00 for some for some qq then then X(t,1)X(t,1) is multifractal time series is multifractal time series•For monofractal time series scaling function For monofractal time series scaling function (q)(q) is linear: is linear: (q)=D(q)=D00(q-1)(q-1)
Remarks:Remarks:
The The spectrum of fractal dimensionsspectrum of fractal dimensions of squared log-returns of squared log-returns X(t,1) X(t,1) is defined asis defined as
( )
1
q
qD
q
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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 The multifractal spectrum function f(f() ) describes thedescribes the distribution of local Holder exponent in multifractal process:distribution of local Holder exponent in multifractal process:
where where NN((t) t) is the number of intervals is the number of intervals of size of size t t characterized by the fixed characterized by the fixed
The multifractal spectrum function The multifractal spectrum function f(f() ) describes thedescribes the distribution of local Holder exponent in multifractal process:distribution of local Holder exponent in multifractal process:
where where NN((t) t) is the number of intervals is the number of intervals of size of size t t characterized by the fixed characterized by the fixed
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Description of major USA market crashes
•Computer tradingComputer trading•Trade & budget deficitsTrade & budget deficits•OvervaluationOvervaluation
October October 19871987
•Oil embargo Oil embargo •Inflation (15-17%)Inflation (15-17%)•High oil pricesHigh oil prices•Declined debt paysDeclined debt pays
Summer Summer 19821982
•Asian crisisAsian crisis•Internationality of Internationality of US corp.US corp.•OvervaluationOvervaluation
Autumn 1998Autumn 1998 September 2001September 2001
•Terror in New YorkTerror in New York•OvervaluationOvervaluation•Economic problemsEconomic problems•High-tech crisisHigh-tech crisis
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Singularity at financial markets
( ) ( )f t h f t const h
Remark:Remark: as as =1=1, , f(x)f(x) becomes a differentiable function becomes a differentiable functionas as =0=0, , f(x) f(x) has a nonremovable discontinuity has a nonremovable discontinuity
( )( ) ( ) tf t h f t const h
- local Holder exponents - local Holder exponents ((tt))
Local Holder exponents are convenientLocal Holder exponents are convenientmeasurement tool of singularitymeasurement tool of singularity
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DJIA 1980-1988
Log-priceLog-price
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DJIA 1995-2002Log-priceLog-price
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Detection of 1987 crash
Log-priceLog-price
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Detection of 2001crashLog-priceLog-price
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Acknowledgements
Professor Yu. Kuperin, Saint-Petersburg State Professor Yu. Kuperin, Saint-Petersburg State UniversityUniversity
Professor S. Slavyanov, Saint-Petersburg State Professor S. Slavyanov, Saint-Petersburg State UniversityUniversity
Professor C. Zenger, Professor C. Zenger, Technische UniversitätTechnische Universität M München ünchen
My family – dad, mom and sisterMy family – dad, mom and sister
My friends – Oleg, Timothy, Alex and otherMy friends – Oleg, Timothy, Alex and other