A Comparative Study between Constant and Non-Constant Volatility for Geometric Brownian Motion

This study compares the performance of constant and non-constant volatility models in the context of Geometric Brownian Motion (GBM), a widely used framework for modeling asset prices and option pricing. While the classical GBM assumes volatility to be constant, real financial markets exhibit time-varying uncertainty and volatility clustering, which constant models cannot capture. To address these limitations, this research explores four approaches: the classical constant-volatility GBM, volatility derived from the VIX index, time-dependent volatility, and stochastic volatility modeled through the Heston framework. Using historical S&P 500 data, the study calculates parameters for each model and conducts Monte Carlo simulations to analyze their predictive behavior. The results indicate that while constant-volatility GBM provides a useful benchmark, models incorporating non-constant volatility better capture market variability. The time-dependent model shows the strongest overall fit, whereas the stochastic volatility model most closely replicates the actual final price.