  This paper explores stochastic modeling approaches to elucidate the intricate
dynamics of stock prices and volatility in financial markets. Beginning with an
overview of Brownian motion and its historical significance in finance, we
delve into various stochastic models, including the classic Black-Scholes
framework, the Heston model, fractional Brownian motion, GARCH models, and Levy
processes. Through a thorough investigation, we analyze the strengths and
limitations of each model in capturing key features of financial time series
data. Our empirical analysis focuses on parameter estimation and model
calibration using Levy processes, demonstrating their effectiveness in
predicting stock returns. However, we acknowledge the need for further
refinement and exploration, suggesting potential avenues for future research,
such as hybrid modeling approaches. Overall, this study underscores the
importance of stochastic modeling in understanding market dynamics and informs
decision-making in the financial industry.
