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

2025-04-04

AI Summary

This paper addresses the limitation of the Black–Scholes model in pricing Google stock options—its neglect of time-varying volatility and interest rates. To overcome this, we propose an extended PDE framework that jointly models stochastic volatility and stochastic interest rates within a single unified partial differential equation—a novel coupling approach. Methodologically, we systematically compare finite difference methods (FDM) and Long Short-Term Memory (LSTM) neural networks in terms of pricing accuracy and computational efficiency: LSTM achieves lower prediction errors in historical backtesting, whereas FDM offers superior numerical stability and faster computation. Empirical results demonstrate that the hybrid modeling strategy significantly enhances robustness and generalization across varying market volatility regimes. The framework thus provides a theoretically rigorous yet computationally practical solution for pricing complex derivatives, bridging the gap between financial theory and engineering implementation.

Technology Category

• Search and Optimization: Mixed Discrete/Continuous Search
• Machine Learning: Learning with Manifolds
• Natural Language Processing: Learning & Optimization for NLP

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Economics, Online Markets and Human Computation: Cost models of using LLMs in production systems
• Web Mining and Content Analysis: Models for Web evolution

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.

Problem

• Extending Black-Scholes model for stochastic volatility and interest rates
• Comparing PDE and LSTM models for Google option pricing
• Evaluating computational efficiency versus predictive accuracy

Innovation

• Extended Black-Scholes model with stochastic volatility
• Finite difference method for PDE solution
• LSTM model for predictive accuracy