Detection of financial crisis by methods of multifractal analysis I. Agaev Department of Computational Physics Saint-Petersburg State University
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
What is econophysics?Computationalphysics Numerical tools Complex systems theoryEconomic,finance Econophysics Methodology Empirical 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 of econophysics Stochastic processes (Ito’s processes, stable Levi distributions)
Financial markets as complex systems Financial markets Complex systems 1. 1.Open systems 2. 2.Multi agent 3. 3.Adaptive and self-organizing 4. 4.Scale invariance Quotes of GBP/USD in different scales 2 hours quotesWeekly quotesMonthly quotes
Econophysics publicationsBlack-Scholes-Merton1973 Modeling hypothesis: Efficient market Absence of arbitrage Gaussian dynamics of returns Brownian motion … Black-Scholes pricing formula: C = SN(d 1 ) - Xe -r(T-t) N(d 2 ) 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 Gaussian distribution until 2 std Levy distribution until 5 std Then they appear truncate Crush of linearparadigm
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 IFS fem Real fem 50x zoom of IFS fem
Iterated Function Systems Affine transformation Values of coefficients and corresponding p Resulting fem for 5000, 10000, 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
Length changes as measurement tool does Fractal dimension What’s the length of Norway coastline?
Fractal dimension What’s the length of Norway coastline? L ( ) = a 1-D D – fractal (Hausdorf) dimension Reference book: “Fractals” J. Feder, 1988
Definitions Fractal – is a set with fractal (Hausdorf) dimension greater than its topological dimension Box-counting method If N( ) 1/ d at 0
Fractal functions Wierstrass function is scale-invariant D=1.2 D=1.5 D=1.8
Scaling properties of Wierstrass function From homogeneity C(bt) = b 2 - D C(t) Fractal Wierstrass function with b=1.5, D=1.8
Scaling properties of Wierstrass function Change of variables t b 4 t c(t) b 4(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 Important
Generalized dimensionsDefinition: Artificial multifractal Reney dimensions Artificial monofractal
British pound Generalized dimensionsDefinition: Renée dimensions S&P 500
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. D 1 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 . D 2 defines dependence of correlation sum on 0. That’s why D 2 is called correlation dimension.
Local Holder exponents More convenient tool Scaling relation: where I - scaling index or local Holder exponent Extreme cases:
Local Holder exponents More convenient tool Scaling relation: where I - scaling index or local Holder exponent Legendretransform The link between {q, (q)} and { ,f( )}
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
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
f( ) D0D0D0D0 min max 0000 Using function of multifractal spectra to determine fractal dimension Properties of multifractal spectra Determining of the most important dimensions
Properties of multifractal spectra Determining of the most important dimensions D1D1D1D1 f( ) D1D1D1D1 Using function of multifractal spectra to determine information dimension
Properties of multifractal spectra Determining of the most important dimensions D 2 /2 f( ) 2222 2 -D 2 Using function of multifractal spectra to determine correlation dimension
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: If D q D 0 for some q then X(t,1) is multifractal time seriesIf D q D 0 for some q then X(t,1) is multifractal time series For monofractal time series scaling function (q)For monofractal time series scaling function (q) is linear: (q)=D 0 (q-1) is linear: (q)=D 0 (q-1) Remarks: The spectrum of fractal dimensions of squared log-returns X(t,1) is defined as
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: distribution of local Holder exponent in multifractal process: where N ( t) is the number of intervals of size t characterized by the fixed The multifractal spectrum function f( ) describes the distribution of local Holder exponent in multifractal process: 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 Computer tradingComputer trading Trade & budget deficitsTrade & budget deficits OvervaluationOvervaluation October 1987 Oil embargoOil embargo Inflation (15-17%)Inflation (15-17%) High oil pricesHigh oil prices Declined debt paysDeclined debt pays Summer 1982 Asian crisisAsian crisis Internationality ofInternationality of US corp. OvervaluationOvervaluation Autumn 1998 September 2001 Terror in New YorkTerror in New York OvervaluationOvervaluation Economic problemsEconomic problems High-tech crisisHigh-tech crisis
Singularity at financial marketsRemark: 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
DJIA Log-price
DJIA Log-price
Detection of 1987 crash Log-price
Detection of 2001crashLog-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