标题： 扩展的布莱克-斯科尔斯框架下的期权定价的数学建模 显示英文标题

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

摘要： 本研究通过扩展布莱克-斯科尔斯模型以纳入偏微分方程（PDE）中的随机波动率和利率可变性，来提升期权定价性能。该PDE使用有限差分法求解。开发并评估了扩展的布莱克-斯科尔斯模型以及基于机器学习的LSTM模型用于对谷歌股票期权进行定价。两种模型均使用历史市场数据进行了回测。虽然LSTM模型表现出更高的预测准确性，但有限差分法展示了更出色的计算效率。这项工作提供了对不同市场条件下模型表现的见解，并强调了混合方法在稳健金融建模中的潜力。 显示英文摘要

摘要： 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.