Uncertainty and Explainability in Deep Rough Volatility: A Neural Information-Theoretic Posterior Approach

Damiano Brigo; Raphaël Huser; Dan Leonte

stat.MLSep 25, 2026

Uncertainty and Explainability in Deep Rough Volatility: A Neural Information-Theoretic Posterior Approach

Authors: Damiano Brigo, Raphaël Huser, Dan Leonte

Abstract

Deep learning has substantially accelerated the calibration of complex stochastic-volatility models, but neural point calibration alone does not capture the uncertainty remaining after an implied-volatility (IV) surface has been observed. We develop a simulation-based inference framework for rough Heston (rHeston) calibration that learns the posterior distribution of the model parameters conditional on an IV surface. Using neural ratio estimation, we obtain calibrated posterior samples that can be propagated through heteroscedastic neural surrogate pricers for path-dependent exotic options. The resulting posterior-predictive distributions combine residual parameter uncertainty with conditional surrogate uncertainty and yield uncertainty-aware price intervals. We further introduce Hellinger-SHAP, an information-theoretic explainability method for posterior inference. Rather than attributing a single parameter point estimate, it applies local-background Kernel SHAP to a posterior-information functional measuring contraction from the prior to the posterior. This identifies maturity--moneyness regions associated with posterior information gain for individual rHeston parameters. In a simulation study, posterior-predictive intervals provide calibrated or conservative coverage across forward-start, barrier, and realized-variance claims, while point plug-in prices can be materially unreliable for selected contract regimes. Together, the UQ and XAI analyses provide a transparent framework for uncertainty-aware neural calibration and downstream exotic pricing under the specified prior-predictive model.

Figures & tables

Parameter	Symbol	Support Range
Mean-reversion Speed	$\kappa$	$(0.10,2.00)$
Long-run Variance	$\theta$	$(0.01,0.25)$
Vol-of-Vol	$\nu$	$(0.05,0.45)$
Leverage correlation	$\rho$	$(-0.95,-0.1)$
Initial Variance	$V_{0}$	$(0.01,0.25)$
Hurst Exponent	$H$	$(0.05,0.45)$

Table 1. Parameter supports for the rough Heston model synthetic dataset.

	$N=504$				$N=1008$				$N=2016$				$N=4032$
$r$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$
$1$	0.03	0.23	0.38	0.80	0.02	0.15	0.23	0.48	0.02	0.10	0.15	0.29	0.02	0.08	0.12	0.21
$2$	0.02	0.10	0.14	0.24	0.02	0.08	0.11	0.19	0.02	0.07	0.10	0.15	0.01	0.07	0.09	0.15
$3$	0.02	0.10	0.13	0.22	0.02	0.08	0.11	0.17	0.02	0.07	0.09	0.15	0.01	0.07	0.10	0.17

Table 2. Quantiles of the absolute difference between the HQE and Lewis implied-volatilities

\Delta_{\mathrm{IV}}^{\mathrm{HQE}}(T,z;\boldsymbol{\theta})

defined in ( 5.1 ) over all pairs

(T,z)

, for different time-grid resolutions

N\in\{504,1008,2016,4032\}

and grading rates

r\in\{1,2,3\}

. We use MC estimators with

M=5\cdot 10^{5}

samples; quantiles are estimated based on

500

pairs

(\boldsymbol{\theta},\boldsymbol{x})

. Non-uniform grids perform better, and we settle on

N=2016

and

r=2

for the dataset generation.

Figure 1. Posterior distribution of the parameter vector

\boldsymbol{\theta}

conditional on an IV surface

\boldsymbol{x}

from the held-out test set. The diagonal panels display the estimated marginal posterior density of each parameter, while the off-diagonal panels show the corresponding pairwise marginal posterior distributions. For each parameter, the ground truth value is shown together with three point summaries of the posterior: the maximum a posteriori (MAP) estimate, the sample posterior mean and the sample posterior median. Parameter recovery is good, while the shape of the posterior distributions reflect the remaining parameter uncertainty and correlation. These plots are generated from

15000

posterior draws.

Figure 2. Empirical highest posterior density (HPD) regions coverage of the learnt TRE conditional posteriors over the held-out test set. We note good coverage results, i.e., the empirical coverage of the HPD regions matches the nominal values almost exactly. This confirms the learnt approximations

\hat{p}(\kappa\mid\boldsymbol{x}),\ldots,\hat{p}(H\mid\boldsymbol{x},\kappa,\theta,\nu,\rho,V_{0})

are neither over- nor under-confident. See Appendix E.2.3 , in particular the second half of this section, for further details.

Panel of Figure 3 — Figure 3. Neural point estimator trained with the MSE loss evaluated over the test set. Each subfigure illustrates the true versus predicted parameter values, with the $y=x$ line representing perfect parameter recovery.

Panel of Figure 5 — Figure 5. Empirical distributions of relative Monte Carlo standard errors for the three exotic contract families. The relative standard error is defined as the estimated Monte Carlo standard error divided by the corresponding estimated option price. For realized-variance options, contracts with estimated prices below $10^{-5}$ are excluded from this relative diagnostic to avoid instability from near-degenerate prices that are not informative. The results are pooled over all retained contracts within each family.

Panel of Figure 6 — Figure 6. Frequency ( $\%$ ) of test samples for which the relative plug-in pricing error exceeds $3\%$ , by exotic-contract specification. The plug-in price evaluates the exotic surrogate at the MSE- or MAE-trained neural point estimator, and the error is measured relative to the held-out MC reference price. Within each family, the left and right panels show rates for the MSE- and MAE-trained estimators, respectively. Grey cells denote excluded near-degenerate realized-variance contracts. Results for the $1\%$ and $5\%$ thresholds are reported in Appendix C .

	Error > 1%		Error > 3%		Error > 5%
Family	MSE	MAE	MSE	MAE	MSE	MAE
Forward-start	6.46	4.86	1.08	0.85	0.46	0.40
Barrier	17.25	14.87	1.03	0.85	0.15	0.11
RV	22.05	18.38	6.83	6.05	4.21	3.81

Table 3. Percentage of held-out test pairs for which the relative plug-in pricing error exceeds the stated threshold, pooled over the retained contract within each exotic family. The plug-in price is obtained by evaluating the exotic surrogate at the MSE- or MAE-trained neural point calibration, and the error is measured relative to the held-out Monte Carlo reference price. Contract-level patterns are reported in Appendix C .

Family	Median (%)	$90\%$ quantile (%)	$95\%$ quantile (%)
Forward-start	0.65	2.46	3.84
Barrier	2.20	4.56	5.82
Realized variance	1.66	8.17	18.18

Table 4. Distribution of the relative width

W_{ij}^{(0.95)}

of the full nominal-

95\%

predictive interval, pooled over all retained contract–test pairs within each exotic family. For each pair,

W_{ij}^{(0.95)}

is defined in ( 5.3 ) as the relative interval length expressed as a percentage of the held-out Monte Carlo reference price. Entries are the median,

90\%

quantile, and

95\%

quantile of the resulting distribution of relative widths. Contract-level results are reported in Tables 14 – 16 .

Family	PO $95\%$ median	PO $95\%$ IQR	Full UQ $95\%$ median	Full UQ $95\%$ IQR
Forward-start	94.60	1.58	97.19	0.33
Barrier	78.83	15.59	95.75	0.90
Realized variance	94.24	1.66	96.89	0.43

Table 5. Median and interquartile range (IQR) of empirical coverage rates for nominal-

95\%

predictive intervals across retained contract specifications within each exotic family. “Posterior only” (PO) propagates draws from the post-training-calibrated TRE posterior through the conditional surrogate mean. “Full UQ” additionally samples from the conditional surrogate-predictive law and therefore incorporates both posterior and conditional surrogate uncertainty. All entries are percentages; IQRs are reported in percentage points. Contract-level results are reported in Tables 17 – 19 .

Family	TRE median	TRE IQR	Surrogate median	Surrogate IQR
Forward-start	81.67	8.01	18.33	8.01
Barrier	43.07	18.40	56.93	18.40
Realized variance	89.69	3.66	10.31	3.66

Table 6. Median and interquartile range (IQR), across retained contract specifications, of the case-wise shares of full predictive variance attributed to TRE posterior uncertainty and conditional surrogate uncertainty. For each held-out test pair, the shares are computed from the law of total variance in ( 5.5 ) and then averaged within its contract specification. All entries are percentages; IQRs are reported in percentage points. Contract-level results are reported in Appendix C .

Panel of Figure 7 — Figure 7. Mean absolute squared-Hellinger Kernel-SHAP attributions for the six rough-Heston parameters, averaged over $500$ held-out IV surfaces from the test set. Each panel displays the average magnitude of the local attribution at a maturity–standardized-moneyness location; it does not display a signed effect.

Appendix figures & tables32 assets

Supplementary material from the paper’s appendix.

Appendix

		$N=504$				$N=1008$				$N=2016$				$N=4032$
Simulation scheme	$r$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$
Hybrid ( $K_{h}=1$ )	1	0.06	0.64	1.14	2.45	0.04	0.46	0.97	2.11	0.03	0.34	0.84	1.80	0.02	0.28	0.72	1.54
	2	0.03	0.40	0.98	1.84	0.02	0.32	0.85	1.59	0.02	0.27	0.73	1.39	0.02	0.22	0.62	1.21
	3	0.03	0.39	0.96	1.78	0.02	0.32	0.83	1.54	0.02	0.26	0.71	1.34	0.02	0.22	0.60	1.17
Hybrid ( $K_{h}=2$ )	1	0.06	0.65	1.13	2.44	0.04	0.44	0.95	2.08	0.03	0.35	0.83	1.78	0.02	0.28	0.71	1.54
	2	0.03	0.40	0.96	1.83	0.02	0.33	0.84	1.59	0.02	0.27	0.72	1.38	0.02	0.22	0.61	1.21

Appendix

Table 7. Quantiles

\Delta^{\textrm{hybrid}}_{\mathrm{IV}}(T,z;\boldsymbol{\theta})

and

\Delta^{\textrm{HQE}}_{\mathrm{IV}}(T,z;\boldsymbol{\theta})

of the absolute difference between implied volatilities derived from Lewis pricing and those produced by inverting prices computed with the hybrid and HQE schemes, pooled over all

(T,z)

. The HQE scheme gives IVs that are substantially closer to the Lewis benchmark than the hybrid scheme, even when

K_{h}=2

for the hybrid scheme, in which both the head-cell and the following cell are treated exactly. We use

M=5\cdot 10^{5}

paths for the MC estimation. Quantiles are estimated based on

500

pairs

(\boldsymbol{\theta},\boldsymbol{x})

drawn from the prior.

$T$	$z$	$q_{50}$	$q_{90}$	$q_{95}$	$q_{99}$
$5/252$	$-2.5$	0.02	0.12	0.16	0.29
	$-2$	0.02	0.10	0.13	0.21
	$-1.5$	0.02	0.08	0.10	0.16
	$-1$	0.02	0.07	0.10	0.16
	$-0.5$	0.02	0.07	0.10	0.15
	$0$	0.01	0.07	0.10	0.15

Appendix

Table 8. Absolute errors

\Delta_{\mathrm{IV}}^{\textrm{HQE}}(T,z;\boldsymbol{\theta})

between HQE and Lewis implied volatilities. This table provides a breakdown by

(T,z)

pairs of the values reported in Table 7 .

	$N=504$		$N=1008$		$N=2016$		$N=4032$
$r$	$q_{50}$	$q_{95}$	$q_{50}$	$q_{95}$	$q_{50}$	$q_{95}$	$q_{50}$	$q_{95}$
$1$	1.57	12.00	1.16	7.06	0.92	4.67	0.82	3.37
$2$	0.99	4.78	0.83	3.43	0.75	2.93	0.71	2.65
$3$	0.95	4.53	0.82	3.35	0.74	2.91	0.72	2.60

Appendix

Table 9. Quantiles of the coefficient of variation for the MC estimators of European options, pooled over all pairs

(T,z)

. The coefficient of variation is defined as the standard error of the MC estimator, divided by the corresponding MC price. We conclude that for our choice of prior,

N\geq 2016

discretisation steps with

r=2

(or

r=3

) yields HQE discretisation bias that is roughly on the same order of magnitude, or lower than that of the randomness inherent to MC estimation with

M=5\cdot 10^{5}

samples. As before, quantiles are estimated based on

500

pairs

(\boldsymbol{\theta},\boldsymbol{x})

drawn from the prior.

$T$	$z$	$q_{50}$	$q_{95}$
$5/252$	$-2.5$	0.68	2.13
	$-2$	0.68	1.87
	$-1.5$	0.66	1.82
	$-1$	0.63	1.69
	$-0.5$	0.59	1.58
	$0$	1.11	3.64

Appendix

Table 10. Quantiles of the coefficient of variation for the MC estimators of European options. This provides a breakdown of the values reported in Table 9 by

(T,z)

pairs. The three largest values at the

95\%

level are

5.12

4.98

4.90

for maturities

T=5/252

1/12

10/252

respectively and

z=1.5

. These correspond to deep out-of-the-money contracts where the MC price is small, thus inflating the relative MC error.

$T_{1}$	$T_{2}$	$k$	min	$q_{01}$	$q_{05}$	$q_{50}$	$q_{95}$	$q_{99}$	$q_{95}$	$q_{99}$
			$\widehat{P}^{\mathrm{MC}}$						$\mathrm{CV}$
0.08	0.17	-0.10	9.51711	9.57	9.68	10.50	11.32	11.46	0.000	0.001
0.08	0.17	-0.05	4.94331	5.20	5.50	6.95	8.06	8.24	0.001	0.001
0.08	0.17	0.00	0.67721	1.49	2.06	4.02	5.31	5.52	0.001	0.001
0.08	0.17	0.05	0.01395	0.16	0.41	1.96	3.20	3.42	0.002	0.003
0.08	0.17	0.10	0.00001	0.01	0.05	0.79	1.75	1.95	0.005	0.010

Appendix

Table 11. Summary statistics for the dataset of forward-start MC prices.

			$\widehat{P}^{\mathrm{MC}}$						$\mathrm{CV}$
$T$	$k_{K}$	$k_{B}$	min	$q_{01}$	$q_{05}$	$q_{50}$	$q_{95}$	$q_{99}$	$q_{95}$	$q_{99}$
0.25	0.00	-0.10	0.0390	0.07	0.08	0.16	0.63	0.91	0.005	0.005
0.25	0.00	-0.15	0.1739	0.28	0.34	0.64	1.56	1.90	0.003	0.003
0.25	0.00	-0.20	0.4599	0.73	0.86	1.44	2.53	2.81	0.002	0.003
0.25	-0.05	-0.10	0.0036	0.01	0.01	0.02	0.07	0.11	0.009	0.010
0.25	-0.05	-0.15	0.0409	0.07	0.09	0.18	0.44	0.54	0.004	0.005

Appendix

Table 12. Summary statistics for the dataset of barrier MC prices.

$T$	$c$	$K$	$=0\,(\%)$	$<10^{-5}\,(\%)$	$q_{01}$	$q_{05}$	$q_{50}$	$q_{95}$	$q_{99}$	$q_{95}$	$q_{99}$
			$\widehat{P}^{\mathrm{MC}}$							$\mathrm{CV}$
0.08	0.25	0.03	0.10	0.43	0.0005	0.006	0.10	0.20	0.21	0.003	0.006
0.08	0.45	0.06	1.13	2.88	0.0001	0.001	0.07	0.17	0.18	0.006	0.020
0.25	0.25	0.03	0.01	0.13	0.0016	0.010	0.10	0.19	0.21	0.002	0.005
0.25	0.45	0.06	0.39	1.27	0.0001	0.002	0.07	0.16	0.18	0.005	0.014
0.50	0.25	0.03	0.00	0.06	0.0027	0.013	0.10	0.19	0.20	0.002	0.004

Appendix

Table 13. Summary statistics for the dataset of realised-variance MC prices; quantiles and the coefficient of variations (CV) are only computed for MC prices above

10^{-5}

Panel of Figure 8 — Figure 8. As Figure 6 in the main text, for the $1\%$ relative plug-in-pricing-error threshold.

Panel of Figure 9 — Figure 9. As Figure 6 in the main text, for the $5\%$ relative plug-in-pricing-error threshold.

T1	T2	k	$95\%$ PI median	$95\%$ PI q90	$95\%$ PI q95
0.08	0.17	-0.10	0.14	0.21	0.24
0.08	0.17	-0.05	0.28	0.43	0.50
0.08	0.17	0.00	0.56	1.34	1.84
0.08	0.17	0.05	1.02	3.54	5.54
0.08	0.17	0.10	1.70	6.59	10.50
0.08	0.33	-0.10	0.27	0.40	0.45

Appendix

Table 14. Forward-start options: median,

90\%

and

95\%

quantiles of the ratio of the length of the

95\%

full predictive intervals to the held-out MC prices. All reported widths are percentages of the reference price.

T	$k_{K}$	$k_{B}$	$95\%$ PI median	$95\%$ PI q90	$95\%$ PI q95
0.25	-0.05	-0.20	1.15	1.75	2.22
0.25	-0.05	-0.15	1.60	2.20	2.64
0.25	-0.05	-0.10	2.86	3.73	4.20
0.25	0.00	-0.20	0.99	1.46	1.73
0.25	0.00	-0.15	1.29	1.82	2.13
0.25	0.00	-0.10	1.87	2.64	3.07

Appendix

Table 15. Barrier options: median,

90\%

and

95\%

quantiles of the ratio of the length of the

95\%

predictive intervals to the held-out MC prices. All reported widths are percentages of the reference price.

T	c	K	$95\%$ PI median	$95\%$ PI q90	$95\%$ PI q95
0.08	0.25	0.03	1.19	7.32	23.04
0.08	0.45	0.06	1.54	13.56	27.52
0.25	0.25	0.03	1.22	5.40	13.03
0.25	0.45	0.06	1.60	9.48	20.70
0.50	0.25	0.03	1.31	4.57	8.08
0.50	0.45	0.06	1.73	7.70	14.49

Appendix

Table 16. Realized-variance options: median,

90\%

and

95\%

quantiles of the ratio of the length of the

95\%

predictive intervals to the held-out MC prices. All reported widths are percentages of the reference price.

$T_{1}$	$T_{2}$	$k$	Posterior only $95\%$	Full UQ $95\%$
0.08	0.17	-0.10	87.66	97.39
0.08	0.17	-0.05	92.95	97.26
0.08	0.17	0.00	93.84	96.62
0.08	0.17	0.05	93.87	96.87
0.08	0.17	0.10	93.80	97.01
0.08	0.33	-0.10	92.95	97.50

Appendix

Table 17. Empirical coverage rates for nominal-

95\%

predictive intervals for forward-start options. All entries as explained in percentages. These results complement Table 5 .

T	$k_{K}$	$k_{B}$	Posterior only $95\%$	Full UQ $95\%$
0.25	-0.05	-0.20	82.92	96.17
0.25	-0.05	-0.15	79.34	95.70
0.25	-0.05	-0.10	63.54	94.72
0.25	0.00	-0.20	87.08	96.56
0.25	0.00	-0.15	86.06	95.78
0.25	0.00	-0.10	80.14	95.01

Appendix

Table 18. Empirical coverage rates for nominal-

95\%

predictive intervals for barrier options. All entries as explained in percentages. These results complement Table 5 .

T	c	K	Posterior only $95\%$	Full UQ $95\%$
0.08	0.25	0.03	91.70	96.42
0.08	0.45	0.06	93.24	96.29
0.25	0.25	0.03	93.10	96.62
0.25	0.45	0.06	94.16	96.73
0.50	0.25	0.03	94.32	96.73
0.50	0.45	0.06	95.11	97.09

Appendix

Table 19. Empirical coverage rates for nominal-

95\%

predictive intervals for realized-variance options. All entries as explained in percentages. These results complement Table 5 .

$T_{1}$	$T_{2}$	$k$	TRE variance share (%)	Surrogate variance share (%)
0.08	0.17	-0.10	49.99	50.01
0.08	0.17	-0.05	69.29	30.71
0.08	0.17	0.00	80.20	19.80
0.08	0.17	0.05	80.18	19.82
0.08	0.17	0.10	78.41	21.59
0.08	0.33	-0.10	64.66	35.34

Appendix

Table 20. Percentage of variance attributed by the law of total variance to the posterior uncertainty and to the surrogate pricer for contracts in the forward start family. These results complement Table 6 .

$T$	$k_{K}$	$k_{B}$	TRE variance share (%)	Surrogate variance share (%)
0.25	0.00	-0.10	48.38	51.62
0.25	0.00	-0.15	56.33	43.67
0.25	0.00	-0.20	55.64	44.36
0.25	-0.05	-0.10	27.84	72.16
0.25	-0.05	-0.15	43.04	56.96
0.25	-0.05	-0.20	47.09	52.91

Appendix

Table 21. Percentage of variance attributed by the law of total variance to the posterior uncertainty and to the surrogate pricer for contracts in the barrier family. These results complement Table 6 .

T	c	K	TRE variance share (%)	Surrogate variance share (%)
0.08	0.25	0.03	87.63	12.37
0.08	0.45	0.06	86.40	13.60
0.25	0.25	0.03	89.05	10.95
0.25	0.45	0.06	89.84	10.16
0.50	0.25	0.03	91.92	8.08
0.50	0.45	0.06	91.63	8.37

Appendix

Table 22. Percentage of variance attributed by the law of total variance to the posterior uncertainty and to the surrogate pricer for contracts in the Realized variance family. These results complement Table 6 .

Panel of Figure 10 — Figure 10. Mean absolute KL Kernel-SHAP attributions for each of the six rough Heston parameters, averaged over 500 explained held-out IV surfaces.

Figure 11. Mean absolute Kernel-SHAP attributions for MSE- and MAE-trained point estimator for the $\kappa,\theta,\nu$ rough Heston parameters. Results are averaged over 500 explained held-out IV surfaces. See Figure 12 for results on the remaining three parameters $\rho,V_{o},H$ . These results complement the Hellinger-SHAP attributions from Figure 10 .

Figure 12. Mean absolute Kernel-SHAP attributions for MSE- and MAE-trained point estimator for the $\rho,V_{o},H$ rough Heston parameters. Results are averaged over 500 explained held-out IV surfaces. See Figure 11 for results on the first three parameters $\kappa,\theta,\nu$ .

Figure 13. Schematic representation of the neural point estimator architecture. The observed IV surface

\boldsymbol{x}

is passed to an MLP network to obtain a set of informative summary statistics, call them

s(\boldsymbol{x})

. Next

s(\boldsymbol{x})

is passed as input to

m

MLP heads to obtain estimates of

\theta^{1},\ldots,\theta^{m}

Panel of Figure 14 — Figure 14. Validation losses for the neural point estimators trained with the scaled MSE (left) and MAE (right) losses. The epoch selected through early stopping is marked in black.

Panel of Figure 15 — Figure 15. Breakdown of the validation losses on the original parameter scale, i.e., unscaled, for the two neural point estimators from Figure 14 . The $y$ axis is displayed on log scale.

Figure 16. Schematic representation of the surrogate pricer architecture. For each family the exotic contracts (forward-start, barrier, realized-variance), the parameter vector

\boldsymbol{\theta}

is passed to an MLP encoder to obtain a set of informative summary statistics

s(\boldsymbol{\theta})

. Next

s(\boldsymbol{\theta})

is passed to

k

MLP heads to obtain estimates of the exotic options prices

P^{\text{exo},1},\ldots,P^{\text{exo},k}

, where

k

is the number of contracts within the exotic family.

Panel of Figure 17 — Figure 17. Validation negative log-likelihoods for the exotic-option surrogate models. The epoch selected through early stopping is marked in black.

Figure 18. Schematic representation of the telescoping ratio estimator (TRE) architecture. The observed IV surface

\boldsymbol{x}

is passed to an MLP network to obtain a set of informative summary statistics. Then, for

i\in\{1,\ldots,m\}

, the

i^{\text{th}}

head inputs

s(\boldsymbol{x})

concatenated with the first

i

components

\boldsymbol{\theta}^{1:i}=(\theta^{1},\ldots,\theta^{i})

\boldsymbol{\theta}

and outputs an estimate of

\log r_{i}(\boldsymbol{x},\boldsymbol{\theta}^{1:i})

. Summing the

m

outputs yields an estimate of

\log r(\boldsymbol{x},\boldsymbol{\theta})

Figure 19. Schematic representation of the marginal neural ratio estimator (NRE) architecture. The observed IV surface

\boldsymbol{x}

is passed to an MLP network to obtain a set of informative summary statistics. Then, for

i\in\{1,\ldots,m\}

, the

i^{\text{th}}

head inputs

s(\boldsymbol{x})

concatenated with

\theta^{i}

\boldsymbol{\theta}

and outputs an estimate of

\log\frac{p(\theta^{i}\mid x)}{p(\theta^{i})}

i=1,\ldots,m.

Panel of Figure 20 — Figure 20. Binary cross-entropy validation losses $\mathcal{L}_{\textrm{TRE}}$ and $\mathcal{L}_{\textrm{mNRE}}$ for the two ratio-estimation models. The epoch selected through early stopping is marked in black.

Panel of Figure 21 — Figure 21. Empirical coverage of the HPD regions of the learnt posterior approximations $\hat{p}(\kappa\mid\boldsymbol{x}),\ldots,\hat{p}(H\mid\boldsymbol{x},\kappa,\theta,\nu,\rho,V_{0})$ . We note that the $\beta$ -calibrated classifiers yield posterior densities with near-perfect coverage, and that $\beta$ -calibration qualitatively improves the coverage properties of the learnt conditional distributions.

Hyperparameter	Values
Shared-encoder widths	$(64,64,64)$ , $(128,64)$ , $(64,64)$ , $(64,32)$ , $(32,32)$ , $(32,16)$
Encoder dropout	$0.1$ , $0$
Head widths	$(16,16,1)$ , $(8,8,1)$ , $(16,1)$ , $(8,1)$
Learning rate	$10^{-3}$ , $5\times 10^{-4}$
AdamW weight decay	$10^{-3}$ , $10^{-4}$

Appendix

Table 23. Hyperparameter grid for the neural point estimator (NPE), marginal NRE and TRE.

Hyperparameter	Values
Shared-encoder widths	$(16,32,64,128)$ , $(16,32,64)$ , $(16,16)$ , $(16,32)$
Encoder dropout	$0.1$ , $0$
Head widths	$(16,8,4,1)$ , $(16,16,1)$ , $(8,8,1)$ , $(8,4,1)$ ,
Learning rate	$10^{-3}$ , $5\times 10^{-4}$
AdamW weight decay	$10^{-3}$ , $10^{-4}$

Appendix

Table 24. Hyperparameter grid for the exotic surrogate pricers.

Quantitative Finance

Quantitative finance uses mathematical models, statistics, and machine learning to value assets, measure risk, and guide trading and portfolio decisions.

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ν

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Shaïn Afzali, Serena Della Corte, Antonis Papapantoleon

Model Calibration Feature Selection Neural Network Interpretability

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Aug 2, 2026

Amortizing the Calibration Triple: A Projection-Consistent Neural Operator for Local-Stochastic Volatility

Local-stochastic volatility (LSV) combines vanilla marginals with richer smile dynamics, but calibration requires a slow, noisy and sequential McKean--Vlasov fixed point. We learn a projection-consistent operator for the calibration triple. Given finite quotes and a stochastic-volatility (SV) backbone, it jointly returns an implied-volatility surface subject to static-arbitrage constraints, its Dupire local volatility, LSV leverage and the conditional moment required by the projection identity. Starting from option-price marginals, we derive a division-free Dupire residual in log-implied-variance coordinates and a quotient Fokker--Planck equation after Gyöngy projection. Deep Operator Network (DeepONet) and Fourier Neural Operator (FNO) implementations enforce quote fit, static-arbitrage, Dupire and projection constraints. For the witness-augmented residual system, we prove conditional identification and empirical consistency under LSV existence and inverse residual stability. In controlled synthetic tests, forward-start and cliquet errors differ from a particle method by 0.1 and 0.2 percentage points, while calibration latency falls from 98.5 to 0.6 ms. Compared with the tested baselines, local-volatility root-mean-square error (RMSE) falls by 36% and leverage RMSE by 7-16%. These results support amortizing the LSV fixed point: the expensive solve moves offline, while online calibration reduces to a single projection-consistent operator evaluation.

Xiaozhen Wang, Anaïs Després, Martin Dureau +1

1CEREMADE, Université Paris Dauphine-PSL · 2LaMME, Université Évry Paris-Saclay · 3AIML Global Markets, Barclays

Quantitative Finance Model Calibration Neural Operators Finance

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Mar 2, 2026

Practical Deep Heteroskedastic Regression

Uncertainty quantification (UQ) in deep learning regression is of wide interest, as it supports critical applications including sequential decision making and risk-sensitive tasks. In heteroskedastic regression, where the uncertainty of the target depends on the input, a common approach is to train a neural network that parameterizes the mean and the variance of the predictive distribution. Still, training deep heteroskedastic regression models poses practical challenges in the trade-off between uncertainty quantification and mean prediction, such as optimization difficulties, representation collapse, and variance overfitting. In this work we identify previously undiscussed fallacies and propose a simple and efficient procedure that addresses these challenges jointly by post-hoc fitting a variance model across the intermediate layers of a pretrained network on a hold-out dataset. We demonstrate that our method achieves on-par or state-of-the-art uncertainty quantification on several molecular graph datasets, without compromising mean prediction accuracy and remaining cheap to use at prediction time.

Mikkel Jordahn, Jonas Vestergaard Jensen, James Harrison +2

Technical University of Denmark · Google DeepMind

Uncertainty Quantification Heteroscedastic Regression

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Abstract

Quantitative Finance

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