CITY UNIVERSITY OF HONG KONG DEPARTMENT OF PHYSICS AND MATERIALS SCIENCE

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1 CITY UNIVERSITY OF HONG KONG DEPARTMENT OF PHYSICS AND MATERIALS SCIENCE BACHELOR OF SCIENCE (HONS) IN APPLIED PHYSICS PROJECT REPORT Price fluctuation in financial markets by Chan Pui Hang March 2012

2 Price fluctuation in financial markets by Chan Pui Hang Submitted in partial fulfilment of the requirements for the degree of BACHELOR OF SCIENCE (HONS) IN APPLIED PHYSICS from City University of Hong Kong March 2012 Project Supervisor : Prof. KS Chan

3 Table of content: Table of content:... i List of figures:... ii List of Table:... iv Abstract:... v 1. Introduction: Literature Review General fact Random matrix theory Correlation matrix Random correlation matrix Eigenvalue of correlation matrix Methodology Data and Result Normalized price return calculation correlation matrices in different business sector Comparison of eigenvalues generated with RMT Conclusion Reference i

4 List of figures and tables Figure Number title Page number Fig.1 Fig.2 Fig.3 Fig.4 Fig.5 Fig.6 Fig.7 Fig.8 Fig.9 Fig.10 Pearson product-moment correlation coefficient between X and Y Sample about the normalized 5-mintues returns distribution of 5 stocks in NSE from the period of January 2003 to March Another sample about the normalized return distribution with 489 stocks in NSE in the same period as Fig.2 The list of table of four bussiness sectors of Hong Kong Hang Seng Index An example of 0001 Cheung Kong stock price fluctuation before applying the normalized price return calculation An example of 0001 Cheung Kong stock price fluctuation after applying the normalized price return calculation The probability distribution of all component stocks Correlation Matrix in Hong Kong Hang Seng Index The probability distribution of Correlation Matrix of component stocks of Hong Kong Hang Seng Index in the same business sectors The probability distribution of Correlation Matrix of component stocks in finance sectors comparing with the other business sectors in Hong Kong Hang Seng Index The probability distribution of Correlation Matrix of component stocks in utilities sectors comparing with ii

5 Fig.11 Fig.12 Fig.13 the other business sectors in Hong Kong Hang Seng Index The probability distribution of Correlation Matrix of component stocks in properties sectors comparing with the other business sectors in Hong Kong Hang Seng Index The probability distribution of Correlation Matrix of component stocks in commerce and industry sector comparing with the other business sectors in Hong Kong Hang Seng Index The probability distribution of the eigenvalues of the correlation matrix C, P(λ), for Hong Kong Hang Seng Index is displaying by blue line and red line is the theoretical curve of P(λ) from random matrix theory iii

6 Abstract The correlation of the component stocks of the Hong Kong Hang Seng Index (HKHSI) can be found out by analyzing the correlation matrix C of the normalized price return of the stock data. The correlation matrices are compared in different groups according to their business sectors. An easy and dominant result can be computed to have a first impression of the correlation of different component stocks. Moreover, the eigenvalues of the correlation matrix are analyzed in order to find out the stock contribution to the entire market and the relation of the value of the largest eigenvalue of all the correlation matrices. If the eigenvalues overlap with the eigenvalue generated by the equation: [1] Then those eigenvalues from the market are therefore in random motion. That means the random stocks price can exist in every market. The largest eigenvalue is correlated with all the other eigenvalues and represented the group correlation. iv

7 Introduction A financial market is a market in which people in different countries and cultures can trade financial capital and commodities according to the law of supply and demand. Such complicated market can be considered as a complex system in which many correlated components are interacting with each other in some noticeable way such as stock or market index. [1, 2] Therefore, the fluctuation or influence of the financial market is highly dependent on those noticeable elements like stocks and external information. Such a complicated economic system is difficult to construct a relation even an equation describing the element inside. In physics, there are a kind of subject called econophysics which is focused on the research in other subjects such as economic and finance, applying physics or mathematical theories and methods originally generated or modified by physicists to find out the solution in economics or finance, which usually includes those relating to uncertainty or stochastic processes and nonlinear dynamics. Its application to the study of financial markets has also been termed statistical finance, referring to its roots in statistical physics. There are many tools people can use in order to construct a relationship on an equation describing the element in a stock market. In this research, matrices and MATLAB are mainly used to find out the correlation and statistical properties of the stock market. 1

8 Objective 1. To analyze the cross-correlation matrix of price return of the Hong Kong Hang Seng Index 2. To compare and analyze the cross-correlation of price return of same business sectors and different business sectors. 3. To investigate the eigenvalues of the cross-correlation matrix and to check the eigenvalues with random matrix theory. 2

9 Literature Review Physicists have investigated and analyzed the statistical properties of stock price fluctuation and their correlation. The fluctuation distribution of stock price is found to obey the power law with exponential α~ 3 and known as inverse cubic law [3, 4] This law is developed very well in emerging market while emerging market are nations with social or business activity in the process of rapid growth and industrialization. From the data of 2010, there are 40 emerging market and Hong Kong is one of them. [5] However, inverse cubic law is not proved that the crosscorrelation of stocks and price fluctuation are having similar behavior and nature. The research on inverse cubic law is mainly focusing on emerging market while lack of research on developed market. A research focused on the New York exchange market (NYSE) and Tokyo stock exchange has verified that the eigenvalue distribution generated matching the prediction by RMT. [6-9] The largest eigenvalue has been verified that affect the entire market and all the stocks, while the other large eigenvalues are related to different business sector. A further research has been carried to reconstruct the structure of NYSE using filtering techniques to implant matrix composition. Filtered correlation matrix can remove common market influence and random noise. This method may provide a better and accurate representation of the intra market. [10,11] 3

10 Random matrix theory In order to investigate the interaction of different elements in the financial market, a common method used by scientists is to focus on the spectrum of the correlation matrix of different stocks. And a well-known random matrix theory (RMT) provides a lot of information and result about the correlation between different stocks price movement. A random matrix is a matrix with random variable. [12] For the fluctuation analyzed, linear statistics N f,h = n 1 f(λ j ), [2] one is also interested in the fluctuations about f(λ) dn(λ). A recently distributed research from the institute of Mathematical Science in India has demonstrated the stocks in emerging market are more correlated than in developed markets. They found that the properties of collective mode in Indian market have important deviations from developed markets. [13] The fluctuation distribution of stocks in both NYSE and Indian market are both obey and follow the inverse cubic law. And the Indian market even has a higher degree of correlation when comparing with developed market. And one of its significant conclusions is that most of their correlation among stocks is due to common market while the correlation among stocks in the same business catalogue is weak. 4

11 Correlation matrix Correlation matrix is a common method to analyze price movement and stocks especially financial risk management. In statistics, dependence means any statistical relationship between two random variables or two sets of data. Correlation refers to any of a broad class of statistical relationships involving dependence. There are many way to measure the dependence of the correlation, and one of the most familiar ways is Pearson product-moment correlation coefficient. [14-16] [3] Where E is the expected value operator, cov is covariance, and, corr is a alternative notation for Pearson's correlation. The correlation matrix consists of n random variables Y 1 to Y n is the n n matrix where i,j (X i, X ji ) entry is two of all the correlation. [17] The correlation matrix is the same as the covariance matrix of the standardized random variables Y i / σ (Y i ) for i = 1 to n if the measures of correlation used are product-moment coefficients. Consequently, each is necessarily a positive-semidefinite matrix which the data should be absolute in order to find out the fluctuation only with including positive and negative variations. Due to the correlation between X ij is the same with the correlation between X ji, therefore, the correlation matrix is definitely symmetric with n X n matrix. 5

12 Fig1 shows that Pearson product-moment correlation coefficient between X and Y. Comparing the range of two set of data are not limited and limited to (0, 1). It found that data between (0, 1) is the best to represent the correlation. Return cross-correlation matrix Cross-correlation is a measure of similarity of two waveforms as a function of a timelag applied to one of them. In order to compare and observe the correlation between price movements of different component stocks, the data taken should be independent of the scale of measurement. [18] If P i (t) is the price of the stock i=1 and to N(t) and the price return of the ith stock over a time interval t is defined as P i (t, t) In P i (t+ t) - In P i (t) [4] 6

13 Due to the level of volatility of different stock are different, the normalized return is defined as [5] Which σ <R i 2 > - <R i > 2, is the standard deviation of R i. The <R i > is represented the average time over the whole period of data processing. With the calculated r i, the equal time cross-correlation matrix C can be calculated mathematically. C ij <r i r j > [6] The above equation can relate the return of the particular stock i and stock j. By the correlation matrix equation, 47component stocks of HKHSI can be constructed 2209 matrix. Those C matrix are symmetric and has a domain vale[-1,1] Fig.2 is the sample about the normalized 5-mintues returns distribution of 5 stocks in NSE from the period of January 2003 to March

14 Fig.3 is another sample about the normalized return distribution with 489 stocks in NSE in the same period as Fig.2. Eigenvalue of correlation matrix In a time series of length T, assume N return time are uncorrelated to each other, the eigenvalue distribution of the uncorrelated random matrix can be found from the equation, Where the matrix is called Wishart matrix. [33] The matrix is set in a limit N -> infinite and T -> infinite so that the Q T/N 1 [7] For, the distribution are bounded in a region of. 8

15 Methodology HSI was firstly started on November 24, 1969, that is currently compiled and maintained by Hang Seng Index Company Limited, which is a wholly owned subsidiary of Hang Seng Bank, one of the largest banks registered and listed in Hong Kong in terms of market capitalization. The Hang Seng Index is a free float-adjusted market capitalization-weighted stock market index in Hong Kong. The HKHSI is dominated by 48 component stocks and occupied about 70% capitalization of the Hong Kong Stock exchange. Those 48 component stocks are divided into four sectors according to their nature, business and majority source of sales revenue. In this research, it is mainly focused on the correlation of component stocks of HKHSI within the same sectors. The research has taken the daily stock data of 47 component stocks of HKHSI from 1 st Jan 1997 to 1 st Jan This data is obtained from the yahoo finance website. However, those daily data taken from the website are not daily matched each other. Some of them may be contained 1258 data while some may be contained 1234 data during the preset period. If a particular stock data is missed in a particular stock, all the other component stock is not compared with that stock on that day. Therefore, when comparing a stock of 1258 data and another stock 1234 data, only the same 1234 data of both stocks in the same period is compared. And there is a special component stock 1299 AIA contained not enough data during the preset period because that stock was out of list of the HKHSI for about two years from 2008 to So that the research is not included AIA stock and the total component stocks 9

16 of HKHSI are 47 not 48. After the selection of the data which ensure the comparison of data of different stocks are in the same day and same period. After that, each day (each data) is calculated by the equation of return cross-correlation matrix [8] so that all the data can be compared in connecting the correlation matrix. C ij <r i r j > [9] And then, All the correlation matrix, n x n matrix, are calculated to find the value of their eigenvalues by MATLAB to check whether it fits the equation below. [10] If the eigenvalues fit the equation above, the random movement of stock prices can be existed in every market. 10

17 Fig.4 The list of table of four bussiness sectors of Hong Kong Hang Seng Index 11

18 Data analyzed Series1 2 per. Mov. Avg. (Series1) Fig5 is a example of 0001 Cheung Kong stock price fluctuation before applying the normalized return price calculation on each day from the period of 1 st january 2007 to 1 st january Series1 2 per. Mov. Avg. (Series1) Fig.6 is a example of 0001 Cheung Kong stock price fluctuation after applying the normalized price return calculation on each day from the period of 1 st january 2007 to 1 st january

19 After applying the normalized price return, all the data can be compared at the same level rather than compared in different level. 400 P(Cij) all correlation matrix Cij Fig.4 The probability distribution of all component stocks Correlation Matrix in Hong Kong Hang Seng Index. The above graph is the overall number of correlation matrices of all the component stocks of Hong Kong Hang Seng Index. It shows that no correlation matrix is lower than 0.4. That means no stock is relatively no or less relation with each other. And the peak of all matrices are located from 0.7 to 0.8. The distribution of all correlation matrices are mainly located within 0.6 to 0.8. Therefore, most of the component stocks of Hong Kong Hang Seng Index are closly related to each other. However, there are few number of correlation matrices are located within 0.9 to 1 which indicates a few of component stocks are very closly correlated to each other. A futher clear correlation within the same business sectors and different business sectors are analyzed by plotting the graph in different extents. 13

20 P(Cij) C(finance-finance) C(utilities-utilities) C(properties-properties) C(c&i-c&i) Cij Fig.5 The probability distribution of Correlation Matrix of component stocks of Hong Kong Hang Seng Index in the same business sectors From the Fig.5, it shows all the component stocks inside their business sectors (finance, utilizes, properties and commerce and industry) having a closed relation as their correlation matrix are highly distributed between Although they are all normal distribution curve, but the correlation matrix of C finance-finance and C properties-properties have the peak from 0.6 to 0.7 while C utilities-utilities and C commerce and industry-commerce and industry have the peak within When the value of correlation matrix is increased, the correlation between the stocks is also increased. And finally, if they reach to 1, it means they are 100% the same. Only focusing on the correlation matrix, the inter-correlation of utilities sector and commerce & industry may higher than that of finance sector and properties sector. 14

21 120 P(Cij) C(finance-utilities) C(finance-properties) C(finance-c&i) Cij Fig6 is the probability distribution of Correlation Matrix of component stocks in finance sectors comparing with the other business sectors in Hong Kong Hang Seng Index From the graph above, when comparing with the correlation matrix, the correlation of properties and commerce and industry with finance is greater than that of utilities because the distribution density of properties and commerce and industry is mainly located within 0.8 to 0.9 while that of utilities is mainly located 0.7 to 0.8. This may be due to the variation in finance may have a greater effect in the economic in properties and industry and commerce sectors while has a less effect in utilities. 15

22 40 P(Cij) C(utilities-finance) C(utilities-properties) C(utilities-c&i) Cij Fig7 is the probability distribution of Correlation Matrix of component stocks in utilities sectors comparing with the other business sectors in Hong Kong Hang Seng Index It sharply points out that there is no any very-closed correlation of utilities with other business sector as no correlation matrix is located within 0.9. However, there is a peak of C utilities-finance and commerce and industry within 0.7 to 0.8 while that of C utilities-properties within 0.6 to 0.7. From the Fig 4-6, there is a commonly peak within while the highest peak in the correlation Matrix of component stocks in utilities sectors comparing with the other business sectors is only That means the utilities may be having a smaller correlation with other business sectors as the different of 0.1 in correlation matrix has already very large. 16

23 70 P(Cij) C(properties-finance) C(properties-utilities) C(properties-c&i) Cij Fig8 is the probability distribution of Correlation Matrix of component stocks in properties sectors comparing with the other business sectors in Hong Kong Hang Seng Index The peak of the correlation matrix of C properties-finance is within 0.8 to 0.9 while that of C properties-commerce and industry is within and that of C properties-utilities is within 0.6 to 0.7. It is clearly arranged the correlation priority from the correlation matrix. Properties sector has a greater correlation with finance and then commerce and industry, finally the utilities. Again, it sharply figure out that there is no any very-closed correlation of utilities with properties sector as no correlation matrix is located within 0.9. And all the correlation matrix C properties-commerce and industry and C properties-finance mainly distributed in 0.7 to 0.9 while C properties-utilities is mainly distributed in 0.6 to

24 120 P(Cij) C(c&i-finance) C(c&i-utilities) C(c&i-properties) Cij Fig9 is the probability distribution of Correlation Matrix of component stocks in commerce and industry sector comparing with the other business sectors in Hong Kong Hang Seng Index The correlation of commerce and industry are almost the same with both three other sectors finance, utilities and properties. Because the peak of C properties-finance, C properties-commerce and industry, C properties-utilities are both located at 0.7 to 0.8. And their distributions are both mainly located from 0.6 to

25 p(λ) 8 HKHSI p(λ) λ Fig.13 The probability distribution of the eigenvalues of the correlation matrix C, P(λ), for Hong Kong Hang Seng Index is displaying by blue line and red line is the theoretical curve of P(λ) from random matrix theory. Fig.13 shows a majority of eigenvalues distribution from the correlation matrices has a clearly agreement with the result from the theory of Random Matrices. Therefore, the random movement of stock prices can be existed in every market. The largest eigenvalue of the correlation matrices is which indicate entire market and all the stocks, while the other large eigenvalues 3.529, are related to different business sector. However, from the bound, there are some eigenvalues deviates from the Wishart matrix and the largest eigenvalue from correlation matrix deviates from all of the other eigenvalues on the graph. 19

26 Conclusion To conclude, the component stocks of Hong Kong Hang Seng Index were analyzed in various catalogues according to their business sectors. Through the correlation matrices, it found that all the component stocks are having closed correlation to each other as the distribution of all correlation are mainly located within 0.6 to 0.8. And no correlation matrix is lower than 0.4 which means no stocks are relatively no relation with each other. Correlation Matrix of component stocks of Hong Kong Hang Seng Index compared in the same business sectors have the same peak position from 0.8 to 0.9. Finance sector, properties sector and commerce & industry sector are more correlated to others while utilities are less correlated to the other three business sectors. The probability distribution of the eigenvalues of the correlation matrix C, P(λ), for Hong Kong Hang Seng Index is relatively match with the probability distribution of the eigenvalues generated by the random matrix theory. Therefore, the random movement of stock prices can be existed in every market. 20

27 Reference [1] R. N. Mantegna and H. E. Stanley, Introduction to Econophysics, Cambridge University Press, Cambridge, UK, [2] J. P. Bouchaud and M. Potters, Theory of Financial Risk and Derivative Pricing, 2nd ed. _Cambridge University Press, Cambridge, UK, [3] Econophysics: An Emerging Science, edited by I. Kondor and J. Kertesz _Kluwer, Dordrecht, [4] Econophysics of Stock and Other Markets, edited by A. Chatterjee and B. K. Chakrabarti Springer, Milan, [5] emerging economies and the transformation of international business By Subhash Chandra Jain. Edward Elgar Publishing, 2006 p.384 [6] L. Laloux, P. Cizeau, J. P. Bouchaud, and M. Potters, Phys. Rev. Lett. 83,1467 (1999). [7] V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. Nunes Amaral, and H. E. Stanley, Phys. Rev. Lett. 83, 1471 (1999). [8] V. Plerou, P. Gopikrishnan, B. Rosenow, Luis A. Nunes Amaral, T. Guhr, and H. E. Stanley, Phys. Rev. E 65,

28 (2002). [9] A. Utsugi, K. Ino, and M. Oshikawa, Phys. Rev. E 70, (2004). [10] R. N. Mantegna, Eur. Phys. J. B 11, 193 _1999_. [11] J. P. Onnela, A. Chakraborti, K. Kaski, and J. Kertesz, Eur. Phys.J.B 30, 285 (2002). [12] Edelman, A.; Rao, N.R (2005). "Random matrix theory". Acta Numer. 14: [13] Sarma M., Eurorandom Report, (2005). [14] Frederick Emory Croxton, Dudley Johnstone Cowden and Sidney Klein; Applied general statistics, page 625 [15] Cornelius Frank Dietrich; Uncertainty, calibration, and probability : the statistics of scientific and industrial measurement, Page 331 [16] Alexander Craig Aitken; Statistical mathematics, Page 95 [17] J. L. Rodgers and W. A. Nicewander. Thirteen ways to look at the correlation coefficient. The American Statistician, 42(1):59 66, February [18] Campbell, Lo, and MacKinlay 1996: The Econometrics of Financial Markets, NJ: Princeton University Press. 22