Beyond Black-Scholes: A Computational Framework for Option Pricing Using Heston, GARCH, and Jump Diffusion Models

Abstract

This research addresses accurate option pricing by employing models beyond the traditional Black-Scholes framework. While Black-Scholes provides a closed-form solution, it is limited by assumptions of constant volatility, no dividends, and continuous price movements. To overcome these limitations, we use Monte Carlo simulation alongside the GARCH model, Heston stochastic volatility model, and Merton jump-diffusion model. The Black-Scholes-Monte Carlo method simulates diverse stock price paths using geometric Brownian motion. The GARCH model forecasts time-varying volatility from historical data. The Heston model incorporates stochastic volatility to capture volatility clustering and skew. The Merton jump-diffusion model adds sudden price jumps via a Poisson process. Results show the Heston model consistently produces estimates closer to market prices, while the Merton model performs well for volatile assets with sudden price movements. The GARCH model provides improved volatility forecasts for future option price prediction. All experiments used live market data from November 2024.