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

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

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.

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.

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

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

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

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.

Pepsi Co

К2-entropy К2 ≈ 0,15 Т ≈ 6-7 days

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

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

EI DuPont de Nemours

К2-entropy К2 ≈ 0,09 Т ≈ 11 days

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

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

HJ Heinz Co

К2-entropy К2 ≈ 0,13 Т ≈ 7-8 days

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

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

Harley-Davidson, Inc.

К2-entropy К2 ≈ 0,12 Т ≈ 8 days

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

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

Results Time horizons for periods before and during the crisis CompanyT, daysT before the crisis, daysT during the crisis, days PepsiCo Inc EI DuPont de Nemours11612 HJ Heinz Co Harley-Davidson Inc Marriott Intl. Inc Microsoft Corp 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

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).

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