Chapter 5
Conclusions
In this thesis we have looked at the Spanish stock market as a complex system of interacting particles, the participant agents. We have applied physics tools to the financial data pulling out some underlying dynamic properties of the system. We have described first of all its general characteristics, focusing the attention on the properties which are in common with others complex systems. We have then analysed the statistical properties of share prices, looking at the thirty most traded stocks of the Spanish stock market, which are part of the Ibex-35 index. We have done this in order to compare some of the main properties of this market with the ones of the New York Stock Exchange studied in the literature, to prove empirically that they belong to the same class of markets.
We have found that the probability distribution function of price returns moves away from a Gaussian distribution, exhibiting an inverse power law behaviour for large returns. The exponent of their cumulative distribution is ) similar to that, , found for the returns of S&P . For the purpose of investigating the cross-correlations between stocks we have calculated the correlation coefficients matrix of the analysed set at different time intervals, showing that for larger time intervals the average of the cross-correlation coefficients increases, due to the fact that the majority of stocks more or less “follow” the same market index, in our case Ibex-35 , and this introduces a kind of underlining positive
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cross-correlation.
To better point out the hierarchical structure among stocks, we have represented the cross-correlation matrix using a minimal spanning tree, which, among the mathematical graphs, provides a topological arrangement of stocks, selecting the most relevant connec-tions among them. We have obtained the well known underlying structure of the selected stocks, without additional assumptions. In fact the stocks in the minimal spanning tree ap-pear divided into different linked groups, according to their market sectors.
For the purpose of understanding which relationships dominate the statistical properties of price stocks, we moved on to look at the correlation between agents, i.e. companies where shareholders are market members. We have chosen the sixty most active agents, meaning for activity the number of transactions that they have. Before looking for correlations, we have looked at the agents ensemble as a group of persons, showing that the distribution of agents’ wealth obeys a Pareto law, with exponent similar to the ones of all societies investigated. Unlike what done for stocks, to build up the agents time series it was necessary to define a new variable, the action variable: this variable on one hand has to contain as much as possi-ble of the information availapossi-ble in the database, while on the other hand it should eliminate the spurious effects due to the rationality of agents, such as a different trading enthusiasm depending on the time of day. After these preliminary operations we have first carried out a cross-correlation analysis using the same techniques of mathematical graphs theory adopted for stocks analysis. In this way, we have extracted the hierarchical structure present among agents at different time intervals. We have shown that there are some little clusters which do not change upon increasing of the time interval, exhibiting a structure stable in time. In order to extract more information about this underlying structure, we have applied the ran-dom matrix theory, to look at the degree of deviation of the agents cross-correlation matrix from the prediction of the theory. We have shown that the probability density function of the eigenvalues does not fulfill the Gaussian random matrix predictions, and in particular the highest eigenvalue is more than six times larger than the predicted maximum random eigenvalue, this eigenvalue reflects the “basic” correlation among agents, i.e., it has roughly equal components on all of the
agents. Another aspect investigated in this thesis has been the presence of time memory in the agent action time series, using the autocorrelation func-tions. We have found that the correlation function exhibits an inverse power law behaviour
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with exponent , which is an index of a long-range correlation. The cross-correlation function among agents at different times is also long range correlated, with a lower exponent , meaning a larger correlation range. It is necessary to note that this exponent is close to that found in the autocorrelation function of the absolute values of price changes in other markets. The absolute values of price changes if often used as an estimate of the volatility, which is the key parameter in measuring the risk of a financial investment, and it can be also related to the amount of information arriving in the market at a given time. This seeming agreement is the result of a process of price formation in which the agents take a dominant role. At the end, we have investigated a possible market model which yields the statistical properties of price dynamics, starting from the microstructure found in this thesis. We have first introduced some market models, discussing their merits and shortcomings. We have described the Bornholdt model for the class of spin models systems, which is based on the Ising model; we have then described the Cont-Bouchaud model for the class of agent based models, which is the basis of the model proposed here. Then we have introduced the model, which assumes that the agents are not homogenous, but they have limited resources according to the Pareto wealth law found for the empirical data. We have shown that the sim-ulated returns price series has a probability distribution function which exhibits an inverse power law behaviour similar to the one found in the real data, with an exponent depending on the linking parameter of the model. We have thought that to introduce an hierarchical structure among agents modelled on the one found in the third chapter, means to fix the value of the parameter and consequently the exponent of power law tail of the price returns pdf. However due to the complexity of how to define the relationship between the correla-tion coefficients and the probability to have a link, because of the both negative and positive signs of the correlation coefficients. In any case, this result proves that the observed fat-tail distribution in price return of financial-data arise from an interaction between heterogenous agents, which form at each time opinion clusters and make decision as a whole (herding behaviour).
In summary, in this thesis we have tried to unravel the heterogeneity and differences present in a stock market starting from the high-frequency data of the Spanish stock market. Our purpose has been to check whether the usual hypothesis in physics that complex be-haviour comes from interaction and correlation of many degrees of freedom can be extended
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to real markets. In fact, many models suggested by the statistical physics community are able to recover the observed stylised facts of price dynamics in stock markets. However, as we have seen, those models could be an oversimplification of the reality, and more fundamental ingredients (such as the distribution of wealth and/or the network of correlation among the agents) should be taken into account when modelling a real stock market.
