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Speaker : Kozlov Alexander A.

Financial Academy under the Government of the Russian Federation. Report. Singular spectrum analysis for a forecasting of financial time series. Speaker : Kozlov Alexander A. Moscow - 2009. Content list:. Introduction to nonlinear dynamics approach

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Speaker : Kozlov Alexander A.

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  1. Financial Academy under the Government of the Russian Federation Report Singular spectrum analysis for a forecasting of financial time series Speaker: Kozlov Alexander A. Moscow - 2009

  2. Content list: • Introduction to nonlinear dynamics approach • Overview of the main methods (including SSA) • Financial time series analysis and forecasting: • Schlumberger Limited • Deutsche Bank • Honda Motor Co., Ltd. • Toyota Motor Corp. • Starbucks • BP plc. • Conclusions

  3. Introduction • Time series is a series of variable values taken in successive periods of time. • Time series analysisis a part of nonlinear dynamics. • Supposition:market of shares is unstable and chaotic. • Objective:analysis and forecasting of stock price time series with nonlinear dynamics methods • In this report the following questions will be considered: • Embedding dimension as “space” characteristic and its estimation • К2-entropy and Lyapunov exponents as “time” charactericsand their estimation • SSA forecasting method

  4. The idea of attractor reconstruction[11]: Satisfactory geometry picture of low-dimensional strange attractorcan be obtained if instead of x-variables from dynamic system equations somebody use k-dimensionaldelay vectors: Overview • Takens theorem [2]: • There is a transformation which can embed to on conditions that . It means that: - k – embedding dimension; - __________________________________________________________________________________________________ [1] Packard N.H., Crutchfield J.P., Farmer J.D., Shaw R.S.,"Geometry from a time series", Phys.Rev.Lett. 45, p.712,1980. [2] F. Takens, "Dynamical Systems and Turbulence", Lect. Notes in Math, Berlin, Springer. №898, 1981, p. 336.

  5. Overview Find , having curves for eachk, starting with k=1; Starting with certain k-number stops growing and stabilizes; This k-numberis embedding dimension ; Maximum valueof is a so-called correlation dimension (or )of the attractor. • Correlation integral on r<<1 and k>>1 [4]: • Grassberger- Procaccia method [3]: • Limitation [4]: ______________________________________________________________________________________ [3]P. Grassberger, I. Procaccia, "Characterization of Strange Attractors",Phys.Rev.Lett.,50,346, 1983 [4] G.G. Malineckiy, A.B. Potapov, “Actual problems of nonlinear dynamics", М: URSS, 2002

  6. Overview • Having fixed r and investigating dependence С(r*,k)fromk (k>>1), somebody can estimateK2-entropy [5]. • K2 defines the time of predictability for the system in “volume” interpretation (growing of the volume in phase space which the system can occupy in the future) • The time of predictability also can be determined fromLyapunov exponents The maximal one is estimated inWolf method[6]. ________________________________________________________________________________________________________ [5] Grassberger, I. Procaccia,"Estimation of the Kolmogorov Entropy from a Chaotic Signal",Phys.Rev.A,vol.28,4,1983,p.2591 [6] Wolf A., Swift J.B., Swinney H.L., "Determining Lyapunov exponents from time series", Physica D, 69 (1985), №3, p.285-317.

  7. SSA forecasting method [7]: 1)Constructionof the delay matrix from time series and preliminary changes in it (centering and normalization) 2)Finding the components (M) and selectionof the most important ones (r) This is equal to search of eigenvectors and и eigenvalues of the matrix . 3)Time series reconstruction with r main components and taking average on each diagonal. 4)Forecast constraction with «caterpillar» method: Equal to constraction of the new delay vector with one unknown coordinate. _____________________________________________________________________________________________________ [20] “The main components of time series: “caterpillar“ method”. Col.articles // ed. D.L. Danilov, А.А. Zhiglyavskiy – St.P.: St.P. University, 1997. - 308 p. Overview

  8. Analysis and forecasting • Criteria for the selected companies: - long time on the market of shares (NYSE) – more than 10 years; - publicity; - from different sectors; • Thus the following companies were chosen: • Schlumberger Limited • Deutsche Bank • Honda Motor Co., Ltd. • Toyota Motor Corp. • Starbucks • BP plc. • Forecasting parameters: - delay number - M=20 - number of the main components – • During forecasting logarithmic profitis taken in to account: - positive in growth - negative in fall

  9. Analysis and forecasting • Periodfrom 31.12.1981 to 31.12.2008 • Time series consists of 6814stock price values (on close). 1. Schlumberger Limited

  10. 1. Schlumberger Limited Analysis and forecasting

  11. Analysis and forecasting • Periodfrom 18.11.1996to31.12.2008. • Time series consists of 3033stock price values (on close). 2. Deutsche Bank

  12. 2. Deutsche Bank Analysis and forecasting

  13. Analysis and forecasting • Periodfrom 11.08.1987to31.12.2008. • Time series consists of 5390stock price values (on close). 3. Honda Motor Co., Ltd.

  14. 3. Honda Motor Co., Ltd. Analysis and forecasting

  15. Analysis and forecasting • Periodfrom 13.04.1993to31.12.2008. • Time series consists of 3956stock price values (on close). 4. Toyota Motor Corp.

  16. 4. Toyota Motor Corp. Analysis and forecasting

  17. Analysis and forecasting • Periodfrom 26.06.1992to31.12.2008. • Time series consists of 4161stock price values (on close). 5. Starbucks

  18. 5. Starbucks Analysis and forecasting

  19. Analysis and forecasting • Periodfrom 03.01.1977to31.12.2008. • Time series consists of 8076stock price values (on close). 6. BP plc.

  20. 6. BP plc. Analysis and forecasting

  21. Analysis and forecasting • Final results of analysis are in the table: • Percentage of coincidence between logarithmic profit signs of forecast and real time series

  22. Conclusions • Nonlinear dynamics methods applied to stock price time series led to a “space” and “time” analysis of the trading system. Thus we determined number of the main components (=embedding dimension) and time of predictability (according to K2-entropy and Lyapunov exponents) for each company. • Obtained results have both fundamental and applied sense for economics. • Complex analysis permitted to make a forecast on the basis of SSA method (“caterpillar”). Forecasted values and logarithmic profit fits the real ones quite well. • Thus SSA forecasting method can be a useful instrument in quantitative analysis of any risks connected with financial time series.

  23. Thank You for attention!

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