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

Paper: arXiv 2604.06068

Authors: Karmanpartap Singh Sidhu, Pranshi Saxena

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.

Complexity vs Empirical Score

• Math Complexity: 6.5/10
• Empirical Rigor: 5.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a solid computational review and comparative implementation of established option pricing models (Heston, GARCH, Merton) using recent market data. While mathematically sound and clear in exposition, it lacks novelty as it largely benchmarks well-known existing frameworks without introducing new theoretical or methodological advancements.

Research Flowchart

  flowchart TD
    A[Goal: Accurate Option Pricing Beyond Black-Scholes] --> B[Input: Live Market Data - Nov 2024]
    B --> C[Monte Carlo Simulation Framework]
    
    C --> D1[GARCH Model: Time-Varying Volatility]
    C --> D2[Heston Model: Stochastic Volatility]
    C --> D3[Merton Model: Poisson Jump-Diffusion]
    
    D1 --> E1[Findings: Improved Future Volatility Forecasts]
    D2 --> E2[Findings: Closest Overall to Market Prices]
    D3 --> E3[Findings: Best for Highly Volatile Assets]