# Deep Learning in Financial Modeling: Predicting European Put Option Prices with Neural Networks
## 1 Introduction  
Option pricing is a cornerstone of quantitative finance, providing tools for risk management and derivative valuation. European put options, in particular, play a critical role in hedging strategies and investment decision-making. Among the most widely used models, the Black-Scholes framework, introduced in 1973 [1], provides an elegant and tractable closed-form solution. However, its reliance on assumptions such as constant volatility, frictionless markets, and normal asset return distributions limits its effectiveness in capturing the nonlinear and stochastic dynamics of modern financial markets [2].  
In recent years, advances in deep learning have shown promise in addressing these challenges. Neural networks, with their ability to model nonlinear relationships and capture complex patterns in large datasets, present a flexible alternative to traditional methods. Unlike the B&S model, neural networks do not require restrictive assumptions about market behavior, making them well-suited for scenarios involving noisy data and dynamic environments. Several studies have applied deep learning to financial modeling, including physics-informed neural networks (PINNs) and Fourier Neural Operators (FNOs), which integrate domain knowledge into neural network architectures to solve partial differential equations (PDEs) and stochastic differential equations (SDEs) [3, 4, 5, 6]. While these methods have demonstrated strong theoretical potential, they often require significant computational resources and can struggle with convergence in high-dimensional problems.  
This study proposes a dense neural network (DNN) framework for pricing European put options. Compared to PINNs and FNOs, our approach emphasizes computational simplicity and scalability, making it suitable for large-scale applications. The key objectiveis to leverage synthetic datasets generated using the B&S analytical solution to train and validate the model, providing a controlled environment for evaluating its performance. The DNN's ability to approximate theoretical prices with high accuracy underpins its potential for extension to real-world financial data. While the reliance on synthetic data is a limitation, it allows for a baseline assessment before integrating real-market datasets.  
Our contributions are as follows:  
1. The development of an optimized DNN architecture for option pricing, incorporating advanced regularization techniques to enhance generalization.
2. A systematic evaluation of the DNN's performance on synthetic datasets, highlighting its scalability and robustness.
3. A discussion on the limitations of the methodology and pathways for future research, including the integration of real-market data and hybrid approaches that combine model-driven and data-driven methods.  
The rest of this paper is organized as follows: Section 2 reviews related works, comparing traditional and AI-based approaches. Section 3 presents the methodology, including the DNN architecture and data generation process. Section 4 discusses the results and their implications, and Section 5 concludes with limitations and future directions.