Detection of financial crisis by methods of multifractal analysis I. Agaev Department of Computational Physics Saint-Petersburg State University e-mail:

$Page 1: Detection of financial crisis by methods of multifractal analysis I. Agaev Department of Computational Physics Saint-Petersburg State University e-mail:$
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]

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

What is econophysics?

ComputationalComputationalphysicsphysics

Numerical toolsNumerical tools

Complex systemsComplex systemstheorytheory

Economic,Economic,financefinance

EconophysicsEconophysics

MethodologyMethodology Empirical dataEmpirical data

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)

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

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

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

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

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

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

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

Iterated Function Systems

IFS femIFS fem

Real femReal fem

50x zoom of IFS 50x zoom of IFS femfem


AAffine transformationffine transformation

Values of coefficientsValues of coefficientsand corresponding and corresponding pp

Resulting fem forResulting fem for5000, 10000, 50000 5000, 10000, 50000 iterationsiterations


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

Length changes asLength changes asmeasurement toolmeasurement tool

doesdoes

Fractal dimension

What’s the length of Norway coastline?What’s the length of Norway coastline?

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

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

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

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

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

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

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

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

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

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

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()})}

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

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

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



dimensions

DD11

f(f())

DD11

Using function of multifractal spectraUsing function of multifractal spectrato determine to determine information dimensioninformation dimension



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

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

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

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

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

DJIA 1980-1988

Log-priceLog-price

DJIA 1995-2002Log-priceLog-price

Detection of 1987 crash

Log-priceLog-price

Detection of 2001crashLog-priceLog-price

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

Documents

Detection of financial crisis by methods of multifractal analysis I. Agaev Department of Computational Physics Saint-Petersburg State University e-mail: