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?Computationalphysics Numerical tools Complex systems theoryEconomic,finance Econophysics Methodology Empirical data

4. 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)

5. 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

6. 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

7. Econophysics publications Mantegna-Stanley Physica A 239 (1997) Experimental data (logarithm of prices) fit to 1. 1. Gaussian distribution until 2 std. 2. 2. Levy distribution until 5 std. 3. 3. Then they appear truncate Crush of linearparadigm

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

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

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

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

19. Scaling properties of Wierstrass function From homogeneity C(bt) = b 2 - D C(t) Fractal Wierstrass function with b=1.5, D=1.8

20. 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

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

22. Generalized dimensionsDefinition: Artificial multifractal Reney dimensions Artificial monofractal

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

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. 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.

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

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

27. 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

28. 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

29. f(  ) D0D0D0D0  min  max 0000 Using function of multifractal spectra to determine fractal dimension Properties of multifractal spectra Determining of the most important dimensions

30. Properties of multifractal spectra Determining of the most important dimensions D1D1D1D1 f(  ) D1D1D1D1  Using function of multifractal spectra to determine information dimension

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

32. 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

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: 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 

34. 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

35. 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

36. DJIA 1980-1988 Log-price

37. DJIA 1995-2002Log-price 

38. Detection of 1987 crash Log-price

39. Detection of 2001crashLog-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