# Deep Learning in Financial Modeling: Predicting European Put Option Prices with Neural Networks
## 2 Related Works  
The Black-Scholes (B&S) model has been a cornerstone in option pricing since its introduction in 1973 [1]. This analytical approach assumes constant volatility and frictionless markets, offering closed-form solutions for European options. While widely used for its simplicity, the B&S model struggles in real-world applications where market conditions deviate from these assumptions, such as stochastic volatility, market anomalies, and sudden jumps in asset prices [2]. To address these limitations, numerical solvers and spectral methods have been employed to provide more accurate solutions under complex conditions. However, these approaches often come with significant computational costs, particularly in high-dimensional settings or when dealing with stochastic dynamics. Recent advancements in deep learning have provided powerful tools to overcome the limitations of traditional methods. Neural networks have been applied to solve partial differential equations (PDEs) and stochastic differential equations (SDEs), which underpin option pricing models. Several key architectures and frameworks have emerged in this domain, each addressing specific challenges in financial modeling. 1. Physics-Informed Neural Networks (PINNs):  
The idea of using neural networks to solve differential equations has its roots in earlier works, such as [3], who utilized artificial neural networks to solve ODEs and PDEs by embedding analytical conditions into the training process. [PERSON] and [PERSON] [3] introduced Deep Galerkin Methods (DGMs), which extended this idea to high-dimensional PDEs using mesh-free approximations. Building on these foundations, [PERSON] et al. [4] formalized the term \"physics-informed neural networks\" (PINNs), integrating physical laws directly into the loss function. While PINNs have demonstrated strong theoretical potential, their computational cost and convergence issues remain challenging, particularly for high-dimensional or noisy datasets. 2. Fourier Neural Operators (FNOs):  
FNOs extend deep learning capabilities by leveraging Fourier transforms to model spatial and temporal dynamics in PDEs [5]. This method reduces computational complexity by operating in the frequency domain, enabling efficient solutions to parametric PDEs. While promising, FNOs often rely on structured, grid-based data, which limits their applicability in unstructured financial datasets. * Residual Neural Networks (ResNets): Residual networks (ResNets) have shown significant success in addressing vanishing gradient issues in deep architectures [6]. In financial modeling, ResNets have been used to improve the stability of neural networks when approximating complex payoff structures or handling high-dimensional data. Their skip-connection mechanism enables deeper architectures without a proportional increase in training difficulty, making them a strong candidate for dynamic market environments. * Transformers in Financial Applications: Originally designed for natural language processing [7], Transformers have recently gained traction in financial modeling for their ability to capture long-range dependencies and interactions in sequential data. Studies have adapted Transformer architectures for option pricing, particularly in predicting time-dependent payoff structures or volatility surfaces [8]. Their scalability and robustness make them a promising alternative to traditional architectures. * Dense Neural Networks (DNNs): Dense neural networks have been widely applied in option pricing for their simplicity and adaptability to large datasets [9]. Compared to PINNs and FNOs, DNNs provide a computationally efficient solution, albeit without explicitly integrating domain-specific knowledge. However, when combined with advanced regularization techniques, DNNs can achieve strong performance in approximating theoretical pricing models. Dense neural networks (DNNs), in contrast, provide a simpler and computationally scalable framework for option pricing.