Detection of financial crisis by methods of multifractal analysis I. Agaev Department of Computational Physics Saint-Petersburg State University e-mail: ilya-agaev@yandex.ru JAAS 2004

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 JAAS 2004

What is econophysics? Econophysics Complex systems theory Economic, Numerical tools Complex systems theory Economic, finance Econophysics Methodology Empirical data Computational physics JAAS 2004

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) JAAS 2004

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 JAAS 2004

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 JAAS 2004

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 JAAS 2004

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 JAAS 2004

Introduction to fractals “Fractal is a structure, composed of parts, which in some sense similar to the whole structure” B. Mandelbrot JAAS 2004

Introduction to fractals “The basis of fractal geometry is the idea of self-similarity” S. Bozhokin JAAS 2004

Introduction to fractals “Nature shows us […] another level of complexity. Amount of different scales of lengths in [natural] structures is almost infinite” B. Mandelbrot JAAS 2004

Iterated Function Systems Real fem IFS fem 50x zoom of IFS fem JAAS 2004

Iterated Function Systems Affine transformation Values of coefficients and corresponding p Resulting fem for 5000, 10000, 50000 iterations JAAS 2004

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 JAAS 2004

Fractal dimension What’s the length of Norway coastline? Length changes as measurement tool does JAAS 2004

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 JAAS 2004

Definitions Box-counting method If N( )  1/ d at   0 Fractal – is a set with fractal (Hausdorf) dimension greater than its topological dimension JAAS 2004

Fractal functions D=1.2 Wierstrass function is scale-invariant D=1.5 JAAS 2004

Scaling properties of Wierstrass function From homogeneity C(bt)=b2-DC(t) Fractal Wierstrass function with b=1.5, D=1.8 JAAS 2004

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 JAAS 2004

Multifractals Important Fractal dimension – “average” all over the fractal Local properties of fractal are, in general, different JAAS 2004

Generalized dimensions Definition: Reney dimensions Artificial multifractal Artificial monofractal JAAS 2004

Generalized dimensions Definition: Renée dimensions S&P 500 British pound JAAS 2004

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. JAAS 2004

Local Holder exponents More convenient tool Scaling relation: where I - scaling index or local Holder exponent Extreme cases: JAAS 2004

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()} JAAS 2004

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 JAAS 2004

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 JAAS 2004

Properties of multifractal spectra Determining of the most important dimensions f() D0 min max 0 Using function of multifractal spectra to determine fractal dimension JAAS 2004

Properties of multifractal spectra Determining of the most important dimensions D1 f()  Using function of multifractal spectra to determine information dimension JAAS 2004

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 JAAS 2004

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: JAAS 2004

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  JAAS 2004

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 JAAS 2004

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 JAAS 2004

DJIA 1980-1988 Log-price  JAAS 2004

DJIA 1995-2002 Log-price  JAAS 2004

Detection of 1987 crash Log-price  JAAS 2004

Detection of 2001crash Log-price  JAAS 2004

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