The main goal of the paper is to show how mutual information can be used as a measure of dependence in financial time series. One major advantage of this approach resides precisely in its ability to account for nonlinear dependencies with no need to specify a theoretical probability distribution or use of a mean-variance model. r

In recent years there has been a closer interrelationship between several scientific areas trying to obtain a more realistic and rich explanation of the natural and social phenomena. Among these it should be emphasized the increasing interrelationship between physics and financial theory. In this field the analysis of uncertainty, which is crucial in financial analysis, can be made using measures of physics statistics and information theory, namely the Shannon entropy. One advantage of this approach is that the entropy is a more general measure than the variance, since it accounts for higher order moments of a probability distribution function. An empirical application was made using data collected from the Portuguese Stock Market.

Long memory and volatility clustering are two stylized facts frequently related to financial markets. Traditionally, these phenomena have been studied based on conditionally heteroscedastic models like ARCH, GARCH, IGARCH and FIGARCH, inter alia. One advantage of these models is their ability to capture nonlinear dynamics. Another interesting manner to study the volatility phenomena is by using measures based on the concept of entropy. In this paper we investigate the long memory and volatility clustering for the SP 500, NASDAQ 100 and Stoxx 50 indexes in order to compare the US and European Markets. Additionally, we compare the results from conditionally heteroscedastic models with those from the entropy measures. In the latter, we examine Shannon entropy, Renyi entropy and Tsallis entropy. The results corroborate the previous evidence of nonlinear dynamics in the time series considered.

Abstract. This paper presents a new test of independence (linear and nonlinear) among distributions, based on the entropy of Shannon. The main advantages of the presented approach are the fact that this measure does not need to assume any type of theoretical probability distribution and has the ability to capture the linear and nonlinear dependencies, without requiring the specification of any kind of dependence model.

When uncertainty dominates understanding stock market volatility is vital. There are a number of reasons for that. On one hand, substantial changes in volatility of financial market returns are capable of having significant negative effects on risk averse investors. In addition, such changes can also impact on consumption patterns, corporate capital investment decisions and macroeconomic variables. Arguably, volatility is one of the most important concepts in the whole finance theory. In the traditional approach this phenomenon has been addressed based on the concept of standard-deviation (or variance) from which all the famous ARCH type models-Autoregressive Conditional Heteroskedasticity Models-depart. In this context, volatility is often used to describe dispersion from an expected value, price or model. The variability of traded prices from their sample mean is only an example. Although as a measure of uncertainty and risk standard-deviation is very popular since it is simple and easy to calculate it has long been recognized that it is not fully satisfactory. The main reason for that lies in the fact that it is severely affected by extreme values. This may suggest that this is not a closed issue. Bearing on the above we might conclude that many other questions might arise while addressing this subject. One of outstanding importance, from which more sophisticated analysis can be carried out, is how to evaluate volatility, after all? If the standard-deviation has some drawbacks shall we still rely on it? Shall we look for an alternative measure? In searching for this shall we consider the insight of other domains of knowledge? In this paper we specifically address if the concept of entropy, originally developed in physics by Clausius in the XIX century, which can constitute an effective alternative. Basically, what we try to understand is, which are the potentialities of entropy compared to the standard deviation. But why entropy? The answer lies on the fact that there is already some research on the domain of Econophysics, which points out that as a measure of disorder, distance from equilibrium or even ignorance, entropy might present some advantages. However another question arises: since there is several measures of entropy which one since there are several measures of entropy, which one shall be used? As a starting point we discuss the potentialities of Shannon entropy and Tsallis entropy. The main difference between them is that both Renyi and Tsallis are adequate for anomalous systems while Shannon has revealed optimal for equilibrium systems.

Abstract:A new general fitting method based on the Self-Similar (SS) organization of random sequences is presented. The proposed analytical function helps to fit the response of many complex systems when their recorded data form a self-similar curve. The verified SS principle opens new possibilities for the fitting of economical, meteorological and other complex data when the mathematical model is absent but the reduced description in terms of some universal set of the fitting parameters is necessary. This fitting function is verified on economical (price of a commodity versus time) and weather (the Earth's mean temperature surface data versus time) and for these nontrivial cases it becomes possible to receive a very good fit of initial data set. The general conditions of application of this fitting method describing the response of many complex systems and the forecast possibilities are discussed.PACS (2008)

a b s t r a c tThis paper analyzes stock market relationships among the G7 countries between 1973 and 2009 using three different approaches: (i) a linear approach based on cointegration, Vector Error Correction (VECM) and Granger Causality; (ii) a nonlinear approach based on Mutual Information and the Global Correlation Coefficient; and (iii) a nonlinear approach based on Singular Spectrum Analysis (SSA). While the cointegration tests are based on regression models and capture linearities in the data, Mutual Information and Singular Spectrum Analysis capture nonlinear relationships in a non-parametric way. The framework of this paper is based on the notion of market integration and uses stock market correlations and linkages both in price levels and returns. The main results show that significant co-movements occur among most of the G7 countries over the period analyzed and that Mutual Information and the Global Correlation Coefficient actually seem to provide more information about the market relationships than the Vector Error Correction Model and Granger Causality. However, unlike the latter, the direction of causality is difficult to distinguish in Mutual Information and the Global Correlation Coefficient. In this respect, the nonlinear Singular Spectrum Analysis technique displays several advantages, since it enabled us to capture nonlinear causality in both directions, while Granger Causality only captures causality in a linear way. The results also show that stock markets are closely linked both in terms of price levels and returns (as well as lagged returns) over the 36 years analyzed.

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