# Kolmogorov-Arnold Finance-Informed Neural Network in Option Pricing
## 6 Experiment and Results
### Experiment Setting and Implementation
The architectural choices were made to strike a balance between computational efficiency, model complexity, and generalizability. The KAFIN model opts for a compact design with fewer hidden units to minimize computational cost while retaining predictive accuracy. In contrast, the FINN DNN, with its deep architecture, is designed to model \begin{table}
\begin{tabular}{l l l} \hline \hline
**Parameter** & **Value** & **Parameter Description** \\ \hline Strike Price (\(K\)) & 60 & Fixed price at which the option can be exercised. \\ \hline Risk-Free Rate (\(r\)) & 0.05 & Theoretical return on a risk-free investment. \\ \hline Volatility (\(\sigma\)) & 0.2 & Price fluctuation of the underlying asset. \\ \hline Maturity (\(T\)) & 1 & Timeframe for option exercise. \\ \hline Asset Price Range (\(S\)) & \(S\in[0,160]\) & Range of asset prices considered. \\ \hline Time Range (\(t\)) & \(t\in[0,T]\) & From current time to maturity. \\ \hline Number of Samples (\(N_{\text{sample}}\)) & 10,000 & Sample points for training and testing. \\ \hline Finance Info Weight (\(\lambda_{financial}\)) & 0.1 & Finance-informed learning rate for training. \\ \hline Initial Value Weight (\(\lambda_{initial}\)) & 1 & Initial constraint penalty for training. \\ \hline Zero Boundary Weight (\(\lambda_{initial}\)) & 1 & Zero boundary penalty for training. \\ \hline Infinity Boundary Weight (\(\lambda_{initial}\)) & 1 & Infinity boundary penalty for training. \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameter settings for option pricing test. more intricate data relationships, albeit at the cost of increased training time and potential overfitting. The FINN GRU, with fewer units and recurrent layers, captures temporal dependencies effectively while optimizing for computational efficiency. The Adam optimizer was selected for its efficiency and adaptability in training deep learning models. Combining the benefits of both Adagrad and RMSprop, Adam adjusts learning rates for each parameter based on first and second moment estimates of the gradient, which accelerates convergence and helps prevent oscillations during training. For the experiments, the learning rate was set to \(1\times 10^{-5}\). This relatively small value, chosen after extensive empirical testing, strikes a balance between convergence speed and model accuracy. A lower learning rate ensures stable convergence and minimizes the risk of overshooting the optimal point, although it may require additional training epochs to reach the desired level of accuracy. This choice was made to accommodate the complexity of carbon emission and sequestration models, ensuring gradual convergence while minimizing overfitting. Results  
In this section, we present the results of our KAFIN-based approach for European option pricing. Figures 1 and 2 show the learning results of KAFIN for both European call and American put pricing. The figure employs a color scheme with a gradient scale, where lighter shades correspond to higher values, and darker shades indicate lower values. After 2300 epochs of training, the KAFIN predictions are close to the analytical solutions of both as a benchmark. We further compare the performance of KAFIN with baselines, and Tables 3 and 4 show the values of the residuals between the analytical solutions and predicted solutions among all the baselines. KAFIN shows the lowest residuals after 2300 training epochs in all residual measures including mean, standard deviation (STD), minimum, absolute minimum, maximum, absolute maximum, and MSE. Figures 3 and 4 show a visualization of the residuals between methods and the analytical benchmark, in which, the residual of KAFIN is bounded within \([-3,+3]\) in European call option pricing, and \([-2,+2]\) in American put option pricing, while the rest of the methods still need more training. Figure 3: Overall residual comparison among methods. Figure 2: Comparison of predicted American put option pricing by KAFIN model and analytical solutions from Black–Scholes model.