Applying Methods of Nonlinear Dynamics for Financial Time Series Analysis

1. Finance Academyunder the Government of the Russian Federation Applying Methods of Nonlinear Dynamics for Financial Time Series Analysis Yuri Khakhanov yurimikha@gmail.com 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 isKolmogorov entropy, • K2is 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. К2-entropy Limit of K2-entropy can be approximated using the following function

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. PepsiCo

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 ≈ 7-8days

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 ≈ 8days

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

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 periodbefore the crisis K2-entropy rises (Time horizon declines). During the crisis K2-entropy declines (Time horizon rises).

27. THANK YOU FOR YOUR ATTENTION!