Bridging Structured Knowledge and Data: A Unified Framework with Finance Applications

Yi Cao; Zexun Chen; Lin William Cong; Heqing Shi

stat.MLApr 1, 2026

Bridging Structured Knowledge and Data: A Unified Framework with Finance Applications

Authors: Yi Cao, Zexun Chen, Lin William Cong, Heqing Shi

Organizations: Department of Financial and Actuarial Mathematics, Xi’an Jiaotong-Liverpool University, Suzhou, Jiangsu 215123, PR China. · Management Science and Business Economics Group, Business School, University of Edinburgh, 29 Buccleuch Place, Edinburgh EH8 9JS, UK. · Nanyang Technological University, ABFER, CEPR, & NBER, 91 Nanyang Avenue, Wee Cho Yaw Plaza, Singapore 639956.

Abstract

We develop Structured-Knowledge-Informed Neural Networks (SKINNs), a unified estimation framework that embeds theoretical, simulated, previously learned, or cross-domain insights as differentiable constraints within flexible neural function approximation. SKINNs jointly estimate neural network parameters and economically meaningful structural parameters in a single optimization problem, enforcing theoretical consistency not only on observed data but over a broader input domain through collocation, and therefore nesting approaches such as functional GMM, Bayesian updating, transfer learning, PINNs, and surrogate modeling. SKINNs define a class of M-estimators that are consistent and asymptotically normal with root-N convergence, sandwich covariance, and recovery of pseudo-true parameters under misspecification. We establish identification of structural parameters under joint flexibility, derive generalization and target-risk bounds under distributional shift in a convex proxy, and provide a restricted-optimal characterization of the weighting parameter that governs the bias-variance tradeoff. In an illustrative financial application to option pricing, SKINNs improve out-of-sample valuation and hedging performance, particularly at longer horizons and during high-volatility regimes, while recovering economically interpretable structural parameters with improved stability relative to conventional calibration. More broadly, SKINNs provide a general econometric framework for combining model-based reasoning with high-dimensional, data-driven estimation.

Figures & tables

Panel of Figure 1 — Figure 1: PSKR by a (semi-)parametric equation. $\mathbf{x}^{\text{SK}}_{1}\cdots\mathbf{x}^{\text{SK}}_{d_{\text{SK}}}$ denote each of the observable features, and $\mathbf{\phi}_{1}\cdots\mathbf{\phi}_{d_{\mathbf{\phi}}}$ denote each of the learnable latent parameters, in the PSKR.

Panel of Figure 2 — Figure 2: SPSKR by an expensive-to-solve target system. The inputs, i.e., the observable features $\mathbf{X}^{\text{SK}}$ and the latent parameters $\mathbf{\phi}$ , are still determined by theories, as in PSKRs. But the structured-knowledge function $g_{\mathbf{\phi}}(\mathbf{X}^{\text{SK}})$ , in this case, is approximated by a pre-trained DSNN.

Panel of Figure 3 — Figure 3: NPSKR by an unknown distribution. The inputs include a high-dimensional vector of probabilities, which serves as the learnable latent parameters in SKINNs. The probabilities are passed to a softmax function to ensure non-negativity and the total probability of one unit.

Panel of Figure 4 — Figure 4: NPSKR by an AE. The inputs include a vector of realizations from an unknown DGP, with size $n$ . The encoder compresses the inputs into a vector of latent parameters of size $d$ ( $d\ll n$ ). The $d$ -dimensional AE latent parameters are then passed to a decoder, which returns an $n$ -dimensional vector that reconstructs the inputs.

Panel of Figure 5 — Figure 5: The architecture of SKINNs. The NN on the top (blue circles) is the base function approximator. The mappings at the bottom (red circles) are structured-knowledge representations of different formats. All of them take observable features $\mathbf{X}^{\text{SK}}_{\text{Colloc}}$ ; the current epoch value of the learnable latent parameters as inputs, and output $g_{\phi}$ .

Panel of Figure 6 — Figure 6: The probability density of the loss gradients with respect to the NN weights and biases, for each loss component in a PINN (with inverse problem approach), including a boundary loss. The PINN is trained, for 5,000 iterations, using a cross-section of simulated European call option prices with noise, according to the BSM model.

Panel of Figure 7 — Figure 7: The probability density of the loss gradients with respect to the NN weights and biases, for each loss component in a SKINN, including a boundary loss. The SKINN is trained, for 5,000 iterations, using a cross-section of simulated European call option prices with noise, with the PSKR of the BSM model.

Model	Structured Knowledge 9 9 9 “No” in the column structured knowledge means that there is no prior knowledge; “Bnd” means that the boundary conditions are incorporated; “Shape” means that the model-free shape constraints are incorporated; and “AE-BSM” means the machine-learned AE-based NPSKR structured-knowledge function from the BSM model with noise; “AE-MIX” means such a structured-knowledge function from the mixture of models with noise.	$\text{dim}(\phi)$ (learnable/total)	Activation function
Benchmark NNs
NN	No	0/0	ReLU
NN+Bnd	Bnd	0/0	ReLU
NN+Shape	Shape	0/0	SiLU
PINN+BSM	BSM	1/1	SiLU
TLNN+HSV	HSV	0/5	ReLU

Table 1: The full list of the NN models we consider in this paper. Only the PINNs, the TLNNs, and the SKINNs introduce the structural model as prior knowledge from option pricing to the NN. Others introduce either the boundary constraints or the model-free shape constraints to the NNs.

Panel of Figure 8 — Figure 8: The forward rolling model training schedule. The first training period covers the daily option panels from 04 Jan 1996 to 29 Mar 1996. The corresponding shorter test horizon (Test 1) covers options from 01 Apr 1996 to 30 Apr 1996, and the longer test horizon (Test 2) covers options from 01 May 1996 to 31 May 1996. For each of the periods from period 2 to period 317, all the Train, Test 1, and Test 2 are rolled one month forward based on the preceding period.

Model	Structural models				Benchmark NNs
Panel (A)	BSM	ABSM	HSV	HSVJ	NN	NN	NN	PINN
	BSM	ABSM	HSV	HSVJ	NN	+Bnd	+Shape	+BSM
BSM	–	1.99**	-2.16**	-9.24***	-1.72**	-2.42***	0.16	0.21
ABSM	–	–	-2.89***	-8.91***	-1.88**	-2.58***	-0.07	-0.02
HSV	–	–	–	-12.95***	-1.49*	-2.22**	0.59	0.64
HSVJ	–	–	–	–	-0.61	-1.35*	1.81**	1.85**

Table 2: This table reports pairwise Diebold-Mariano test statistics comparing the out-of-sample option pricing performance, measured in terms of RMSE, of different models. We compare 8 variants of SKINNs with different specifications of structured-knowledge representations, with 4 classical structural models, and 4 benchmark neural networks. The out-of-sample option pricing performance is evaluated in the 317 shorter prediction horizons. We square all the pricing errors to penalize large errors.

Model	Structural models				Benchmark NNs
Panel (A)	BSM	ABSM	HSV	HSVJ	NN	NN	NN	PINN
	BSM	ABSM	HSV	HSVJ	NN	+Bnd	+Shape	+BSM
BSM	–	2.49***	0.62	-5.23***	2.67***	1.36*	2.51***	2.54***
ABSM	–	–	-0.57	-5.83***	2.56***	1.26	2.40***	2.42***
HSV	–	–	–	-12.95***	2.63***	1.33*	2.45***	2.47***
HSVJ	–	–	–	–	3.09***	1.74**	2.94***	2.97***

Table 3: This table reports pairwise Diebold-Mariano test statistics comparing the out-of-sample option pricing performance, measured in terms of RMSE, of different models. We compare 8 variants of SKINNs with different specifications of structured-knowledge representations, with 4 classical structural models, and 4 benchmark neural networks. The out-of-sample option pricing performance is evaluated in the 317 longer prediction horizons. We square all the pricing errors to penalize large errors.

Model	Structural models				Benchmark NNs
Panel (A)	BSM	ABSM	HSV	HSVJ	NN	NN	NN	PINN
	BSM	ABSM	HSV	HSVJ	NN	+Bnd	+Shape	+BSM
BSM	–	-3.62***	6.63***	6.56***	2.64***	2.82***	2.99***	2.48***
ABSM	–	–	6.92***	6.83***	2.90***	3.19***	3.17***	2.66***
HSV	–	–	–	-1.31*	-1.65**	-2.59***	-0.12	-0.31
HSVJ	–	–	–	–	-1.55*	-2.49***	0.01	-0.21

Table 4: This table reports pairwise Diebold-Mariano test statistics comparing the out-of-sample option hedging performance, measured in terms of MHE, of different models. We compare 8 variants of SKINNs with different specifications of structured-knowledge representations, with 4 classical structural models, and 4 benchmark neural networks. The out-of-sample option hedging performance is evaluated in the 317 shorter prediction horizons. We square all the hedging errors to penalize large errors.

Model	Structural models				Benchmark NNs
Panel (A)	BSM	ABSM	HSV	HSVJ	NN	NN	NN	PINN
	BSM	ABSM	HSV	HSVJ	NN	+Bnd	+Shape	+BSM
BSM	–	-4.30***	7.69***	7.67***	3.44***	3.48***	3.52***	3.20***
ABSM	–	–	7.83***	7.81***	3.68***	3.83***	3.72***	3.43***
HSV	–	–	–	-1.42*	0.25	-1.15	1.11	0.20
HSVJ	–	–	–	–	0.39	-0.99	1.21	0.32

Table 5: This table reports pairwise Diebold-Mariano test statistics comparing the out-of-sample option hedging performance, measured in terms of MHE, of different models. We compare 8 variants of SKINNs with different specifications of structured-knowledge representations, with 4 classical structural models, and 4 benchmark neural networks. The out-of-sample option hedging performance is evaluated in the 317 longer prediction horizons. We square all the hedging errors to penalize large errors.

Model	Frozen/Trainable layers	Learning rate	Epochs	Inputs	$\phi$
DSNN+HSV	6/0	2E-4	500	$\mathbf{X},\phi$	Simulated
TLNN+HSV1	0/6	6E-6	6	$\mathbf{X},\phi$	SKINN-learned
TLNN+HSV2	4/2	6E-6	6	$\mathbf{X},\phi$	SKINN-learned
TLNN+HSV3	4/2	6E-6	100	$\mathbf{X},\phi$	SKINN-learned
TLNN+HSV4	4/2	6E-6	500	$\mathbf{X},\phi$	SKINN-learned
SKINN+HSV	0/3	1E-3	500	$\mathbf{X}$	–

Table 6: This table summarizes the network configurations applied to train TLNNs and SKINN+HSV. We employ a smaller network size in order to allow TLNNs (6 hidden layers, 64 neurons each) to be comparable to our SKINNs (3 hidden layers, 32 neurons each). The different number of frozen layers for TLNNs corresponds to the different compliance with the theory.

Model	DSNN	TLNN	TLNN	TLNN	TLNN	SKINN
Model	+HSV	+HSV1	+HSV2	+HSV3	+HSV4	+HSV
DSNN+HSV	–	-10.48***	-10.88***	-10.48***	-11.39***	-13.87****
TLNN+HSV1	–	–	9.12***	-10.37***	-13.71***	-11.30***
TLNN+HSV2	–	–	–	-10.59***	-13.70***	-11.60***
TLNN+HSV3	–	–	–	–	-22.46***	-6.63***
TLNN+HSV4	–	–	–	–	–	-2.80***

Table 7: The pair-wise Diebold-Mariano test statistics, which compare the out-of-sample option pricing errors (measured by RMSE) across all TLNNs and SKINN+HSV. The out-of-sample option pricing errors are evaluated using the options within the 317 periods of longer prediction horizons. A negative number indicates the column model outperforms the row model, and vice versa.

Figure 9: The out-of-sample option pricing performance of TLNNs and SKINN+HSV, over the 317 periods of longer prediction horizons. Among all the TLNNs, TLNN+HSV4 performs the best.

Model	BSM	ABSM	HSV	HSVJ	NN	–	–	–
Panel (A)	BSM	ABSM	HSV	HSVJ	+Bnd
$\beta_{1}$	0.0843 ∗∗∗	0.0891 ∗∗∗	0.0563 ∗∗∗	0.0720 ∗∗∗	-0.0116 ∗	–	–	–
	(0.0141)	(0.0140)	(0.0146)	(0.0143)	(0.0069)	–	–	–
$\beta_{0}$	-0.0134 ∗∗∗	-0.0141 ∗∗∗	-0.0082 ∗∗	-0.0151 ∗∗∗	0.0005	–	–	–
	(0.0031)	(0.0031)	(0.0032)	(0.0032)	(0.0015)	–	–	–
Obs	317	317	317	317	317	–	–	–

Table 8: We run the regression ( 80 ) for each model. A statistically significant negative

\beta_{1}

implies that the model performs better compared with a plain vanilla neural network when the market is more volatile. The results are based on all 317 test periods of the shorter prediction horizons.

Model	BSM	ABSM	HSV	HSVJ	NN	–	–	–
Panel (A)	BSM	ABSM	HSV	HSVJ	+Bnd
$\beta_{1}$	-0.0317	-0.0274	-0.0483 ∗∗	-0.0340	-0.0303 ∗∗∗	–	–	–
	(0.0214)	(0.0214)	(0.0215)	(0.0216)	(0.0096)	–	–	–
$\beta_{0}$	0.0019	0.0013	0.0053	-0.0008	0.0027	–	–	–
	(0.0047)	(0.0047)	(0.0047)	(0.0048)	(0.0021)	–	–	–
Obs	317	317	317	317	317	–	–	–

Table 9: We run the regression ( 80 ) for each model. A statistically significant negative

\beta_{1}

implies that the model performs better compared with a plain vanilla neural network when the market is more volatile. The results are based on all 317 test periods of the longer prediction horizons.

Model	BSM	ABSM	HSV	HSVJ	NN	–	–	–
Panel (A)	BSM	ABSM	HSV	HSVJ	+Bnd
$\beta_{1}$	-0.1979	-0.1172	-0.1354	-0.1288	-0.0346	–	–	–
	(0.1657)	(0.1644)	(0.1646)	(0.1671)	(0.1192)	–	–	–
$\beta_{0}$	0.0230	0.0125	0.0167	0.0119	0.0024	–	–	–
	(0.0205)	(0.0204)	(0.0204)	(0.0207)	(0.0148)	–	–	–
Obs	64	64	64	64	64	–	–	–

Table 10: We run the regression ( 80 ) for each model. A statistically significant negative

\beta_{1}

implies that the model performs better compared with a plain vanilla neural network when the market is more volatile. The results are based on the 64 low volatility testing periods in the longer prediction horizons, where

\text{AvgVIX}(t)

is below the 20% quantile of AvgVIX across all the longer horizon testing periods.

Model	BSM	ABSM	HSV	HSVJ	NN	–	–	–
Panel (A)	BSM	ABSM	HSV	HSVJ	+Bnd
$\beta_{1}$	-0.0708	-0.0630	-0.1299 ∗∗	-0.0974 ∗	-0.0686 ∗∗	–	–	–
	(0.0525)	(0.0525)	(0.0524)	(0.0532)	(0.0275)	–	–	–
$\beta_{0}$	0.0086	0.0075	0.0202 ∗∗	0.0104	0.0098 ∗	–	–	–
	(0.0102)	(0.0103)	(0.0102)	(0.0104)	(0.0054)	–	–	–
Obs	189	189	189	189	189	–	–	–

Table 11: We run the regression ( 80 ) for each model. A statistically significant negative

\beta_{1}

implies that the model performs better compared with a plain vanilla neural network when the market is more volatile. The results are based on the 189 medium volatility testing periods in the longer prediction horizons, where

\text{AvgVIX}(t)

is between the 20% quantile and the 80% quantile of AvgVIX across all the longer horizon testing periods.

Model	BSM	ABSM	HSV	HSVJ	NN	–	–	–
Panel (A)	BSM	ABSM	HSV	HSVJ	+Bnd
$\beta_{1}$	-0.0364	-0.0416	-0.0345	-0.0272	-0.0466	–	–	–
	(0.0783)	(0.0787)	(0.0788)	(0.0791)	(0.0287)	–	–	–
$\beta_{0}$	0.0051	0.0077	0.0025	-0.0013	0.0094	–	–	–
	(0.0262)	(0.0264)	(0.0264)	(0.0265)	(0.0096)	–	–	–
Obs	64	64	64	64	64	–	–	–

Table 12: We run the regression ( 80 ) for each model. A statistically significant negative

\beta_{1}

implies that the model performs better compared with a plain vanilla neural network when the market is more volatile. The results are based on the 64 high volatility testing periods in the longer prediction horizons, where

\text{AvgVIX}(t)

is above the 80% quantile of AvgVIX across all the longer horizon testing periods.

BSM	$\sigma$	–	–	–	–	–
$\beta_{1}$	1.1181***	–	–	–	–	–
	(18.781)	–	–	–	–	–
$\beta_{0}$	0.0255**	–	–	–	–	–
	(2.543)	–	–	–	–	–
Obs	317	–	–	–	–	–
Adj. $R^{2}$	52.7%	–	–	–	–	–

Table 13: The predictive information contained in each of the parameters calibrated according to the standard procedure, for the lagged averaged AvgVIX over one month.

\{\sigma\}

is from the BSM model,

\{a_{1},a_{2},a_{3},a_{4},a_{5},a_{6}\}

is from the ABSM model,

\{\bar{v},v_{0},\sigma_{v},\rho,\kappa\}

is from the HSV model, and

\{\bar{v},v_{0},\sigma_{v},\rho,\kappa,\lambda,\eta_{0},\eta_{1},p_{up}\}

is from the HSVJ model.

BSM	$\sigma^{*}$	–	–	–	–	–
$\beta_{1}$	1.0896***	–	–	–	–	–
	(17.329)	–	–	–	–	–
$\beta_{0}$	0.0323***	–	–	–	–	–
	(3.087)	–	–	–	–	–
Obs	317	–	–	–	–	–
Adj. $R^{2}$	48.6%	–	–	–	–	–

Table 14: The predictive information contained in each of the parameters learned from the SKINNs, for the lagged averaged AvgVIX over one month.

\{\sigma^{*}\}

is from

g_{\phi}^{\text{BSM}}

\{a_{1}^{*},a_{2}^{*},a_{3}^{*},a_{4}^{*},a_{5}^{*},a_{6}^{*}\}

is from

g_{\phi}^{\text{ABSM}}

\{\bar{v}^{*},v_{0}^{*},\sigma_{v}^{*},\rho^{*},\kappa^{*}\}

is from

g_{\phi}^{\text{HSV}}

, and

\{\bar{v}^{*},v_{0}^{*},\sigma_{v}^{*},\rho^{*},\kappa^{*},\lambda^{*},\eta_{0}^{*},\eta_{1}^{*},p_{up}^{*}\}

is from

g_{\phi}^{\text{HSVJ}}

Figure 10: The risk-neutral variances over 10 different tenors. Each risk-neutral variance $\mathbb{V}\text{ar}[\phi^{(T_{k})}_{t}]$ is translated from the learned risk-neutral probabilities by SKINN+MOPA for the $k$ -th tenor, $1\leq k\leq 10$ . The upper panel illustrates the learned risk-neutral variances, and the lower panel illustrates the AvgVIX over the subsequent months after the SKINN+MOPA training periods.

MOPA	$\mathbb{V}\text{ar}[\phi^{(T_{1})}]$	$\mathbb{V}\text{ar}[\phi^{(T_{2})}]$	$\mathbb{V}\text{ar}[\phi^{(T_{3})}]$	$\mathbb{V}\text{ar}[\phi^{(T_{4})}]$	$\mathbb{V}\text{ar}[\phi^{(T_{5})}]$
$\beta_{1}$	26.2949 ∗∗∗	16.3890 ∗∗∗	10.2331 ∗∗∗	6.6412 ∗∗∗	5.9800 ∗∗∗
	(1.6483)	(0.9353)	(0.5712)	(0.4056)	(0.3548)
$\beta_{0}$	0.1292 ∗∗∗	0.1217 ∗∗∗	0.1098 ∗∗∗	0.1041 ∗∗∗	0.0884 ∗∗∗
	(0.0058)	(0.0057)	(0.0062)	(0.0070)	(0.0076)
Obs	317	317	317	317	317
Adj. R 2	44.5%	49.2%	50.3%	45.8%	47.3%

Table 15: The predictive information contained in the risk-neural variances converted from the latent parameters (i.e., the 2,000 risk-neutral probabilities over 10 tenors) learned from SKINN+MOPA, for the lagged averaged VIX over one month.

\mathbb{V}\text{ar}[\phi^{(T_{k})}_{t}]

is the risk-neutral variance converted according to the learned risk-neutral probabilities for the

k

-th tenor.

Panel of Figure 11 — Figure 11: The latent structural parameters learned by SKINNs with PSKRs.

Panel of Figure 12 — Figure 12: The latent structural parameters (risk-neutral probabilities) learned by SKINN+MOPA, which uses an NPSKR with an unknown distribution. We select two training periods to demonstrate the learned probabilities, from the 2007-2009 global financial crisis and the COVID-19 crisis, respectively.

Panel of Figure 13 — Figure 13: The histograms of the variation of the SKINN-learned (orange) and the calibrated (blue) latent structural parameters, between two consecutive training periods over time. The SKINN-learned latent structural parameters exhibit a higher density around zero, indicating higher numerical stability compared to those from the conventional calibration.

Panel of Figure 14 — Figure 14: The latent structural parameters learned from the SKINNs with two SPSKRs and one NPSKR, all of which employ an NN to proxy the structured-knowledge function $g_{\phi}$ . Specifically, the SKINNs are SKINN+DSNN-HSV, SKINN+DSNN-HSVJ, and SKINN+AE-BSM.

DSNN-HSV	$\bar{v}$	$v_{0}$	$\sigma_{v}$	$\rho$	$\kappa$	–
$\beta_{1}$	-7.8261 ∗∗	167.5138 ∗∗∗	-12.7938 ∗∗∗	1.9625 ∗	6.7324 ∗∗∗	–
	(3.6186)	(10.7461)	(3.1635)	(1.0485)	(2.5814)
$\beta_{0}$	21.1113 ∗∗∗	16.3179 ∗∗∗	23.6611 ∗∗∗	21.8788 ∗∗∗	18.8241 ∗∗∗	–
	(0.5378)	(0.4312)	(0.9039)	(0.8771)	(0.7726)	–
Obs	317	317	317	317	317	–
Adj. R 2	1.15%	43.37%	4.63%	0.79%	1.80%	–

Table 16: The predictive information contained in the latent parameters learned from SKINN+DSNN-HSV, SKINN+DSNN-NASV, and SKINN+AE-BSM, for the lagged averaged AvgVIX over one month.

\{\bar{v},v_{0},\sigma_{v},\rho,\kappa\}

is from

g_{\phi}^{\text{DSNN-HSV}}

\{\kappa_{v},\theta_{v},\sigma_{v},\gamma_{v},\rho_{sv},v_{0}\}

is from

g_{\phi}^{\text{DSNN-NASV}}

, and

\{\phi_{\text{AE}}^{(1)},\phi_{\text{AE}}^{(2)}\}

is from

g_{\phi}^{\text{AE-BSM}}

Model		NN			SKINN ( $\eta=0$ )			SKINN ( $\eta=10$ )
	Group	Avg %	SD %	SR	Avg %	SD %	SR	Avg %	SD %	SR
1-32	H	60.73	130.07	0.47	68.22	109.78	0.62	70.83	102.91	0.69
1-32	L	59.29	136.54	0.43	54.78	133.41	0.41	57.77	134.48	0.43
2-32	H	57.01	127.73	0.45	71.86	119.78	0.60	65.32	99.71	0.66
2-32	L	72.12	145.66	0.50	70.98	141.31	0.50	57.19	140.62	0.41
3-32	H	56.57	127.98	0.44	68.94	122.60	0.56	59.45	100.10	0.59

Table 17: The comparison of the SKINN asset pricing models and the plain-vanilla NN asset pricing models. All portfolio performance metrics are evaluated with the out-of-sample stock data in the testing periods. We train asset pricing models using 5 different architectures (1-5 hidden layers with 32 neurons). The results are averaged across 20 different random initializations of an NN. Group H denotes the top-decile portfolio, and Group L denotes the bottom-decile.

Appendix figures & tables7 assets

Supplementary material from the paper’s appendix.

Appendix

Model	Structural models				Benchmark NNs
Panel (A)	BSM	ABSM	HSV	HSVJ	NN	NN	NN	PINN
	BSM	ABSM	HSV	HSVJ	NN	+Bnd	+Shape	+BSM
BSM	–	22.48**	23.43	5.68***	17.71***	14.13***	25.18	25.34
ABSM	–	–	22.58*	5.20***	17.15***	13.82***	24.53	25.12
HSV	–	–	–	2.94***	19.13***	16.19***	25.89	26.53
HSVJ	–	–	–	–	25.74	23.54	35.30***	35.12***

Appendix

Table 18: The Wilcoxon (1945) test results for the out-of-sample option pricing accuracy comparisons in shorter prediction horizons. We square all the pricing errors to penalize large errors.

Model	Structural models				Benchmark NNs
Panel (A)	BSM	ABSM	HSV	HSVJ	NN	NN	NN	PINN
	BSM	ABSM	HSV	HSVJ	NN	+Bnd	+Shape	+BSM
BSM	–	25.23	25.75	9.68***	27.71*	22.11**	27.52*	28.20**
ABSM	–	–	24.24	8.48***	27.44*	21.81**	27.11	27.78*
HSV	–	–	–	3.79***	27.75*	22.40**	27.20	28.15**
HSVJ	–	–	–	–	32.41***	27.46*	33.16***	34.08***

Appendix

Table 19: The Wilcoxon (1945) test results for the out-of-sample option pricing accuracy comparisons in longer prediction horizons. We square all the pricing errors to penalize large errors.

Model	Structural models				Benchmark NNs
Panel (A)	BSM	ABSM	HSV	HSVJ	NN	NN	NN	PINN
	BSM	ABSM	HSV	HSVJ	NN	+Bnd	+Shape	+BSM
BSM	–	5.16***	46.50***	47.13***	22.27**	21.40***	25.08	24.21
ABSM	–	–	46.79***	47.31***	23.79	23.39	27.21	25.96
HSV	–	–	–	20.19**	9.80***	7.81***	12.32***	10.56***
HSVJ	–	–	–	–	9.59***	8.12***	12.60***	10.81***

Appendix

Table 20: The Wilcoxon (1945) test results for the out-of-sample option hedging accuracy comparisons in shorter prediction horizons. We square all the hedging errors to penalize large errors.

Model	Structural models				Benchmark NNs
Panel (A)	BSM	ABSM	HSV	HSVJ	NN	NN	NN	PINN
	BSM	ABSM	HSV	HSVJ	NN	+Bnd	+Shape	+BSM
BSM	–	6.19***	46.34***	46.83***	22.52*	21.78**	24.95	24.75
ABSM	–	–	46.59***	47.12***	23.76	23.64	27.32*	26.56
HSV	–	–	–	20.55**	12.36***	10.91***	14.20***	12.84***
HSVJ	–	–	–	–	12.51***	11.17***	14.49***	13.24***

Appendix

Table 21: The Wilcoxon (1945) test results for the out-of-sample option hedging accuracy comparisons in longer prediction horizons. We square all the hedging errors to penalize large errors.

	Shorter Prediction Horizons		Longer Prediction Horizons
Model	SKINN+SABR	SKINN+AE-MIX	SKINN+SABR	SKINN+AE-MIX
BSM	-1.22	-2.74***	-1.59*	0.51
ABSM	-1.36*	-2.93***	-1.81**	0.37
HSV	-0.96	-2.58***	-1.76*	0.45
HSVJ	-0.17	-1.51*	-0.81	1.03
NN	0.30	-1.65**	-3.68***	-3.23***

Appendix

Table 22: The Diebold-Mariano test of the out-of-sample option pricing performance, measured in terms of RMSE, of the two SKINNs. We square all the pricing errors to penalize large errors.

	All		Low Volatility
Model	SKINN+SABR	SKINN+AE-MIX	SKINN+SABR	SKINN+AE-MIX
AvgVIX	-0.0835***	-0.0821***	-0.2368	-0.0765
	(0.019)	(0.012)	(0.161)	(0.118)
Constant	0.0096**	0.0139***	0.0259	0.0097
	(0.004)	(0.003)	(0.020)	(0.015)
Obs	317	317	64	64

Appendix

Table 23: The marginal outperformance of SKINN against the plain-vanilla NN given a one-unit increase in AvgVIX. We consider different volatility regimes from the 317 longer prediction horizons.

Paradigm	Configuration	Reference
Plain neural network	$\lambda=0$	Gu et al. (2020)
PINNs	$g_{\phi}=\mathbb{D}[f_{\theta}]$ , $\lambda\to\infty$	Raissi et al. (2019)
Transfer learning	$\phi$ fixed, two-stage training	Pan and Yang (2010)
Bayesian MAP estimation	$\mathcal{L}_{\text{SK}}$ as neg. log-prior	Neal (1996)
Regularized / functional GMM	$\mathcal{L}_{\text{data}},\mathcal{L}_{\text{SK}}$ as moment conditions	Hansen (1982)
Sieve GMM / Deep GMM	$f_{\theta}$ as neural sieve, optimal $\mathbf{W}$	Ai and Chen (2003)

Appendix

Table 24: SKINNs as a unifying framework: limiting configurations of the composite objective.

Parameter Estimation

Parameter estimation studies how to infer unknown quantities in statistical, physical, and machine-learning models from observed data.

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Beyond Time Induction: Structured Conditional Neural Solvers for Parametric Stochastic-Volatility Pricing

Jose Miguel Lara Rangel, Mixtli Vazquez Mendoza

University College London, UK · Aicraft, Mexico

Neural PDE Solvers Physics-Informed ML Quantitative Finance

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Physics-informed neural networks provide a flexible framework for learning parametric PDE solution families without labeled data, yet preserving accurate derivatives across heterogeneous regimes remains challenging. Structured neural solvers offer compact representations, but restrictive conditional parameterizations can limit adaptation across the solution family. We introduce BeTINN, a structured conditional architecture extending time-induced modulation through multi-directional time-parameter conditioning and a joint residual pathway. We evaluate BeTINN on the Heston stochastic-volatility PDE, a demanding parametric problem with mixed derivatives and economically relevant first- and second-order sensitivities. BeTINN improves joint accuracy across normalized option value and key sensitivity metrics relative to our baselines. Controlled ablations characterize the roles of its architectural components. We further find that validation physics loss is informative but does not fully determine differential fidelity, and that accurate solution values can coexist with localized derivative errors. These results support structured conditional adaptation and motivate derivative-aware evaluation when sensitivities are downstream quantities of interest.

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Parameter Estimation

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