Detection of financial crisis by methods of multifractal analysis

1. 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

2. 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

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

4. 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, stableLevi distributions)

5. Financial markets as complex systems Quotes of GBP/USD in different scales 2 hours quotes Weekly quotes Monthly quotes Financial markets Complex systems • Open systems • Multi agent • Adaptive and self-organizing • Scale invariance

6. 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

7. 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

8. 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

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

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

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

12. Iterated Function Systems Real fem IFS fem 50x zoom of IFS fem

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

14. 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

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

16. Fractal dimension L( ) = a 1-D D – fractal (Hausdorf) dimension Reference book: “Fractals” J. Feder, 1988 What’s the length of Norway coastline?

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

18. Fractal functions Wierstrass function is scale-invariant D=1.5 D=1.8 D=1.2

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

20. 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

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

22. Generalized dimensions Artificial monofractal Definition: Reney dimensions Artificial multifractal

23. Generalized dimensions S&P 500 British pound Definition: Renée dimensions

24. 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.

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

26. Local Holder exponents The link between {q,(q)} and { ,f()} Legendre transform More convenient tool Scaling relation: where I - scaling index or local Holder exponent

27. Function of multifractal spectra For monofractals: For multifractals: Non-homogeneous Cantor’s set Homogeneous Cantor’s set Distribution of scaling indexes What is number of cells that have a scaling index in the range between  and  + d ?

28. Function of multifractal spectra For monofractals: For multifractals: S&P 500 index British pound Distribution of scaling indexes What is number of cells that have a scaling index in the range between  and  + d ?

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

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

31. Properties of multifractal spectra Using function of multifractal spectra to determine correlation dimension f() 2-D2 D2/2  2 Determining of the most important dimensions

32. Multifractal analysis 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 qthen X(t,1) is multifractal time series • For monofractal time series scaling function (q) • is linear: (q)=D0(q-1) Remarks: Definitions

33. 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 

34. Description of major USA market crashes • Oil embargo • Inflation (15-17%) • High oil prices • Declined debt pays • Computer trading • Trade & budget deficits • Overvaluation Summer 1982 October 1987 • Asian crisis • Internationality of • US corp. • Overvaluation Autumn 1998 September 2001 • Terror in New York • Overvaluation • Economic problems • High-tech crisis

35. 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

36. DJIA 1980-1988 Log-price 

37. DJIA 1995-2002 Log-price 

38. Detection of 1987 crash Log-price 

39. Detection of 2001crash Log-price 

40. 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