We analyze the Standard & Poor's 500 stock market index from the last 22 years. The probability density function of price returns exhibits two well-distinguished regimes with self-similar structure: the first one displays strong super-diffusion together with short-time correlations, and the second one corresponds to weak super-diffusion with weak time correlations. Both regimes are well-described by q-Gaussian distributions. The porous media equation is used to derive the governing equation for these regimes, and the Black-Scholes diffusion coefficient is explicitly obtained from the governing equation.
The Lévy-stable distribution is the attractor of distributions which hold power laws with infinite variance. This distribution has been used in a variety of research areas; for example, in economics it is used to model financial market fluctuations and in statistical mechanics it is used as a numerical solution of fractional kinetic equations of anomalous transport. This function does not have an explicit expression and no uniform solution has been proposed yet. This paper presents a uniform analytical approximation for the Lévy-stable distribution based on matching power series expansions. For this solution, the trans-stable function is defined as an auxiliary function which removes the numerical issues of the calculations of the Lévy-stable distribution. Then, the uniform solution is proposed as a result of an asymptotic matching between two types of approximations called "the inner solution" and "the outer solution." Finally, the results of analytical approximation are compared to the numerical results of the Lévy-stable distribution function, making this uniform solution valid to be applied as an analytical approximation.
Our study presents the analysis of stock market data of S&P500 before and after been detrended. The analysis is based on two types of returns, simple return and log-return respectively. Both of them are non-stationary time series. This means that their statistical distribution change over time. Consequently a detrended process is made to neutralize the non-stationary effects. The detrended process is obtained by decomposing the financial time series into a deterministic trend and random fluctuations. We present an alternative method on detrending time series based on the classical moving average (MA) models, where Kurtosis is used to determine the windows size. Then, the dentrending fluctuation analysis (DFA) is use to show that the detrended part is stationary. This is done by considering the autocorrelation of detrended price return and the power spectrum analysis of detrended price.
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