# Kolmogorov-Arnold Finance-Informed Neural Network in Option Pricing
## 3 Methodology  
### Modeling  
The elements listed in Table 1 provide essential components for the financial modeling framework, each serving a specific function in the overall structure. The univariate functions, denoted by \(g_{i}\), are foundational to the Kolmogorov-Arnold representation, which allows for the decomposition of complex multivariate functions into simpler one-dimensional components. This representation is particularly valuable in financial modeling, as it facilitates the capture of intricate relationships between variables while maintaining computational efficiency. The mappings \(h_{i}\) further define the interactions between these univariate functions and the financial variables at play, elucidating how each function contributes to the model's predictive capacity. This interaction is crucial, as it informs how changes in one variable might propagate through the system, affecting the overall outcome. \begin{table}
\begin{tabular}{c l} \hline \hline
**Symbol** & **Description** \\ \hline \(g_{i}\) & Univariate functions in Kolmogorov–Arnold representation \\ \(h_{i}\) & Mappings that define how each univariate function interacts with each variable \\ \(N\) & Number of terms in the superposition, which can be finite or infinite \\ \(\mathcal{F}\) & Financial differential operator \\ \(\mathbf{u}(\mathbf{x},t)\) & Financial variable of interest (e.g., option price, asset value) \\ \(\mathbf{u}_{\theta}(\mathbf{x},t)\) & Neural network representation of the financial variable of interest \\ \(\lambda\) & Parameters of the financial model \\ \(\Omega\) & Domain of financial variables \\ \([0,T]\) & Time period of interest \\ \hline \hline \end{tabular}
\end{table} 
Table 1: List of symbols used in the financial modeling context. The symbol \(N\) indicates the number of terms in the superposition of functions, which can be finite or infinite. The choice of \(N\) directly impacts the model's complexity and accuracy. A larger \(N\) allows for a more nuanced representation of financial phenomena but may also lead to overfitting if not managed properly. Conversely, a smaller \(N\) might simplify the model at the expense of capturing essential dynamics. Therefore, determining an optimal \(N\) is critical for balancing complexity and interpretability. The financial differential operator, \(\mathcal{F}\), is employed to analyze the dynamics of financial variables, enabling the exploration of their behavior over time under various market conditions. This operator plays a vital role in expressing the continuous evolution of financial instruments, allowing the model to incorporate essential features such as volatility and drift. The variable \(\mathbf{u}(\mathbf{x},t)\) represents the financial variable of interest, such as an option price or asset value, serving as the model's primary output. Its accurate estimation is crucial for effective decision-making in finance. In contrast, \(\mathbf{u}_{\theta}(\mathbf{x},t)\) denotes the neural network representation of this variable, leveraging deep learning techniques to enhance the model's predictive capabilities. The use of neural networks allows for capturing nonlinear relationships that traditional methods may overlook, thereby improving the robustness of predictions. Parameters \(\lambda\) govern the dynamics and relationships among the variables within the financial model. Proper calibration of these parameters is essential for aligning the model with real-world data, ensuring that predictions are both reliable and relevant. Additionally, \(\Omega\) defines the domain of financial variables, setting boundaries for the model's applicability. This ensures that the model operates within a realistic range, enhancing its practical utility. The time interval \([0,T]\) specifies the period of interest for the financial analysis, establishing a temporal framework for evaluating the dynamics of the modeled variables. This temporal aspect is critical, as financial markets are inherently dynamic, and the impact of events can vary significantly over time.