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REVIEW 5 major objections 5 minor 1 cited by

Option Pricing with Convolutional Kolmogorov-Arnold Networks

T0 review · 5 major / 5 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read A convolutional Kolmogorov-Arnold network trained on raw CSI 300 option data prices options with lower error than Black-Scholes-Merton, LSTM, and Conv-LSTM baselines.

desk verdict First KAN/Conv-KAN option-pricing application with a sensible B-S-M baseline, but the undefined 'theoretical price' feature and contract-level split make the headline MSE advantage uninterpretable. read the letter →

arxiv 2412.01224 v1 pith:4EFLOLWY submitted 2024-12-02 cs.CE

Zeyuan Li , Qingdao Huang This is my paper · ORCID
classification cs.CE MSC 91G2068T07
keywords optionpricingConvolutionalKolmogorov-ArnoldNetworksBlack-Scholes-MertonmodelCSI300indexoptionsLSTMdatageneralizationneuralnetworkcomparison
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

This paper tests whether modern neural architectures can beat classical parametric option-pricing formulas when trained on raw market data rather than carefully filtered data. It builds a feature set around the dividend-adjusted Black-Scholes-Merton formula and compares six models on 2020 CSI 300 Index options, using the first eight months for training and the last four for testing. The central claim is that Convolutional Kolmogorov-Arnold Networks (Conv-KANs) achieve the lowest mean squared error, RMSE, and mean absolute error of all six models, including the B-S-M parametric baseline. The authors take this as evidence that KAN-style nonlinear layers are a viable alternative to MLP-based networks for option pricing, and that their chronological train/test split better reflects how a trader actually encounters option data.

What carries the argument

Two mechanisms carry the argument. The first is the Conv-KAN layer, in which each kernel entry is a learnable one-dimensional function $\phi(x)=w_1\,\mathrm{spline}(x)+w_2\,\mathrm{silu}(x)$ rather than a scalar weight. For an input image $a$ and a $N\times M$ KAN kernel $K$, the convolution output is $(i\ast K)_{i,j}=\sum_{k=1}^{N}\sum_{l=1}^{M}\phi_{kl}(a_{i+k,j+l})$, so the nonlinearity happens inside the kernel. The paper keeps a conventional fully connected output layer to isolate the effect of the nonlinear convolution. The second mechanism is the data-selection strategy: options are split by calendar date rather than filtered on moneyness, so the training set retains raw noisy observations and some options appear for only one day, forcing the network to generalize.

What would settle it

Retrain the Conv-KAN and B-S-M comparison on the same 2020 CSI 300 samples with the 'theoretical price' input removed or replaced by a neutral feature such as moneyness. If Conv-KAN's MSE no longer beats B-S-M's 0.01552, the reported advantage is an artifact of the setup rather than a property of the architecture.

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Extended reading notes

Core claim

On 2020 CSI 300 Index options, the paper reports that a Conv-KAN trained for 50 epochs reaches MSE 0.00790, RMSE 0.08890, and MAE 0.07098, the best among Black-Scholes, Black-Scholes-Merton, LSTM, Conv-LSTM, KAN, and Conv-KAN. It also reports that the dividend-adjusted B-S-M formula with dividend rate $q$ improves on the basic B-S formula (MSE 0.01552 versus 0.01612). These numbers are used to argue that KAN-based architectures are a new and effective option-pricing method and that training on deliberately noisy, unfiltered data improves generalization.

Load-bearing premise

The central claim rests on the assumption that the 'theoretical price' variable used as a network input is not itself generated by the Black-Scholes or Black-Scholes-Merton baseline formula; if it is, the neural models are simply receiving the parametric answer as a feature.

Editorial extensions

If this is right

  • If the reported errors hold, Conv-KANs can price options from raw data without pre-filtering on moneyness, lowering the data-preparation burden.
  • KAN and Conv-KAN can join the standard option-pricing benchmark set, giving future work a comparison point against MLP, LSTM, and parametric baselines.
  • The B-S-M feature set with dividend rate q is confirmed as a useful input encoding for dividend-paying underlyings.
  • Conv-KANs reach the best error in 50 epochs versus 200 for LSTM, suggesting the advantage is architectural rather than a matter of training budget.
  • The convolutional models' high MAPE shows the error reduction is not uniform across all option prices, pointing to uneven fitting across the price range.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • A separate ablation that removes the undocumented 'theoretical price' input would test whether the Conv-KAN advantage is intrinsic or inherited from the parametric baseline; this check is not reported in the paper.
  • Because the train/test split is chronological, the results also capture any market-regime shift between the first eight and last four months of 2020; a rolling-window or shuffled split on the same raw data would isolate the architecture's contribution.
  • The high MAPE of the convolutional models (15.12 for Conv-KANs versus 0.61 for B-S-M) suggests the improved MSE comes from the bulk of prices while small or near-zero prices are poorly predicted; stratifying errors by moneyness or time to maturity would show where the gains and losses are.
  • The same architecture and data strategy could be moved to other index-option markets with dividend-paying underlyings, which would indicate whether the result is specific to CSI 300 market microstructure.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

5 major / 5 minor

Summary. The paper applies Kolmogorov-Arnold Networks (KANs) and Convolutional KANs (Conv-KANs) to price CSI 300 index options, comparing them with B-S, B-S-M, LSTM, and Conv-LSTM models. It proposes a data-processing strategy that splits each option's life across the training/test boundary and deliberately keeps raw noisy data to simulate a real trading environment. The headline empirical claim is that Conv-KANs achieve the lowest MSE/RMSE/MAE in Table 1 (0.00790, 0.08890, 0.07098), outperforming the parametric B-S-M baseline (0.01552, 0.12457, 0.08442). The text also claims that the B-S-M formula with a dividend-yield variable improves on the classical B-S formula.

Significance. The application of KANs and Conv-KANs to option pricing is novel and could be of interest to the computational-finance community. The proposed data-splitting strategy, if properly implemented, addresses a real concern about training and evaluating under conditions closer to actual trading. However, the current manuscript does not provide trustworthy evidence for its central claim: the undefined 'theoretical price' input feature is likely model-derived, the test split overlaps with training contracts, there are no repeated runs or error bars, and the MAPE results for the convolutional models are two orders of magnitude worse than those of the alternatives. No code or data is provided. These issues are load-bearing and cannot be fixed by minor editing.

major comments (5)
  1. [§4.1, Table 1] The input feature 'theoretical price' is listed among the nine variables but is never defined. In the cited RESSET dataset, this field is conventionally the Black-Scholes or Black-Scholes-Merton formula price. If that is the case, every neural network receives the same quantity that the B-S and B-S-M baselines output, so the comparison in Table 1 is circular and the reported Conv-KAN advantage is an artifact of residual learning on top of the baseline rather than independent superiority. The authors must define how 'theoretical price' is computed and report results with and without this feature; the same concern applies to 'Delta', which is also model-dependent.
  2. [§4.1, temporal split] The proposed data-processing strategy splits individual option contracts across the training/test boundary: the text states that an option issued on July 1st and exercised on September 15th has its July 1-August 31 observations placed in training and the remainder in the test set. This means the same contract appears in both sets, so temporal patterns learned from the training portion can directly assist prediction of the test portion. Consequently, the test set is not an independent out-of-sample evaluation, and the claim of simulating a real trading environment is unsupported. A clean chronological split with no contract overlap should be used, or at least results under both split regimes should be reported.
  3. [§4.2, Table 1] All results correspond to a single run, with no repeated seeds, no confidence intervals, and no statistical tests. The differences between B-S and B-S-M (MSE 0.01612 vs. 0.01552) are small and could easily be within run-to-run variation, and the claimed advantages of Conv-KANs over Conv-LSTM also lack error bars. Given that the comparison involves different epoch counts and model architectures, the reported numerical ranking is not credible without repeated trials and a measure of variability.
  4. [Table 1, MAPE row] The MAPE values for Conv-LSTM (12.36620) and Conv-KANs (15.11880) are roughly two orders of magnitude larger than those of B-S, B-S-M, LSTM, and KANs (0.61-0.79). The text dismisses this as due to 'the nature of the convolutional structures', but such extreme relative errors contradict the claim that the convolutional models are practically superior. The authors should explain this discrepancy and reconcile it with the headline MSE/RMSE/MAE results.
  5. [§4.2, §3.4] The model comparison is not controlled: KANs and Conv-KANs are trained for 50 epochs while LSTM and Conv-LSTM are trained for 200, and the batch size and learning rate are fixed without any hyperparameter search. No details are given for the number of KAN layers, grid size, or the configuration of the fully connected layers in Conv-KANs. Without these details, the claimed 'superior fitting performance' of Conv-KANs cannot be separated from the choice of training budget and model capacity.
minor comments (5)
  1. [§4.2, Eq. (16)] Equation (16) labels a column 'MAP' with the formula 1/N sum |y_i - yhat_i|, which is the mean absolute error; the column should be renamed 'MAE' or the formula corrected.
  2. [§6] The competing-interest declaration ends mid-sentence: 'could have appeared to influence' should be completed (e.g., 'the work reported in this paper').
  3. [References] The reference 'Ivas, cu C F' contains a spacing error; it should be 'Ivașcu C F'.
  4. [Figures 1, 6] Several figures, including Figures 1 and 6, lack axis labels or adequate captions to interpret the plotted quantities.
  5. [§4.1] The 'white-noise / data-splitting rule' mentioned in the Introduction is not formally defined; Section 4.1 describes the split qualitatively but does not specify the exact rule for options that mature near the boundary, such as the treatment of the final day of an option in the training set.

Circularity Check

1 steps flagged · score 8.0 of 10

Undefined 'theoretical price' feature: if it is the B-S/B-S-M formula output, the Conv-KAN versus parametric baseline comparison is circular by construction.

  1. self definitional [Section 4.1, Data and Data Processing Method; Section 4.2, Table 1 discussion]
    "Nine variables were deployed in the study, namely time to maturity, option type, Delta, strike price, spot price, theoretical price, monthly dividend rate, risk-free rate, and volatility calculated using the GARCH model."

    The 'theoretical price' feature is never defined or derived in the paper. The paper's own parametric baseline is the B-S-M formula (Eq. 14), and the cited RESSET data source is the standard source for Black-Scholes theoretical prices. If this input feature is the B-S/B-S-M formula output, then every neural network receives the exact quantity the parametric baselines output as one of its inputs. The target is the observed market option price, so the network's task reduces to residual learning on top of the baseline, and the text's claim that 'its convolutional version outperforms the parametric model' is not an independent architectural result. No ablation without 'theoretical price' or 'Delta' is reported, so the central comparison is unfalsifiable as written.

full rationale

The paper's central empirical claim is that Conv-KANs outperform B-S and B-S-M on CSI 300 option pricing. The feature set in Section 4.1 includes 'theoretical price' and 'Delta' but never defines how either is computed. In the cited RESSET dataset, and given the paper's own derivation of the B-S-M formula (Eq. 14), the only model-based definition of 'theoretical price' is the Black-Scholes-Merton price. If that is the input, the network is trained to map, among other things, the baseline model's output to the market price, i.e., to learn a residual. The favorable comparison against B-S/B-S-M is then not evidence of independent superiority; it is an artifact of giving the network the baseline's answer as a feature. The absence of an ablation without these model-derived features, and the absence of any definition, makes the headline result unfalsifiable. This is a self-definitional circularity in the comparison, not a statistical fit to a subset, so score 8 reflects that the central comparison is forced by the unstated construction.

Assumptions & free parameters 5 free parameters · 4 assumptions · 1 invented entities

The central comparison depends on a handful of hand-chosen hyperparameters, the B-S/M assumptions, and two unverified data-handling choices: the 'theoretical price' input and the option-life split. The latter two are the main sources of circularity and leakage.

free parameters (5)
  • Train/test temporal split boundary = first 8 months (train) / last 4 months (test) of 2020
    Chosen by the authors; no justification or sensitivity analysis, and it interacts with the overlapping option-life split.
  • Learning rate = 0.00001
    Set without reported tuning; central to convergence of all networks.
  • Number of training epochs = 50 for KAN/Conv-KAN, 200 for LSTM/Conv-LSTM
    Chosen by hand; unequal epochs make comparisons between model families difficult.
  • Network architecture hyperparameters = not reported in full (layer widths, KAN grid size, spline order, conv kernel size)
    Required to reproduce the architecture but omitted; the paper only gives schematic figures.
  • White-noise / data-splitting rule = split each option's life at August 31
    The decision to place early observations in training and late observations in test is arbitrary and creates overlap.
assumptions (4)
  • domain assumption Black-Scholes-Merton assumptions: no arbitrage, continuous trading, lognormal stock price, constant volatility and interest rate.
    Invoked in Section 3.1 and Appendix B as the basis for formulas (13) and (14).
  • standard math Kolmogorov-Arnold representation theorem justifies KAN architecture for the option-pricing function.
    Cited to Liu et al. [7]; the theorem is not proved and its practical applicability to noisy financial data is assumed.
  • ad hoc to paper The 'theoretical price' input feature is independent of the baseline formulas and does not leak the target.
    Section 4.1 lists 'theoretical price' among nine inputs but never defines how it is computed; if it comes from B-S/M or similar, the network is given the baseline output as input.
  • ad hoc to paper Splitting each option's life across the train/test boundary preserves temporal validity.
    Section 4.1's new data selection strategy places July-August data in training and September onward in test for the same option contract, so the same contract appears in both sets.
invented entities (1)
  • None
    purpose: No new physical or mathematical entities are introduced.
    The paper uses existing models (KAN, Conv-KAN, LSTM, Conv-LSTM) and a data-splitting rule; nothing new is postulated.

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Cite this review

Pith. "Pith review of Option Pricing with Convolutional Kolmogorov-Arnold Networks." pith.science (2026). https://pith.science/paper/4EFLOLWY

@misc{pith2026241201224,
  author       = {Pith},
  title        = {Pith review of: Option Pricing with Convolutional Kolmogorov-Arnold Networks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4EFLOLWY}},
  note         = {Machine review of arXiv:2412.01224}
}
read the original abstract

With the rapid advancement of neural networks, methods for option pricing have evolved significantly. This study employs the Black-Scholes-Merton (B-S-M) model, incorporating an additional variable to improve the accuracy of predictions compared to the traditional Black-Scholes (B-S) model. Furthermore, Convolutional Kolmogorov-Arnold Networks (Conv-KANs) and Kolmogorov-Arnold Networks (KANs) are introduced to demonstrate that networks with enhanced non-linear capabilities yield superior fitting performance. For comparative analysis, Conv-LSTM and LSTM models, which are widely used in time series forecasting, are also applied. Additionally, a novel data selection strategy is proposed to simulate a real trading environment, thereby enhancing the robustness of the model.

Figures

Figures reproduced from arXiv: 2412.01224 by the authors.

Figure 1
Figure 1. Feature representation From figure1, it can be concluded that the input for a single day’s option is (C, N, D), where C = 1 and N, representing the numbers of channels and observations, respectively, while D = 9 denotes the number of data features. In the original paper [6], Conv-LSTM is intended to capture spatiotempo￾ral correlations. The entire dataset for input has the shape of (T, C, N, D, E), T is the whole qu… view at source ↗
Figure 2
Figure 2. The architecture of the Conv-LSTM model tions. This can also be understood as the combination of all activation functions [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. The structures of MLPs and KANs In this paper, our aim is to capture the nonlinear nature of the formula in option pricing, and Nonlinear formulas from theoretical finance, such as the B-S-M formula (14), are listed, which also hold significant guiding value for pricing in real markets. Liu Z et al. noted that, for accuracy, smaller KANs can achieve comparable or even better performance than larger MLPs 9 [PITH_FUL… view at source ↗
Figures from the paper (11 more)
Figure 4
Figure 4. Figure 4: An example of convolutional neural network [PITH_FULL_IMAGE:figures/full_fig_p010_4.png]
Figure 5
Figure 5. Figure 5: Which one to choose [PITH_FULL_IMAGE:figures/full_fig_p011_5.png]
Figure 6
Figure 6. Figure 6: Architectures used in the experiments 11 [PITH_FULL_IMAGE:figures/full_fig_p011_6.png]
Figure 7
Figure 7. Figure 7: The results of the strategies 12 [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]
Figure 8
Figure 8. Figure 8: Strategies presentation This leads to the extreme situation where the option offered on August 31st becomes a point of only one day of data in the training set, that is, white noise. The effects of increased noise and direct use of raw data are to sacrifice the accurac…
Figure 9
Figure 9. Figure 9: Forecasting result of B-S model The results shown by the pictures are similar to other studies [3] in that the two lines of the figures from the B-S model match very well. The differ￾ence is that in this study, the correspondence of the two curves of the B-S-M model an…
Figure 10
Figure 10. Figure 10: Forecasting result of B-S-M model [PITH_FULL_IMAGE:figures/full_fig_p015_10.png]
Figure 11
Figure 11. Figure 11: Forecasting result of LSTM model 15 [PITH_FULL_IMAGE:figures/full_fig_p015_11.png]
Figure 12
Figure 12. Figure 12: Forecasting result of Conv-LSTM model [PITH_FULL_IMAGE:figures/full_fig_p016_12.png]
Figure 13
Figure 13. Figure 13: Forecasting result of KANs model 16 [PITH_FULL_IMAGE:figures/full_fig_p016_13.png]
Figure 14
Figure 14. Figure 14: Forecasting result of Conv-KANs model To provide a more comprehensive evaluation, four performance metrics are selected: mean squared error (MSE), root mean squared error (RMSE), mean absolute error (MAE), and mean absolute percentage error (MAPE). The following are t…

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Option Pricing Using Ensemble Learning

    cs.LG 2025-06 reject novelty 2.0 of 10

    On CSI 300 index options, gradient boosting ensembles (LGBM, XGBoost, NGBoost) achieve the lowest RMSE in most experiments, but the training set includes data from after the test period, invalidating the temporal real...

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