1. Applying Methods of Nonlinear Dynamics for Financial Time Series Analysis Yuri Khakhanov yurimikha@gmail.com Finance Academy under the Government of the Russian Federation 17 September 2009, Moscow

2. Contents Time Series Entropy Definition of K2-entropy Estimating nonlinear parameteres of financial time series Final results Conclusions

3. Time series A Time Series is a sequence of data points, measured typically at successive times, spaced at uniform time intervals. Examples: stock indices, share prices, electrocardiogram, seismogram, etc.

4. Metric entropy Kolmogorov entropy: t – time, d(0) – initial distance (time t=0) d(t) – distance at time “t”. h=0 – regular dynamic (ordered system), 0<h<∞ - deterministic chaos, h→∞ - randomness.

5. Generalized entropy: - joint probability that attractor trajectory visits cubes at times. К1 is Kolmogorov entropy, K2 is a lower bound for Kolmogorov entropy.

6. K2 entropy, where. - correlation integral. m – current embedding dimension ∆t=1 (day). K2-entropy is a limit of correlation integrals ratio.

7. Limit of K2-entropy can be approximated using the following function К2-entropy

8. Time horizon ‘T’ Time horizon refers to a maximum time period, when chaotic system behavior forecasting is possible. where ‘l’ – accuracy of задания initial position When t>T only statistical forecasts are possible.

9. Pepsi Co

10. К2-entropy К2 ≈ 0,15 Т ≈ 6-7 days

11. К2-entropy (1,5 year before the crisis) К2 ≈ 0,17-0,18 Т ≈ 5-6 days

12. К2-entropy (1,5 year during the crisis) К2 ≈ 0,13-0,14 Т ≈ 7-8 days

13. EI DuPont de Nemours

14. К2-entropy К2 ≈ 0,09 Т ≈ 11 days

15. К2-entropy (1,5 year before the crisis) К2 ≈ 0,17 Т ≈ 6 days

16. К2-entropy (1,5 year during the crisis) К2 ≈ 0,08 Т ≈ 12 days

17. HJ Heinz Co

18. К2-entropy К2 ≈ 0,13 Т ≈ 7-8 days

19. К2-entropy (1,5 year before the crisis) К2 ≈ 0,17 Т ≈ 6 days

20. К2-entropy (1,5 year during the crisis) К2 ≈ 0,12 Т ≈ 8 days

21. Harley-Davidson, Inc.

22. К2-entropy К2 ≈ 0,12 Т ≈ 8 days

23. К2-entropy (1,5 year before the crisis) К2 ≈ 0,15 Т ≈ 6-7 days

24. К2-entropy (1,5 year during the crisis) К2 ≈ 0,09 Т ≈ 11 days

25. Results Time horizons for periods before and during the crisis CompanyT, daysT before the crisis, daysT during the crisis, days PepsiCo Inc.6-75-67-8 EI DuPont de Nemours11612 HJ Heinz Co.7-868 Harley-Davidson Inc.86-711 Marriott Intl. Inc.5-6 6-7 Microsoft Corp.107-814 CompanyK2-entropyT, days PepsiCo Inc.0,156-7 EI DuPont de Nemours0,0911 HJ Heinz Co.0,137-8 Harley-Davidson Inc.0,128 Marriott Intl. Inc.0,185-6 Microsoft Corp.0,110

26. Conclusions К2-entropy defines time horizon. К2-entropy for analyzed financial time series gives a green light to reliable 5-10 days forecast. In the period before the crisis K2-entropy rises (Time horizon declines). During the crisis K2-entropy declines (Time horizon rises).

27. THANK YOU FOR YOUR ATTENTION!