# Kolmogorov-Arnold Finance-Informed Neural Network in Option Pricing
## 6 Experiment and Results
### Experiment Setting and Implementation  
In this section, we detail the experimental setup for evaluating the performance of our proposed KAFIN framework in the context of European call option pricing and American put option pricing. The key parameters and settings used in the experiments are outlined below in Table 2.  
The strike price for the option is set to \(K=40\). This is the fixed price at which the option holder can buy the underlying asset at maturity. The risk-free interest rate is set to \(r=0.05\). This rate represents the theoretical return on an investment with no risk of financial loss. The volatility of the underlying asset is \(\sigma=0.2\). Volatility measures the price fluctuations of the underlying asset over time. The time to maturity of the option is \(T=1\) year. This is the period over which the option can be exercised. The range of asset prices considered in the experiments spans from \(S=0\) to \(S=160\). This range is chosen to cover a broad spectrum of possible asset prices. The time range considered in the experiments extends from the current time (\(t=0\)) to the maturity (\(t=T\)). This is expressed as \(t\in[0,T]\). A total of 10,000 sample points (\(N_{\text{sample}}=10,000\)) are used for training and testing the neural network model. These samples are uniformly distributed across the asset price and time ranges to ensure comprehensive coverage of the domain. To further validate the effectiveness of the KAFIN model, we compare its performance with FINN REG [4; 30], FINN FNN [1; 11], FINN DNN [6; 12], and FINN MLP [2; 29], as baselines under the same experiment setting. The following results show the performance after training with 2300 epochs. These models were selected not only for their demonstrated effectiveness in financial contexts but also because they represent state-of-the-art approaches capable of addressing the key challenges of nonlinearity, constraint handling, and computational efficiency in financial modeling. We selected multi-layer perceptrons (MLPs) and deep neural networks (DNNs) for their proven ability to model complex, nonlinear relationships in financial data, which is a key requirement for capturing the intricate patterns and interactions inherent in financial systems. Feed-Forward Networks (FFNs) were included due to their effectiveness in processing multi-dimensional financial data and handling complex constraints within Finance-Informed Neural Networks (FINNs). Despite potential challenges like gradient vanishing, these can be mitigated with techniques such as gradient clipping. Additionally, the residual gain (REG) strategy was chosen for its ability to enhance learning efficiency without increasing computational complexity, making it particularly valuable in financial applications where computational feasibility is crucial. All networks in this study share a common input layer width of 2 and an output layer width of 1. However, the architectures vary in the number of hidden layers and units, which are optimized to balance model complexity and computational efficiency. The KAFIN architecture consists of a single hidden layer with 3 units. This streamlined design ensures computational efficiency while capturing the key dynamics of carbon emission and sequestration processes. The FINN DNN model employs 4 hidden layers, each with 20 units, to capture complex relationships in high-dimensional data. While this deep architecture enhances the model's ability to learn intricate patterns, it also increases the risk of overfitting due to its complexity. The FINN GRU model consists of 4 hidden layers with 4 units, utilizing recurrent layers to capture temporal dependencies in carbon dynamics. This compact design maintains computational efficiency while leveraging the advantages of recurrent networks for modeling sequential data. For comparison, other models use 4 hidden layers with 20 units each, providing a baseline for evaluating the performance of the KAFIN model against more traditional feed-forward architectures.