# Kolmogorov-Arnold Finance-Informed Neural Network in Option Pricing
## 7 Discussion
### Discussion on European Option Pricing
#### 7.1.1 Performance
\begin{table}
\begin{tabular}{l r r r r r r} \hline MEAN & \(\mathcal{L}_{total}\) & \(\mathcal{L}_{initial}\) & \(\mathcal{L}_{financial}\) & \(\mathcal{L}_{boundary}\) & \(\mathcal{L}_{boundary,0}\) & \(\mathcal{L}_{boundary,0}\) \\ \hline KAFIN & 28.8352 & 18.6403 & 2.3630 & 7.8319 & 1.3675 & 6.4644 \\ FINN REG & 8156.2915 & 8156.1657 & 0.1257 & 1203.5108 & 0.0915 & 6952.5634 \\ FINN FFN & 11,421.6493 & 11,421.6481 & 0.0012 & 1861.5178 & 0.0004 & 9560.1299 \\ FINN DNN & 10,732.5727 & 10,732.5727 & 0.0000 & 1722.6049 & 0.1503 & 9009.8176 \\ FINN MLP & 12,110.2974 & 12,110.2974 & 0.0000 & 2020.4457 & 0.0017 & 10,089.8499 \\ \hline STD & \(\mathcal{L}_{total}\) & \(\mathcal{L}_{initial}\) & \(\mathcal{L}_{financial}\) & \(\mathcal{L}_{boundary}\) & \(\mathcal{L}_{boundary,0}\) & \(\mathcal{L}_{boundary,0}\) \\ \hline KAFIN & 151.5633 & 79.9292 & 2.4999 & 91.8692 & 5.0428 & 91.6668 \\ FINN REG & 1868.4424 & 1868.5568 & 0.1521 & 378.4857 & 0.0621 & 1490.7575 \\ FINN FFN & 410.3859 & 410.3870 & 0.0015 & 96.8601 & 0.0010 & 317.0152 \\ FINN DNN & 725.1539 & 725.1539 & 0.0001 & 157.5038 & 0.3906 & 569.4847 \\ FINN MLP & 180.6481 & 180.6481 & 0.0000 & 51.5401 & 0.0011 & 136.8783 \\ \hline MIN & \(\mathcal{L}_{total}\) & \(\mathcal{L}_{initial}\) & \(\mathcal{L}_{financial}\) & \(\mathcal{L}_{boundary}\) & \(\mathcal{L}_{boundary,0}\) & \(\mathcal{L}_{boundary,0}\) \\ \hline KAFIN & 0.9411 & 0.4308 & 0.3437 & 0.0558 & 0.0066 & 0.0485 \\ FINN REG & 5849.0020 & 5848.5151 & 0.0003 & 727.2389 & 0.0287 & 5121.2471 \\ FINN FFN & 10,825.3271 & 10,825.3223 & 0.0000 & 1670.5862 & 0.0000 & 9140.9287 \\ FINN DNN & 9933.3311 & 9933.3311 & 0.0000 & 1481.9844 & 0.0001 & 8407.6611 \\ FINN MLP & 11,744.3564 & 11,744.3564 & 0.0000 & 1879.0813 & 0.0002 & 9850.9521 \\ \hline MAX & \(\mathcal{L}_{total}\) & \(\mathcal{L}_{initial}\) & \(\mathcal{L}_{financial}\) & \(\mathcal{L}_{boundary}\) & \(\mathcal{L}_{boundary,0}\) & \(\mathcal{L}_{boundary,0}\) \\ \hline KAFIN & 5934.0322 & 1675.9518 & 10.5056 & 4255.2658 & 27.0373 & 4255.2524 \\ FINN REG & 12,850.6650 & 12,850.6641 & 0.5156 & 2249.6992 & 0.2395 & 10,644.5205 \\ FINN FFN & 12,270.0977 & 12,270.0977 & 0.0055 & 2103.2632 & 0.0096 & 10,183.2969 \\ FINN DNN & 12,458.5586 & 12,458.5586 & 0.0007 & 2175.2749 & 2.9416 & 10,298.2354 \\ FINN MLP & 12,493.1035 & 12,493.1035 & 0.0000 & 2188.5798 & 0.0034 & 10,312.2305 \\ \hline \end{tabular}
\end{table} 
Table 5: Loss values in European call option pricing with 2300 epoch training. and more erratic residual distributions, suggesting that they require further training to approach the performance level of KAFIN. The performance of KAFIN can be attributed to several factors. Its architecture is likely optimized for learning the complex relationships inherent in financial data, and its training methodology effectively minimizes loss across diverse metrics. The robust integration of features, combined with advanced regularization techniques, likely contributes to KAFIN's ability to stabilize residuals and improve generalization. The results indicate that KAFIN significantly outperforms its baseline counterparts in the task of European option pricing. Its ability to minimize residuals and maintain a close approximation to the analytical solutions highlights its potential as a reliable tool in financial modeling. Future work may focus on further refining KAFIN's architecture and exploring its applicability across different financial derivatives to enhance its utility in real-world scenarios.