Mathematical Modeling of Option Pricing with an Extended Black-Scholes Framework

ArXiv ID: 2504.03175 “View on arXiv”

Authors: Unknown

Abstract

This study investigates enhancing option pricing by extending the Black-Scholes model to include stochastic volatility and interest rate variability within the Partial Differential Equation (PDE). The PDE is solved using the finite difference method. The extended Black-Scholes model and a machine learning-based LSTM model are developed and evaluated for pricing Google stock options. Both models were backtested using historical market data. While the LSTM model exhibited higher predictive accuracy, the finite difference method demonstrated superior computational efficiency. This work provides insights into model performance under varying market conditions and emphasizes the potential of hybrid approaches for robust financial modeling.

Keywords: Option Pricing, Stochastic Volatility, Finite Difference Method, LSTM, Partial Differential Equation (PDE), Equity Derivatives

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail
• Why: The paper uses advanced mathematics including stochastic differential equations (Heston and Vasicek models), partial differential equations, and numerical methods (finite difference), indicating high mathematical complexity. It also demonstrates empirical rigor through backtesting on historical Google options data and comparing models (extended PDE vs. LSTM) on accuracy and computational efficiency.

  flowchart TD
    A["Research Goal: Enhance Option Pricing by Extending<br>Black-Scholes Framework"] --> B["Data: Historical Google Stock Options Data"]
    B --> C{"Key Methodology"}
    C --> D["Method 1: Extended Black-Scholes PDE<br>with Stochastic Volatility & Rates"]
    C --> E["Method 2: LSTM Machine Learning Model"]
    D --> F["Computational Process:<br>Finite Difference Method"]
    E --> G["Computational Process:<br>Deep Learning Training"]
    F --> H["Key Findings & Outcomes"]
    G --> H
    subgraph H [" "]
        H1["LSTM: Higher Predictive Accuracy"]
        H2["Finite Difference: Superior Computational Efficiency"]
        H3["Insight: Hybrid Approaches offer<br>Robust Financial Modeling"]
    end
    H --> I["End"]