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One of the major issues studied in finance that has always intrigued, both scholars and practitioners, and to which no unified theory has yet been discovered, is the reason why prices move over time. Since there are several well-known traditional techniques in the literature to measure stock market volatility, a central point in this debate that constitutes the actual scope of this paper is to compare this common approach in which we discuss such popular techniques as the standard deviation and an innovative methodology based on Econophysics. In our study, we use the concept of Tsallis entropy to capture the nature of volatility. More precisely, what we want to find out is if Tsallis entropy is able to detect volatility in stock market indexes and to compare its values with the ones obtained from the standard deviation. Also, we shall mention that one of the advantages of this new methodology is its ability to capture nonlinear dynamics. For our purpose, we shall basically focus on the behaviour of stock market indexes and consider the CAC 40, MIB 30, NIKKEI 225, PSI 20, IBEX 35, FTSE 100 and SP 500 for a comparative analysis between the approaches mentioned above., Comment: 15 pages, 2 figures, 2 tables, presented at SigmaPhi2008 Conference

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., Comment: 8 pages; 2 figures; paper presented in APFA 6 conference

The investor is interested in the expected return and he is also concerned about the risk and the uncertainty assumed by the investment. One of the most popular concepts used to measure the risk and the uncertainty is the variance and/or the standard-deviation. In this paper we explore the following issues: Is the standard-deviation a good measure of risk and uncertainty? What are the potentialities of the entropy in this context? Can entropy present some advantages as a measure of uncertainty and simultaneously verify some basic assumptions of the portfolio management theory, namely the effect of diversification?, Comment: 9 pages, 2 figures, paper presented in APFA 6 conference

In this paper we apply the techniques of symbolic dynamics to the analysis of a labor market which shows large volatility in employment flows. In a recent paper, Bhattacharya and Bunzel \cite{BB} have found that the discrete time version of the Pissarides-Mortensen matching model can easily lead to chaotic dynamics under standard sets of parameter values. To conclude about the existence of chaotic dynamics in the numerical examples presented in the paper, the Li-Yorke theorem or the Mitra sufficient condition were applied which seems questionable because they may lead to misleading conclusions. Moreover, in a more recent version of the paper, Bhattacharya and Bunzel \cite{BB1} present new results in which chaos is completely removed from the dynamics of the model. Our paper explores the matching model so interestingly developed by the authors with the following objectives in mind: (i) to show that chaotic dynamics may still be present in the model for standard parameter values; (ii) to clarify some open questions raised by the authors in \cite{BB}, by providing a rigorous proof of the existence of chaotic dynamics in the model through the computation of topological entropy in a symbolic dynamics setting., Comment: 3 figures

There is by now a large consensus in modern monetary policy. This consensus has been built upon a dynamic general equilibrium model of optimal monetary policy as developed by, e.g., Goodfriend and King (1997), Clarida et al. (1999), Svensson (1999) and Woodford (2003). In this paper we extend the standard optimal monetary policy model by introducing nonlinearity into the Phillips curve. Under the specific form of nonlinearity proposed in our paper (which allows for convexity and concavity and secures closed form solutions), we show that the introduction of a nonlinear Phillips curve into the structure of the standard model in a discrete time and deterministic framework produces radical changes to the major conclusions regarding stability and the efficiency of monetary policy. We emphasize the following main results: (i) instead of a unique fixed point we end up with multiple equilibria; (ii) instead of saddle--path stability, for different sets of parameter values we may have saddle stability, totally unstable equilibria and chaotic attractors; (iii) for certain degrees of convexity and/or concavity of the Phillips curve, where endogenous fluctuations arise, one is able to encounter various results that seem intuitively correct. Firstly, when the Central Bank pays attention essentially to inflation targeting, the inflation rate has a lower mean and is less volatile; secondly, when the degree of price stickiness is high, the inflation rate displays a larger mean and higher volatility (but this is sensitive to the values given to the parameters of the model); and thirdly, the higher the target value of the output gap chosen by the Central Bank, the higher is the inflation rate and its volatility., Comment: 11 pages

Recent studies show that a negative shock in stock prices will generate more volatility than a positive shock of similar magnitude. The aim of this paper is to appraise the hypothesis under which the conditional mean and the conditional variance of stock returns are asymmetric functions of past information. We compare the results for the Portuguese Stock Market Index PSI 20 with six other Stock Market Indices, namely the S&P 500, FTSE100, DAX 30, CAC 40, ASE 20, and IBEX 35. In order to assess asymmetric volatility we use autoregressive conditional heteroskedasticity specifications known as TARCH and EGARCH. We also test for asymmetry after controlling for the effect of macroeconomic factors on stock market returns using TAR and M-TAR specifications within a VAR framework. Our results show that the conditional variance is an asymmetric function of past innovations raising proportionately more during market declines, a phenomenon known as the leverage effect. However, when we control for the effect of changes in macroeconomic variables, we find no significant evidence of asymmetric behaviour of the stock market returns. There are some signs that the Portuguese Stock Market tends to show somewhat less market efficiency than other markets since the effect of the shocks appear to take a longer time to dissipate., Comment: 11 pages, 3 figures

This paper analyses the behaviour of volatility for several international stock market indexes, namely the SP 500 (USA), the Nikkei (Japan), the PSI 20 (Portugal), the CAC 40 (France), the DAX 30 (Germany), the FTSE 100 (UK), the IBEX 35 (Spain) and the MIB 30 (Italy), in the context of non-stationarity. Our empirical results point to the evidence of the existence of integrated behaviour among several of those stock market indexes of different dimensions. It seems, therefore, that the behaviour of these markets tends to some uniformity, which can be interpreted as the existence of a similar behaviour facing to shocks that may affect the worldwide economy. Whether this is a cause or a consequence of market globalization is an issue that may be stressed in future work., Comment: 10 pages, 3 figures. Paper presented in the APFA 5 conference

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., Comment: 8 pages, 2 figures, presented in the conference Next Sigma-Phi 2005

The main purpose of this paper is to present a kneading theory for two-dimensional triangular maps. This is done by defining a tensor product between the polynomials and matrices corresponding to the one-dimensional basis map and fiber map. We also define a Markov partition by rectangles for the phase space of these maps. A direct consequence of these results is the rigorous computation of the topological entropy of two-dimensional triangular maps. The connection between kneading theory and subshifts of finite type is shown by using a commutative diagram derived from the homological configurations associated to $m-$modal maps of the interval., Comment: 22 pages,6 figures

Let $X,Y$ be topological vector spaces or metric spaces, and let {$f:X\times Y \to \Re $} be a real function lower semicontinuous in the first variable and upper semicontinuous in the second one. It is proved that $f$ is globally measurable. Sierpinski (1925) has been raised this question in the case $X=Y=\Re $. This particular case was solved by Kempisty (1929). The actual result has applications in Calculus of Variations., Comment: 8 pages Latex