Geometric Brownian Motion (GBM) of Stock Indexes and Financial Market Uncertainty in the Context of Non-Crisis and Financial Crisis Scenarios

Abstract

The present article proposes a methodology for modeling the evolution of stock market indexes for 2020 using geometric Brownian motion (GBM), but in which drift and diffusion are determined considering two states of economic conjunctures (states of the economy), i.e., non-crisis and financial crisis. Based on this approach, we have found that the GBM proved to be a suitable model for making forecasts of stock market index values, as it describes quite well their future evolution. However, the model proposed by us, modified geometric Brownian motion (mGBM), brings some contributions that better describe the future evolution of stock indexes. Evidence in this regard was provided by analyzing the DAX, S&P 500, and SHANGHAI Composite stock indexes. Throughout the research, it was also found that the entropy of these markets, analyzed in the periods of non-crisis and financial crisis, does not differ significantly for DAX—German Stock Exchange (EU) and S&P 500—New York Stock Exchange (US), and insignificant differences for SHANGHAI Composite—Shanghai Stock Exchange (Asia). Given the fact that there is a direct link between market efficiency and their entropy (high entropy—high efficiency; low entropy—low efficiency), it can be deduced that the analyzed markets are information-efficient in both economic conjunctures, and, in this case, the use of GBM for forecasting is justified, as the prices have a random evolution (random walk).