Methods for forecasting stock markets

Abstract

This thesis presents a forecasting method for stock market indexes by considering the effects of deterministic factors and market volatility. The method has been developed by analysing the S&P500 index over the past 25 years. The approach decomposes the time series into a deterministic trend and stochastic fluctuations. The deterministic trend is the slowly varying component describing the overall direction of the index to capture financial market tendencies. For the stochastic fluctuations, two distribution functions were considered in the modelling process: the Levy-stable distribution and the q-Gaussian distribution. The Levy-stable distribution models random processes with independent and identically distributed increments, whereas the q-Gaussian distribution describes correlated processes with finite variance. The thesis demonstrates that price return decays with a power-law that rules out the Levy-stable regime. In contrast, the q-Gaussian diffusion provides a better fit for describing the weakly-correlated fluctuations of price returns. A special case of the porous media equation (PME) is used to describe the q-Gaussian diffusion. This variant of the PME is used as the governing equation of the correlated stochastic fluctuations with finite variance. The modification of the PME was carried out by applying the anomalous scaling to the forecasting time variable. Different regimes are identified in the analysed time range: a strong superdiffusion regime for the first 30 minutes, a weak superdiffusion regime between 30 minutes and 30 days, and a normal diffusion process beyond 30 days that is consistent with the classical central limit theorem. Finally, the forecasting method couples the external environment and the market. The forecast is presented by dressing the trend with a q-Gaussian diffusion process. This framework is a novel approach for modelling complex time series that includes nonlinear fluctuations and external deterministic factors to the market.