Multifractality approach of a generalized Shannon index in financial time series.

Multifractality is a concept that extends locally the usual ideas of fractality in a system. Nevertheless, the multifractal approaches used lack a multifractal dimension tied to an entropy index like the Shannon index. This paper introduces a generalized Shannon index (GSI) and demonstrates its application in understanding system fluctuations. To this end, traditional multifractality approaches are explained. Then, using the temporal Theil scaling and the diffusive trajectory algorithm, the GSI and its partition function are defined. Next, the multifractal exponent of the GSI is derived from the partition function, establishing a connection between the temporal Theil scaling exponent and the generalized Hurst exponent. Finally, this relationship is verified in a fractional Brownian motion and applied to financial time series. In fact, this leads us to proposing an approximation called local fractional Brownian motion approximation, where multifractal systems are viewed as a local superposition of distinct fractional Brownian motions with varying monofractal exponents. Also, we furnish an algorithm for identifying the optimal q-th moment of the probability distribution associated with an empirical time series to enhance the accuracy of generalized Hurst exponent estimation.

Deciphering equity style returns: An analysis of size and value anomalies in the Pakistani stock exchange.

This study aims to identify the underlying causes of variation in the time series and cross-sectional equity style returns in the emerging stock market of Pakistan. We use asset pricing models and incorporate variables reflecting business cycle fluctuations to assess the time-varying size and value premiums. The methodology of this paper involves constructing style portfolios based on firm-specific characteristics such as market capitalization, price to earnings ratio, book-to-market equity ratio, momentum, and asset growth. We find that the style portfolios earn abnormal returns consistently which cannot be explained either through asset pricing models or business cycles variables. However, the size and value premiums are found to be subsided during the troughs of economic cycles. The results suggest that the abnormal returns for style portfolios are likely driven by firm-specific characteristics rather than macroeconomic factors. Overall, this study contributes to the literature on style investing by providing insights into the profitability of equity style portfolios in the Pakistani equity market. Our findings have implications for stock picking, investment management and risk factor analysis.

Role of green innovation, trade and energy to promote green economic growth: a case of South Asian Nations.

The objective of this study is to contribute to the existing debate of green economic growth by empirically investigating the role of cleaner energy production, green innovation, and green trade in green economic growth in the context of South Asian countries. For this purpose, the study collects the data of South Asian Economies for 2000-2018 from different sources such as world development indicators (WDI), International Energy Statistics (IES), and Organization for Economic Co-operation and Development (OECD) statistics. The study applied Pesaran's (2007) second-generation unit root test to test the stationarity of the data. Wasteland's (2007) test of cointegration was applied to examine the long-run association among modeled variables. The study confirmed the long-run association among modeled variables that turn to be stationary at the first differences. Moreover, the study applied fully modified least square (FMOLS) and dynamic least square (DOLS) to estimate the empirical results of the study. Results of the study show that the production of clean energy, green innovation, and green trade positively contributes to the green economic growth of South Asian Economies Graphical abstract.

Linear response theory in stock markets.

Linear response theory relates the response of a system to a weak external force with its dynamics in equilibrium, subjected to fluctuations. Here, this framework is applied to financial markets; in particular we study the dynamics of a set of stocks from the NASDAQ during the last 20 years. Because unambiguous identification of external forces is not possible, critical events are identified in the series of stock prices as sudden changes, and the stock dynamics following an event is taken as the response to the external force. Linear response theory is applied with the log-return as the conjugate variable of the force, providing predictions for the average response of the price and return, which agree with observations, but fails to describe the volatility because this is expected to be beyond linear response. The identification of the conjugate variable allows us to define the perturbation energy for a system of stocks, and observe its relaxation after an event.

Stock markets: A view from soft matter.

Different attempts to describe financial markets, and stock prices in particular, with the tools of statistical mechanics can be found in the literature, although a general framework has not been achieved yet. In this paper we use the physics of many-particle systems and the typical concepts of soft matter to study two sets of US and European stocks, comprising the biggest and most stable companies in terms of stock price and trading. Upon correcting for the center-of-mass motion, the structure and dynamics of the systems are studied (in the European set, the structure is studied for the UK subset only). The pair distribution of the stocks, corrected to account for the nonuniform distribution of prices, is close to 1, indicating that there is no direct interaction between stocks, similar to an ideal gas of particles. The dynamics is studied with the mean-squared price displacement (MSPD); the price correlation function, equivalent to the intermediate scattering function; the price fluctuation distribution; and two parameters for collective motions. The MSPD grows linearly and the velocity autocorrelation function is zero, as for isolated Brownian particles. However, the intermediate scattering function follows a stretched exponential decay, the fluctuation distributions deviate from the Gaussian shape, and strong collective motions are identified. These results indicate that the dynamics is much more complex than an ideal gas of Brownian particles, and similar, to some extent, to that of undercooled systems. Finally, two physical systems are discussed to aid in the understanding of these results: a low density colloidal gel, and a dense system of ideal, infinitely thin stars. The former reproduces the dynamical properties of stocks, linear mean-squared displacement (MSD), non-Gaussian fluctuation distribution, and collective motions, but also has strong structural correlations, whereas the latter undergoes a glass transition with the structure of an ideal gas, but the MSD has the typical two-step growth of undercooled systems.