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This paper explores the significant role of stochastic processes in financial modeling, tracing the evolution from basic Brownian motion to sophisticated stochastic differential equations used in modern financial markets. Beginning with the historical development of Brownian motion, identified by Robert Brown and later mathematically modeled by Louis Bachelier for stock price fluctuations, the paper outlines its foundational influence on the Efficient Market Hypothesis and the random walk theory. The extension of these concepts in the Black-Scholes model for option pricing highlights the practical applications of these theories in predicting financial outcomes. The discussion progresses to geometric Brownian motion (GBM) and its crucial role in stock price modeling, emphasizing its use in Monte Carlo simulations for option pricing. The limitations of the Black-Scholes model and GBM in dealing with real market conditions such as stochastic volatility and jump-diffusion processes are addressed, showcasing the evolution of more complex models like the Heston model and GARCH. Interest rate models like the Vasicek and Cox-Ingersoll-Ross models are evaluated for their real-world applicability, particularly in scenarios of low and negative interest rates. This comprehensive review not only underscores the theoretical advancements in financial modeling but also its practical implications in contemporary financial markets.
Zou et al. (Thu,) studied this question.