Financial Modeling with Geometric Brownian Motion

This project evaluates Brownian Motion’s effectiveness compared to historical stock market data. This paper analyzes the application and limitations of this stochastic model, focusing on the Dow Jones Industrial Average (DJIA) to evaluate its accuracy in predicting stock market trends. The paper begins by tracing the historical context of stochastic calculus, highlighting the contributions of Louis Bachelier and Albert Einstein in laying the foundation for modern financial modeling. Geometric Brownian Motion (GBM) is then introduced with options pricing and then examined through the lens of the Markov property, emphasizing its “memoryless” nature. To test this, simulations were coded using Python simulations of the DJIA Index, based on Brownian Motion across the periods 1900-2000 and 2000-2015. These models were then compared to the actual historical data of the DJIA to evaluate predictive validity. Stochastic elements reflect factors that influence the value of a derivative, like time, volatility of the underlying asset, interest rates, and other market conditions. The research also critically examines the model’s inherent limitations, aiming to provide insights into the extent to which GBM can be a reliable tool for financial forecasting. This comparison will enable a nuanced understanding of the utility and shortcomings of this model for economic predictions and stock market analysis.