A Comparative Study Between Constant and Non-Constant Volatility for Geometric Brownian Motion | RISE Research

This research investigates how different assumptions about volatility influence the accuracy and realism of Geometric Brownian Motion (GBM) — a foundational model in asset pricing and option valuation. While traditional GBM assumes constant volatility, real financial markets exhibit dynamic fluctuations, making it necessary to test non-constant approaches. The study evaluates four distinct models: (1) classical constant-volatility GBM, (2) volatility derived from the VIX index, (3) time-dependent or local volatility models, and (4) stochastic volatility modeled through the Heston framework.

Using historical S&P 500 data, each model is simulated via Monte Carlo methods to assess predictive behavior and statistical fit. Quantitative metrics such as Root Mean Squared Error (RMSE) and R² are used to evaluate accuracy. Results show that while constant-volatility GBM performs reasonably well (R² ≈ 0.89), incorporating time-varying volatility leads to a stronger fit (R² up to 0.91). The time-dependent model achieves the highest accuracy, while the stochastic volatility model most closely replicates the actual final price. These findings reveal that allowing volatility to evolve over time captures market variability more effectively than static models.

By systematically comparing model complexity with performance, the study highlights when more sophisticated volatility structures are justified. It bridges theoretical modeling with empirical market behavior, offering insights for traders, quantitative analysts, and researchers interested in improving risk assessment and option pricing accuracy under realistic market conditions.