Furthermore, the FMH claims that information is valued based on each investor’s time horizon. Because information is viewed as a generic item in classical finance theory, the EMH implies that various information affects investors in the same way. The aim of this paper is to highlight the practical implications of our findings on the capital markets under consideration. The main contribution of this paper is to determine whether these markets are efficient (as defined by the efficient market hypothesis), in which case the appropriate dynamics equation of stock indexes is the GBM, or fractal (as defined by the fractal market hypothesis), in which case the appropriate dynamics equation of stock indexes is the GFBM. In this article, we will test which of the dynamics equations, represented by geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM), best de-scribes the evolution of the S&P 500 and Stoxx Europe 600 stock indexes for the US market and the EU market, respectively. Because the EMH is based on standard Brownian motion processes that assume prices evolve through random walk, one obvious consequence is that forecasting future price movements is impossible because market movements are independent and lack autocorrelation, rendering technical analysis useless to investors. The randomness of prices makes it difficult for investors to outperform the market and earn abnormal returns. The random walk model is a financial theory that says that stock market prices cannot be anticipated because they combine the information and expectations of all the market participants. The efficient financial market, according to EMH theory, is the market that fully reflects the available information, and one of the models of the efficient financial market is the random walk model.
The EMH theory’s premises can be summarized as follows : (a) investors are rational (in the sense that they correctly update their beliefs about the value of financial securities when new information becomes available in the market) (b) individual investment decisions satisfy the arbitrage condition (arbitrage leads to price equilibrium) and (c) the market is characterized by collective rationality (the different errors in the values of financial securities determined by individual investors cancel each other out in the market). However, while these findings cannot be generalized, they are verisimilar. The simulation results demonstrate that the GFBM is better suited for forecasting stock market indexes than the GBM when the analyzed markets display fractality. To determine which of the dynamics (GBM, GFBM) is more appropriate, we employed the mean absolute percentage error (MAPE) method. In this paper, we consider two methods for calculating the Hurst exponent: the rescaled range method (RS) and the periodogram method (PE). The main contribution of this work is determining whether these markets are efficient (as defined by the EMH), in which case the appropriate stock indexes dynamic equation is the GBM, or fractal (as described by the FMH), in which case the appropriate stock indexes dynamic equation is the GFBM. Daily stock index data were acquired from the Thomson Reuters Eikon database during a ten-year period, from January 2011 to December 2020.
The article’s major goal is to show how to appropriately model return distributions for financial market indexes, specifically which geometric Brownian motion (GBM) and geometric fractional Brownian motion (GFBM) dynamic equations best define the evolution of the S&P 500 and Stoxx Europe 600 stock indexes. In this article, we propose a test of the dynamics of stock market indexes typical of the US and EU capital markets in order to determine which of the two fundamental hypotheses, efficient market hypothesis (EMH) or fractal market hypothesis (FMH), best describes market behavior.