## Abstract and Applied Analysis

### Nonlinear Fluctuation Behavior of Financial Time Series Model by Statistical Physics System

#### Abstract

We develop a random financial time series model of stock market by one of statistical physics systems, the stochastic contact interacting system. Contact process is a continuous time Markov process; one interpretation of this model is as a model for the spread of an infection, where the epidemic spreading mimics the interplay of local infections and recovery of individuals. From this financial model, we study the statistical behaviors of return time series, and the corresponding behaviors of returns for Shanghai Stock Exchange Composite Index (SSECI) and Hang Seng Index (HSI) are also comparatively studied. Further, we investigate the Zipf distribution and multifractal phenomenon of returns and price changes. Zipf analysis and MF-DFA analysis are applied to investigate the natures of fluctuations for the stock market.

#### Article information

Source
Abstr. Appl. Anal., Volume 2014, Special Issue (2014), Article ID 806271, 11 pages.

Dates
First available in Project Euclid: 3 October 2014

https://projecteuclid.org/euclid.aaa/1412364369

Digital Object Identifier
doi:10.1155/2014/806271

Mathematical Reviews number (MathSciNet)
MR3212449

Zentralblatt MATH identifier
07023111

#### Citation

Cheng, Wuyang; Wang, Jun. Nonlinear Fluctuation Behavior of Financial Time Series Model by Statistical Physics System. Abstr. Appl. Anal. 2014, Special Issue (2014), Article ID 806271, 11 pages. doi:10.1155/2014/806271. https://projecteuclid.org/euclid.aaa/1412364369

#### References

• F. Black and M. Scholes, “The pricing of options and corporate liabilities,” Journal of Political Economy, vol. 81, pp. 637–654, 1973.
• K. Ilinski, Physics of Finance: Gauge Modeling in Non-Equilibrium Pricing, John Wiley & Sons, New York, NY, USA, 2001.
• D. Lamberton and B. Lapeyre, Introduction to Stochastic Calculus Applied to Finance, Chapman & Hall/CRC, London, UK, 2000.
• G. Iori, “A threshold model for stock return volatility and trading volume,” International Journal of Theoretical and Applied Finance, vol. 3, pp. 467–472, 2000.
• T. C. Mills, The Econometric Modeling of Financial Time Series, Cambridge University Press, Cambridge, UK, 2nd edition, 1999.
• S. M. Ross, An Introduction to Mathematical Finance, Cambridge University Press, Cambridge, UK, 1999.
• M.-F. Chen, From Markov Chains to Non-Equilibrium Particle Systems, World Scientific Publishing, Singapore, 1992.
• R. Durrett, Lecture Notes on Particle Systems and Percolation, Wadsworth & Brooks, Pacific Grove, Calif, USA, 1988.
• G. Grimmett, Percolation, Springer, Berlin, Germany, 2nd edition, 1999.
• T. M. Liggett, Interacting Particle Systems, Springer, New York, NY, USA, 1985.
• T. M. Liggett, Stochastic Interacting Systems: Contact, Voter and Exclusion Processes, Springer, Berlin, Germany, 1999.
• W. Y. Cheng and J. Wang, “Dependence phenomenon analysis of the stock market,” Europhysics Letters, vol. 102, Article ID 18004, 2013.
• W. Fang and J. Wang, “Statistical properties and multifractal behaviors of market re-turns by Ising dynamic systems,” International Journal of Modern Physics C, vol. 23, Article ID 1250023, 14 pages, 2012.
• H. Niu and J. Wang, “Volatility clustering and long memory of financial time series and financial price model,” Digital Signal Processing, vol. 23, no. 2, pp. 489–498, 2013.
• D. Stauffer, “Can percolation theory be applied to the stock market?” Annalen der Physik, vol. 7, no. 5-6, pp. 529–538, 1998.
• A. Pei and J. Wang, “Nonlinear analysis of return time series model by oriented percolation dynamic system,” Abstract and Applied Analysis, vol. 2013, Article ID 612738, 12 pages, 2013.
• F. Wang and J. Wang, “Statistical analysis and forecasting of return interval for SSE and model by lattice percolation system and neural network,” Computers and Industrial Engineering, vol. 62, no. 1, pp. 198–205, 2012.
• J. Wang and S. Deng, “Fluctuations of interface statistical physics models applied to a stock market model,” Nonlinear Analysis: Real World Applications, vol. 9, no. 2, pp. 718–723, 2008.
• J. Wang, Q. Y. Wang, and J. G. Shao, “Fluctuations of stock price model by statistical physics systems,” Mathematical and Computer Modelling, vol. 51, no. 5-6, pp. 431–440, 2010.
• W. Fang and J. Wang, “Fluctuation behaviors of financial time series by a stochastic Ising system on a Sierpinski carpet lattice,” Physica A, vol. 392, pp. 4055–4063, 2013.
• Y. Yu and J. Wang, “Lattice-oriented percolation system applied to volatility behavior of stock market,” Journal of Applied Statistics, vol. 39, no. 4, pp. 785–797, 2012.
• J. Zhang and J. Wang, “Modeling and simulation of the market fluctuations by the finite range contact systems,” Simulation Modelling Practice and Theory, vol. 18, no. 6, pp. 910–925, 2010.
• J. Zhang and J. Wang, “Fractal detrended fluctuation analysis of chinese energy markets,” International Journal of Bifurcation and Chaos, vol. 20, no. 11, pp. 3753–3768, 2010.
• T. E. Harris, “Contact interactions on lattice,” The Annals of Probability, vol. 2, no. 6, pp. 969–988, 1974.
• M. Ausloos and K. Ivanova, “Precise (m, k)-Zipf diagram analysis of mathematical and financial time series when $m = 6$, $k = 2$,” Physica A: Statistical Mechanics and its Applications, vol. 270, no. 3, pp. 526–542, 1999.
• Y. L. Guo and J. Wang, “Simulation and statistical analysis of market return fluctuation by zipf method,” Mathematical Problems in Engineering, vol. 2011, Article ID 253523, 13 pages, 2011.
• N. Vandewalle and M. Ausloos, “the n-Zipf analysis of financial data series and biased data series,” Physica A: Statistical Mechanics and its Applications, vol. 268, no. 1, pp. 240–249, 1999.
• G. K. Zipf, Human Behavior and the Principle of Least Effort, Addison-Wesley Press, Cambridge, UK, 1949.
• G. K. Zipf, The Psycho-Biology of Language: An Introduction to Dynamic Psychology, Addison-Wesley Press, Cambridge, UK, 1968.
• Z. Zheng, Matlab Programming and the Applications, China Railway Publishing House, Beijing, China, 2003.
• L. E. Calvet and A. J. Fisher, Multifractal Volatility: Theory, Forecasting, and Pricing, Academic Press Advanced Finance, Academic Press, 2008.
• J. W. Kantelhardt, S. A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde, and H. E. Stanley, “Multifractal detrended fluctuation analysis of nonstationary time series,” Physica A: Statistical Mechanics and its Applications, vol. 316, no. 1–4, pp. 87–114, 2002.
• C.-K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley, and A. L. Goldberger, “Mosaic organization of DNA nucleotides,” Physical Review E, vol. 49, no. 2, pp. 1685–1689, 1994.
• J. Kwapień and S. Drożdż, “Physical approach to complex systems,” Physics Reports, vol. 515, no. 3-4, pp. 115–226, 2012.
• P. Oswiecimka, J. Kwapien, and S. Drozdz, “Wavelet versus detrended fluctuation analysis of multifractal structures,” Physical Review E, vol. 74, Article ID 016103, 2006.
• P. Oświęcimka, J. Kwapień, and S. Drozdz, “Multifractality in the stock market: price increments versus waiting times,” Physica A: Statistical Mechanics and its Applications, vol. 347, pp. 626–638, 2005.
• S. Drod, J. Kwapień, P. Oświecimka, and R. Rak, “Quantitative features of multifractal subtleties in time series,” Europhysics Letters, vol. 88, no. 6, Article ID 60003, 2009.
• S. Drozdz, J. Kwapień, P. Oświęcimka, and R. Rak, “The foreign exchange market: return distributions, multifractality, anomalous multifractality and the Epps effect,” New Journal of Physics, vol. 12, Article ID 105003, 2010.
• J. Kwapień, P. Oświęcimka, and S. Drozdz, “Components of multifractality in high-frequency stock returns,” Physica A: Statistical Mechanics and its Applications, vol. 350, no. 2-4, pp. 466–474, 2005. \endinput