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The main purpose of this work is to reproduce the method used for U.S. market which consists in the approach of random matrices to crossed correlation matrices built with financial data taken from a Mexican stock market database. First we built a cross correlation empirical matrix with these financial data. Eigenvalue spectrum was obtained from this matrix. We made the same spectrum analysis for a random matrix, and finally we compared both eigenvalue sets, and we tried to set up a hypothesis of how risk was related to this random matrix-correlation matrix approach. We used financial data over a period of six months and time series where made upon three hours measures for crossed correlation matrix.

Random Matrix Theory has been one of those mathematical discoveries or inventions from physicists due to their need to solve problems for physics. This theory has been a useful tool to recognize chaotic patterns in com- plex systems, and this tool is useful in the procedure of null hypothesis when it is compared with empirical data matrices for which correlation is clear [

We will refer to Random Matrix Theory as RMT from now on. RMT has been a great eye-opener since its development for nuclear physics applications. As it is well known in literature, atomic and nuclear complex systems were studied successfully with RMT’s help. RMT is one of branches in mathematics for which its development is owned by physics due to its need. A random matrix is defined as a (square or non-square) matrix with random entries. The most interesting part for analysis in RMT is the eigenvalue spectrum of matrices with random entries. This led us to analyze the distributions of eigenvalues and its reciprocal eigenvalues and then focus our attention on integration over these eigenvalue distributions. On the other hand, a cross correlation matrix is a matrix for which its entries are made of with time series of a certain set of a probably correlated data. Eigenvalues of cross correlated data matrices can be analyzed in the same way as for random matrices. High technology developments on statics and complex systems analysis, chaos theory and random matrices, lead us forward in the possible innovation of these tools inside a lot of knowledge fields. These fields vary widely; there have been developed applications for quantum and skew-quantum chaos theory, biological complex systems, weather and finally economy and finance. Applications of this field are made in this article. Just as it was comment at the abstract, we took a database of financial data from Mexican stock market taken from [

Applications on financial engineering have been made of in this article. Just as it was comment at the abstract, we took a database of financial data of Mexican stock market taken from [

Besides time of functioning, there are several reasons for which U.S. trading market is more stable than Mexican. Mexican stock market is weak because of some political and economic reasons such as the existence of old monopolies in communications industry. These companies have been absorbing small companies so there is no balance at all inside economical finance and trading stock markets. Besides the enormous complexity due to a stock trading system, social and political facts are a motive of a difficult problem to solve in terms of continuity of stock prices. It’s our purpose to analyze and apply a former study done in U.S. to Mexico and understands results in Mexican specific context.

This article is structured as follows: We describe the origin of our financial data, then we fix a cross correlation matrix for these last set of data. Simultaneously to this process, we build random matrix of dimensions equal to cross correlation data. In the next section we show which is the analysis comparing both matrix eigenvalue spectra. We name some applications about risk optimization theory and finally we set up our conclusions in terms of our work done (compared with those from references); hence we set up ideas for future work in this area. One can find whole data information, time series, plots of time series and correlation matrix entries from a special package of [

We took empirical data from [

Until this point we have presented where did we take financial data and how do we form financial data base [

Because [

We introduce now the construction process to build a cross correlation matrix from financial data empirical measures. From now on we will call C to cross correlation empirical matrix and it is built as follows:

Bracket notation is to point out time average over the period of analysis. The components of this average are g_{i} and g_{j} in the general case we define:

It is clear that dispersion is given by s_{i}.

Speaking with economic terms, s_{i} is seen as a level of volatility level of a stock market system.

This index is found in financial analysis named “VIMEX” [_{i} empirically with financial data from [

Construction of cross correlated matrices requires for each entry a quite acceptable description of financial states for each trading company.

We used time series for each company to be represented as entries. As we can see in

and _{i} belong to the prices or costs of each one of the actions gathered in [_{ij} are only valid for interval [-1,1]. Then C_{ij} = 1 belong to perfect correlation and C_{ij} = -1 belong to perfect anti-correlation. For C_{ij} = 0 case we have no correlation [

We now present a necessary constant to compute which is Q = L/N hence, if this number is bigger than one, then C would be positive definite. We need this condition to be held for N ® ¥, y L ®¥ for matrix C, as well as for a random matrix. Computing this constant we have: N = 35 and L = 131, therefore, Q = 3:685714286, which is bigger than one. Because both matrices are positive definite, then, computing C is quite possible. As well as in section 4 for random matrix, positive definite is required. This is in order to satisfy Marcencko-Pastur Theorem conditions [

Several kinds of time series have been used for modeling a lot of natural and social phenomena, although, experience in advanced statistics showed that the best approach to stock market behavior is logarithmic time series [

Computing time series is quite simple. We gather every column of stock prices for each company with software (STATGRAPHICS). Then we set up transformation following logarithmic formula for each value of a single company. Finally we plot this last function with time series form directly of software. All the

values for this companies once we made transformation are found in [_{i}(t + Dt) we took maximum as IPC daily return G_{i}, and opening value as lnS_{i}(t). We computed returns G_{i}, normalized with average quantities and standard deviations of each stock as well as it was described at the beginning. Once we obtained the g_{i} we generated entries for C. One can find this new data in [_{i} and g_{j} are vectors, we took the whole base in terms of g_{i} and scalar products we made.

Now, we define the random matrices Wigner and Wishart [_{tn},_{mn} = åX_{tm}X_{tn}, then eigenvalues of W are the elements l_{n}. Hence we used definition above to take advantage of random matrix features and the possibility to relate them to correlation matrices approach. Building random matrix with MATLAB, we generate a random matrix with random entries and with parameters m = 0 and s^{2} = 1. Taking distribution as before is not only helpful to satisfy definition of random matrix but to follow conditions for Marcencko-Pastur Theorem [

We now present the mechanism for which we can compare random matrices eigenvalue bulk versus cross correlation matrices eigenvalue distribution. This is ought to be a representation of Marcencko-Pastur theorem for financial data. It is introduced as follows:

where N ® ¥, T ® ¥ and the quotient satisfies (N/T) ® ¥. Then

Therefore

And it holds that

among eigenvalues of random matrices to correlated matrices.

As one can see (

x = 1.25, something interesting that has to do with it, is that they intersect curve at an specific point (1.25 for this case) to separate useful information than useless one. It is relevant to point out that density mentioned before presents random correlation, although financial data is normalized, and there are normalized and independent such as building of matrix A and displacement is in other direction. From statistical matrix theory, we know that spectral analysis is the analysis due to eigenvalue and eigenvector distribution as probability distributions [

And making a logarithmic approach such as a way to get eigenvalue distribution closer to Marcencko-Pastur requirements we have plot in

Now we can make random matrix eigenvalue analysis for A.

One can have full access to numerical version of distribution in [

Following Marcencko-Pastur Theorem, we see a qualitative similarity from one distribution to another such that make us think in chaotic motion inside MSM because of its isomorphic relation to RMT. Although, study of correlation can lead us to find stability points into this complex system.

One of the reasons for which we decided to work with the comparative method between eigenvalue spectrum of RM-Cross correlated matrix, was to reproduce work done in [

Juan Martín CasillasGonzález,Antonio AlatorreTorres, (2015) Random Matrix Approach to Correlation Matrix of Financial Data (Mexican Stock Market Case). Modern Economy,06,1033-1042. doi: 10.4236/me.2015.69099