# Deep Learning in Financial Modeling: Predicting European Put Option Prices with Neural Networks
## 4 Results
### Comparative Analysis--DNN vs. PINN  
To assess the performance of the proposed deep neural network (DNN) in comparison to alternative deep learning-based approaches, a benchmark analysis was conducted against physics-informed neural networks (PINNs). While both methods leverage neural network architectures to approximate option prices, they differ in their underlying formulation and computational requirements.  
The PINN model was designed with an architecture comprising five fully connected layers, each containing 64 neurons, and utilizes the tanh activation function. Unlike standard DNNs, which optimize purely on data-driven loss functions, the PINN incorporates additional physics-informed constraints, minimizing a weighted sum of the Mean Squared Error (MSE) and residuals of the governing partial differential equation (PDE). This formulation ensures that predictions remain consistent with financial theory but introduces additional computational complexity. The PINN was trained for 200 epochs with a learning rate of \(10^{-4}\) and a batch size of 64, significantly increasing computational demands.  
A performance comparison between the two models reveals that the PINN achieves a slightly lower MSE (0.0017 vs. 0.0021) and a marginally higher R\({}^{2}\) score (0.9620 vs. 0.9533), indicating a minor advantage in predictive accuracy. However, these improvements come at the cost of substantially higher computational requirements. The PINN requires 25 min of training time, compared to just 7 min for the DNN, highlighting the efficiency of the proposed approach. Additionally, the Mean Absolute Error (MAE) suggests that while  
Figure 4: 3D visualization of European put option prices predicted by the trained deep neural network.  
the PINN performs well in capturing theoretical consistency, the DNN exhibits stronger generalization across different parameter configurations, with an MAE of 0.0299 compared to 0.0416 for the PINN.  
Despite its computational efficiency, the DNN does present certain limitations when compared to the PINN. While it performs well across a broad range of input conditions, it tends to struggle with boundary scenarios where PINNs excel due to their explicit incorporation of domain knowledge. The PINN's reliance on PDE residuals allows it to maintain theoretical consistency in extreme pricing conditions, such as deep out-of-the-money options or near-expiry contracts. However, for practical applications that require real-time predictions and scalability, the DNN offers a clear advantage in terms of speed and computational feasibility.  
These findings reinforce the notion that while PINNs offer strong theoretical consistency, their computational expense limits their practicality for large-scale financial applications. In contrast, the DNN provides a highly efficient alternative, achieving comparable accuracy while significantly reducing training time. The trade-offs between these approaches suggest that hybrid models, which integrate both data-driven learning and physics-informed constraints, may provide an optimal balance between accuracy and efficiency in future research.