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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?
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We investigate the large-volatility dynamics in financial markets, based on the minute-to-minute and daily data of the Chinese Indices and German DAX. The dynamic relaxation both before and after large volatilities is characterized by a power law, and the exponents $p_pm$ usually vary with the strength of the large volatilities. The large-volatility dynamics is time-reversal symmetric at the time scale in minutes, while asymmetric at the daily time scale. Careful analysis reveals that the time-reversal asymmetry is mainly induced by exogenous events. It is also the exogenous events which drive the financial dynamics to a non-stationary state. Different characteristics of the Chinese and German stock markets are uncovered.
Multifractality is ubiquitously observed in complex natural and socioeconomic systems. Multifractal analysis provides powerful tools to understand the complex nonlinear nature of time series in diverse fields. Inspired by its striking analogy with hydrodynamic turbulence, from which the idea of multifractality originated, multifractal analysis of financial markets has bloomed, forming one of the main directions of econophysics. We review the multifractal analysis methods and multifractal models adopted in or invented for financial time series and their subtle properties, which are applicable to time series in other disciplines. We survey the cumulating evidence for the presence of multifractality in financial time series in different markets and at different time periods and discuss the sources of multifractality. The usefulness of multifractal analysis in quantifying market inefficiency, in supporting risk management and in developing other applications is presented. We finally discuss open problems and further directions of multifractal analysis.
We investigated financial market data to determine which factors affect information flow between stocks. Two factors, the time dependency and the degree of efficiency, were considered in the analysis of Korean, the Japanese, the Taiwanese, the Canadian, and US market data. We found that the frequency of the significant information decreases as the time interval increases. However, no significant information flow was observed in the time series from which the temporal time correlation was removed. These results indicated that the information flow between stocks evidences time-dependency properties. Furthermore, we discovered that the difference in the degree of efficiency performs a crucial function in determining the direction of the significant information flow.
We introduce simplicial persistence, a measure of time evolution of network motifs in subsequent temporal layers. We observe long memory in the evolution of structures from correlation filtering, with a two regime power law decay in the number of persistent simplicial complexes. Null models of the underlying time series are tested to investigate properties of the generative process and its evolutional constraints. Networks are generated with both TMFG filtering technique and thresholding showing that embedding-based filtering methods (TMFG) are able to identify higher order structures throughout the market sample, where thresholding methods fail. The decay exponents of these long memory processes are used to characterise financial markets based on their stage of development and liquidity. We find that more liquid markets tend to have a slower persistence decay. This is in contrast with the common understanding that developed markets are more random. We find that they are indeed less predictable for what concerns the dynamics of each single variable but they are more predictable for what concerns the collective evolution of the variables. This could imply higher fragility to systemic shocks.
A perspective is taken on the intangible complexity of economic and social systems by investigating the underlying dynamical processes that produce, store and transmit information in financial time series in terms of the textit{moving average cluster entropy}. An extensive analysis has evidenced market and horizon dependence of the textit{moving average cluster entropy} in real world financial assets. The origin of the behavior is scrutinized by applying the textit{moving average cluster entropy} approach to long-range correlated stochastic processes as the Autoregressive Fractionally Integrated Moving Average (ARFIMA) and Fractional Brownian motion (FBM). To that end, an extensive set of series is generated with a broad range of values of the Hurst exponent $H$ and of the autoregressive, differencing and moving average parameters $p,d,q$. A systematic relation between textit{moving average cluster entropy}, textit{Market Dynamic Index} and long-range correlation parameters $H$, $d$ is observed. This study shows that the characteristic behaviour exhibited by the horizon dependence of the cluster entropy is related to long-range positive correlation in financial markets. Specifically, long range positively correlated ARFIMA processes with differencing parameter $ dsimeq 0.05$, $dsimeq 0.15$ and $ dsimeq 0.25$ are consistent with textit{moving average cluster entropy} results obtained in time series of DJIA, S&P500 and NASDAQ.
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