# Kolmogorov-Arnold Finance-Informed Neural Network in Option Pricing
###### Abstract  
Finance-Informed Neural Networks (FINNs), inspired by Physical Information Neural Networks (PINNs) and computational finance, aim to enhance risk assessment and support regulatory decision-making. Despite their promising potential, existing FINNs face significant challenges related to learning efficiency and overall performance. This paper examines the KAFIN framework, based on the Kolmogorov-Arnold representation, which aims to improve financial modeling and analytical calculations. To evaluate the performance of FINNs, this study uses generated data instead of real market data. This choice enables controlled simulations of various financial scenarios while avoiding the complexities and noise inherent in actual market data. By relying on generated data, we are able to isolate and assess the core capabilities of KAFIN under well-defined theoretical conditions, facilitating a clearer analysis of its performance. This study focuses on European option pricing as a case study, using generated input data that simulate a range of market scenarios, including typical regulatory conditions in financial markets. Testing KAFIN under these controlled conditions allows for a rigorous evaluation of its ability to handle the complexities of pricing options across different market assumptions and regulatory constraints. Empirical results demonstrate that KAFIN significantly outperforms the baseline method, improving pricing accuracy by minimizing residual errors and aligning closely with analytical solutions. The architecture of KAFIN effectively captures the inherent complexities of option pricing, integrating financial principles with advanced computational techniques. Performance indicators reveal that KAFIN achieves lower average total losses and reduces the variability in loss components, highlighting its advantage in modeling and analyzing complex financial dependencies while ensuring compliance with regulatory standards. Finance-Informed Neural Networks (FINNs); Physics-Informed Neural Networks (PINNs); Kolmogorov-Arnold representation (KAR); option pricing; computational financial modeling; machine learning; risk assessment 2024 14 11618 1224 14 1224 14 1224 14 1224 14 1224 14 1224 14 1224 14 1224 1224 124 1224 124 122  
By integrating financial theories and constraints, FINNs align outputs with established financial knowledge. This departure, however, complicates their training compared to traditional neural networks. The intricate nature of their loss functions and heightened computational requirements pose challenges to their efficiency and interpretability, necessitating larger structures and training scale to maintain accuracy [13; 14]. Studies in Kolmogorov-Arnold representation in machine learning [15; 16; 17; 18; 19] show promise in overcoming the challenges of FINNs with complexity deduction, representation simplification, and improvement in interpretability and performance in scientific computation. Thus, in this study, we propose the Kolmogorov-Arnold Finance-Informed Neural Network (KAFIN), a framework that integrates Kolmogorov-Arnold representation (KAR) with an FINN to address the challenges above. Option pricing is employed as a case study to showcase KAFIN's feasibility and efficiency. Key contributions of this work include the following:  
* Integration of KAR with FINN: KAFIN integrates KAR within the FINN framework, enhancing its capability to accurately capture complex financial dynamics by decomposing multivariate functions into simpler components. * Incorporation of Financial Rules and Constraints: KAFIN incorporates specific financial rules and constraints directly into its architecture, ensuring adherence to fundamental financial principles and improving model reliability and interpretability. * Case Study Validation: We validate KAFIN through a practical case study, demonstrating its effectiveness in finance practices.