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

• A single parameter vector does not capture residual uncertainty from finite, potentially noisy IV surfaces, limiting what calibration alone provides for downstream pricing and explainability.The uncertainty is conditional on the observed surface under the specified model and prior.Source: §1 · p. 1
• Repeated rough Heston evaluations are numerically demanding: short-maturity, deep out-of-the-money European pricing is unstable, while path-dependent pricing requires costly simulation.Non-Markovian dynamics and non-negativity corrections add further computational challenges.Source: §1 · p. 2
• Telescoping neural ratio estimation learns the parameter posterior from an IV surface, whose samples are propagated through heteroscedastic surrogate pricers to retain parameter and surrogate uncertainty.Separate MSE- and MAE-based inverse networks provide point-estimation benchmarks.Source: §1 · p. 2
• Hellinger-SHAP uses local-background Kernel SHAP to attribute posterior information gain across the IV surface for individual parameters, rather than explaining only a point estimate.The paper also presents squared Hellinger distance as a bounded complement to KL information gain.Source: §1 · pp. 2–3

2.1. Rough volatility (and rough Heston).

• Rough Heston pricing is numerically fragile in short-maturity, deep out-of-the-money regions, and repeated path simulation for exotic pricing or inference can become prohibitive.Path-dependent claims require Monte Carlo, while the non-Markovian variance process is difficult to simulate repeatedly.Source: §2.1 · p. 5

2.2. Simulation algorithm.

• The hybrid quadratic-exponential (HQE) scheme combines a hybrid treatment of the rough-kernel singularity with a nonnegative QE sampler.The QE step addresses negative Gaussian variance updates and reduces truncation bias.Source: §2.2 · pp. 5–6

2.2.1. The hybrid construction and head cell decomposition.

• The hybrid construction treats cells nearest the kernel singularity specially and approximates the smoother history cells, leaving variance freezing as a source of approximation error.The standard left-point discretization poorly captures the steep kernel near the current time.Source: §2.2 · p. 6

2.2.2. The HQE scheme.

• HQE interpolates between the current variance and its one-step conditional mean to improve local moment approximation, then applies QE to enforce nonnegative variance updates.The weighted average alone does not ensure nonnegativity; the QE construction preserves local mean-variance information while enforcing positivity.Source: §2.2.2 · pp. 8–9

2.2.3. Martingale correction, Romano-Touzi and other improvements.

• The simulation scheme adds an analytic martingale correction and variance-reduction methods, including integrating out orthogonal Brownian motion for European and forward-start options.For path-dependent contracts, repeated orthogonal Brownian draws reuse each simulated variance path; graded grids and antithetic sampling are also employed.Source: §2.2.3 · pp. 10–11

3.2. Propagating posterior uncertainty through exotic surrogates.

• Propagating posterior draws through surrogate predictive distributions combines residual parameter uncertainty with conditional surrogate uncertainty in exotic-price predictions.Predictive quantiles can be interpreted as model-implied bid and ask levels.Source: §3 · pp. 11–12, §3.2 · pp. 15–16

3. Simulation-Based Inference methodology

• Offline simulations train inverse estimators and surrogate pricers, while online IV surfaces yield point calibrations or posterior-predictive exotic-price distributions.The training cost is incurred once and amortized across subsequent calibrations.Source: §3 · p. 12

3.1. The neural point and density estimation landscape.

• Telescoping ratio estimation learns the conditional parameter posterior through sequential coordinate-wise classification, supporting fast MCMC-free posterior sampling.The factorization alleviates classifier-loss saturation that can arise when estimating the joint ratio directly.Source: §3.1 · pp. 12–14
• Unlike a single point calibration, the posterior represents residual parameter uncertainty after observing a finite, numerically generated IV surface under the specified prior-predictive simulator.The learned ratio can also be reweighted for another prior whose support lies within the training prior’s support.Source: §3.1 · pp. 12, 14, 15

3.2. Propagating posterior uncertainty through exotic surrogates.

• The surrogate predicts a location and scale in log space, with its learned scale absorbing both approximation error and Monte Carlo noise in the training targets.The stated predictive construction uses a shifted log-normal family; more flexible conditional families are left for future work.Source: §3.2 · pp. 15–16
• When ε > 0, the shifted log-normal construction permits prices in [−ε, 0], a numerical artefact that more flexible predictive families could remove.A deterministic surrogate is the degenerate limit as its predictive scale approaches zero.Source: §3.2 · p. 16

4. A Novel XAI approach: Hellinger-SHAP

• Hybrid surfaces can remain off the rough-Heston IV-surface manifold, so attributions are local relative scores under the stated background distribution, not strictly manifold-preserving counterfactual effects.Using simulated surfaces for both the explained input and background reduces, but does not eliminate, this issue.Source: §4 · p. 17
• Local-background Kernel SHAP attributes surface regions whose inclusion most increases posterior information about a parameter, relative to nearby empirical background surfaces.The background uses 100 nearest simulated IV surfaces, with distance measured in retained principal-component space.Source: §4 · pp. 16–17
• The squared-Hellinger and Kullback–Leibler constructions provide complementary posterior-contraction summaries, and agreement between their attribution maps is an empirical robustness finding.That agreement does not follow merely from correct model specification.Source: §4 · p. 17
• The same local-background Shapley construction can extend from single-parameter recovery to analysis of joint posterior information.The passage gives posterior dependence between two components as an example.Source: §4 · p. 18
• The posterior-information functional and its Shapley attributions are prior- or regime-specific, rather than prior-free statements about IV-surface informativeness.Changing the prior changes both the measured posterior contraction and the appropriate local reference distribution.Source: §4 · p. 18

5. Uncertainty quantification results.

• The study used HQE paths to generate European-option IV surfaces for posterior inference and held-out forward-start, barrier, and realized-variance prices for downstream uncertainty propagation.Given an observed IV surface, the inferred parameter posterior is propagated through exotic pricing maps.Source: §5 · p. 18

5.1.1. Parameter and IV surface design.

• The 63-dimensional IV input standardizes moneyness across maturities and retains dependence among surface errors induced by shared simulation paths.Implied-volatility inversion used the forward convention consistent with the martingale-corrected pricing scheme, reducing spurious inversion noise, especially for deep in- or out-of-the-money contracts.Source: §5.1.1 · pp. 19–20, §5.1.2 · p. 20

5.1.3. Numerical validation.

• With grading rate r = 2, HQE achieved sub-10bp accuracy up to the 95th percentile; the final dataset used N = 2016.Graded grids substantially improved on uniform grids, and remaining upper-tail discrepancies appeared dominated by benchmark instability and Monte Carlo noise rather than HQE discretization.Source: §5.1.3 · p. 21

5.1.4. Surface-quality checks.

• No generated IV surface violated the qualitative convexity- and wing-behavior diagnostics used as surface-quality checks.Source: §5.1.4 · p. 21

5.2. Parameter calibration and posterior recovery.

• TRE conditional posterior HPD regions showed near-perfect empirical coverage across nominal levels, indicating neither overconfidence nor underconfidence.Calibration was assessed separately for the sequentially estimated conditional densities over the held-out test set.Source: Figure 2 · p. 23, §5.2 · p. 24

5.3.2. Variance analysis of Monte Carlo exotic-price targets.

• Forward-start and barrier Monte Carlo targets were sufficiently resolved, while two near-degenerate realized-variance contracts were excluded from subsequent analysis.The excluded contracts were (c, T) = (0.65, 1/12) and (0.65, 1/4); variability for retained contracts was small enough for uncertainty decompositions to reflect mainly posterior and surrogate effects.Source: §5.3.2 · p. 26

5.3.3. Predictive intervals and decomposition of uncertainty.

• MSE/MAE plug-in prices exceeded 1% relative error in 6.46%/4.86% of forward-start, 17.25%/14.87% of barrier, and 22.05%/18.38% of realized-variance cases.Errors clustered in out-of-the-money forward starts, barriers near the prevailing level, and higher-strike realized-variance claims; the difference between MSE and MAE estimators was secondary.Source: Table 3 · p. 27, §5.3.3 · p. 29
• Full 95% interval median widths are 0.65% of reference price for forward starts, 1.66% for realized-variance claims, and 2.20% for barriers.The 95% quantiles are 3.84%, 18.18%, and 5.82%, respectively; contract-level intervals are especially wide for out-of-the-money forward starts, near-barrier claims, and several higher-strike realized-variance claims.Source: §5.3.3 · p. 29, Table 4 · p. 30
• Full UQ median coverage is 97.19% for forward starts, 95.75% for barriers, and 96.89% for realized-variance claims; posterior-only barrier coverage is 78.83%.Full predictive coverage is substantially more stable across retained contract specifications; posterior-only coverage remains contract-dependent for forward starts and realized-variance claims.Source: Table 5 · p. 30, §5.3.3 · p. 30
• The evidence supports calibration of the full predictive construction, not a claim that posterior uncertainty alone is always sufficient.Some specifications modestly over-cover, consistent with conservative intervals and finite-sample estimation; residual departures may reflect the approximate log-normal conditional surrogate law.Source: §5.3.3 · p. 30
• Posterior uncertainty explains approximately 86%–92% of predictive variance for realized-variance claims, while surrogate uncertainty exceeds 80% for some near-barrier, longer-maturity claims.Forward starts are predominantly posterior-driven away from the most in-the-money specifications, although their surrogate share can remain material; both uncertainty layers matter across contracts.Source: §5.3.3 · p. 31
• Point calibration can remain accurate on average yet produce economically consequential errors for selected downstream claims, which wider full predictive intervals flag.The full construction retains uncertainty from the surrogate-pricing stage and attains well-calibrated or conservative coverage.Source: §5.3.3 · pp. 31–32

6. XAI results

• The maps show attribution magnitude, not whether an IV entry increases or decreases posterior contraction for an individual surface.They use standardized-coordinate surfaces and local background completion under the specified prior-predictive experiment, not raw-strike wings or prior-free, manifold-preserving counterfactuals.Source: §6 · pp. 32, 34, Figure 7 · p. 33
• Positive standardized-moneyness regions, especially at medium and long maturities, repeatedly contribute strongly to posterior information gain across the parameter maps.The attributions are local Shapley-coalition contributions, not evidence that one quote identifies every parameter.Source: §6 · p. 32, Figure 7 · p. 33
• For κ and θ, the strongest attributions cluster in the positive wing at the longest maturities; ν is more localized, while ρ matters across a broader maturity range.These profiles are associated respectively with mean-reversion speed and long-run variance, volatility-of-volatility, and smile asymmetry in this simulation design.Source: §6 · p. 32
• V0 retains positive-wing contributions at intermediate and longer maturities, while H has substantial positive-wing contributions at both shorter and longer horizons.On the standardized 63-point surface, roughness is associated with cross-maturity smile information rather than one isolated short-dated quote.Source: §6 · p. 32
• The KL-based maps reproduce the main maturity–moneyness patterns, providing qualitative sensitivity evidence without making the Hellinger and point-estimator estimands equivalent.Point-estimator SHAP explains variation in a conditional point summary; Hellinger–SHAP explains marginal posterior contraction relative to the prior.Source: §6 · pp. 32, 34

7. Conclusions

• Held-out coverage diagnostics show that TRE posterior approximations are well calibrated in the simulation setting under the specified prior-predictive simulator.The method learns a conditional parameter distribution rather than reducing an observed IV surface to one calibrated vector.Source: §7 · p. 34
• Posterior-predictive exotic-price distributions retain residual parameter uncertainty and conditional surrogate uncertainty, and point calibration does not ensure reliable plug-in prices for every contract.Posterior uncertainty leads for most realized-variance claims; conditional surrogate uncertainty is often material and sometimes dominant for barriers.Source: §7 · p. 34
• Hellinger-SHAP attributes posterior contraction through local relative contributions to information gain, rather than explaining a single parameter point prediction.Its attributions are prior-specific and not strictly manifold-preserving counterfactuals; KL comparison provides sensitivity analysis of the information-gain target.Source: §7 · p. 34
• Applying the framework to observed market surfaces requires an observation model for bid–ask spreads, missing quotes, liquidity, and potential model misspecification.The paper also identifies comparison with forward-surrogate optimization and application to other volatility models as future directions.Source: §7 · pp. 34–35

A.1. The QE step and martingale property.

• The QE construction is non-negative and matches the specified mean and variance, while the analytic correction preserves the martingale property of the discretized log-asset increment.The parameterization remains well defined as the relative variance ψ decreases to zero.Source: §A.1 · pp. 39–40

A.2. The HQE scheme with non-uniform grids.

• For non-uniform HQE grids, the paper reports substantial empirical accuracy improvement from graded meshes despite their additional computational cost.The mesh parameter r≥1, with r=1 recovering the uniform mesh.Source: §A.2 · p. 40

A.3. The HQE scheme applied for option pricing and further variance reduction.

• Option-pricing variance reduction combines QE antithetic draws with analytic conditioning where available, and reuses variance paths across asset paths for barriers.For down-and-out puts, the authors use R=3 paired asset paths per variance path and a Brownian-bridge continuity correction for between-grid barrier crossings.Source: §A.3 · p. 41, §A.3.1 · p. 42, §A.3.3 · pp. 42–43

Appendix B. Extended numerical validation of the synthetic dataset

• Appendix B extends validation with diagnostics comparing hybrid and HQE implied-volatility approximations against Lewis pricing, alongside exotic-contract dataset summaries.The appendix outline assigns these topics to its European-surface and exotic-dataset subsections.Source: Appendix B · p. 44

B.1. European option prices and IV surfaces.

• N = 1008 with r = 2 gives better HQE accuracy than N = 4032 with r = 1, a 16-fold reduction in O(N^2) complexity with only minor additional computation.The study consequently uses HQE with N = 2016 steps, r = 2, and M = 5 · 10^5 paths for the remainder of the study.Source: §B.1 · p. 44
• The HQE scheme produces implied volatilities substantially closer to the Lewis benchmark than the hybrid scheme, including the hybrid variant with Kh = 2.Table 7 compares the pooled absolute differences; the reported errors reflect approximation error in both Lewis and HQE pricing.Source: §B.1 · pp. 44–45, Table 7 · p. 45
• Most 95% and 99% quantiles of HQE implied-volatility absolute errors remain below 10 and 15 basis points, respectively, with slightly larger errors in wings at very short maturities.The table breaks down the absolute errors by maturity–moneyness pair.Source: §B.1 · p. 44
• For the selected prior, the authors conclude that HQE discretization bias with at least 2016 steps and r = 2 or 3 is roughly on the order of, or below, MC randomness at 5 · 10^5 samples.The pooled coefficient-of-variation table summarizes MC estimator variability; deep out-of-the-money short-maturity contracts can have especially large relative MC error because their prices are small.Source: Table 9 · p. 47, Table 10 · p. 48

B.2. Exotic contracts.

• Forward-start and barrier MC prices remain bounded away from zero with estimator standard deviations far below the estimated prices; realized-variance evaluation excludes two contracts with frequent zero prices.For retained realized-variance contracts, quantiles and coefficients of variation are computed only for prices above 10^-5, with filtering performed separately by contract.Source: §B.2 · p. 49

Appendix C. Extended uncertainty quantification results

• Contract-level results complement pooled summaries by showing heterogeneity across maturities, moneyness or strike, barrier location, and forward-start dates.All results use the retained contract set.Source: Appendix C · p. 52

C.1. Contract-level plug-in pricing errors.

• Plug-in errors above 1% and 5% vary across contract grids, concentrating in out-of-the-money forward starts, near-price barriers, and higher-strike realized-variance claims.The 1% threshold identifies potential economic imprecision even when average parameter recovery appears strong; 5% errors are not confined to negligible deviations.Source: §C.1 · p. 52, Figure 8 · p. 54, Figure 9 · p. 55

C.2. Relative widths of full predictive intervals.

• Wider full predictive intervals occur for out-of-the-money forward starts, near-barrier contracts, and several higher-strike realized-variance claims.Tables 14–16 report contract-level median and upper quantiles of interval width as percentages of held-out Monte Carlo reference prices.Source: §C.2 · p. 52, Table 14 · p. 56, Table 15 · p. 57

C.3. Coverage of posterior-only and full predictive intervals.

• For barriers, posterior uncertainty alone need not provide adequate coverage; adding conditional surrogate uncertainty restores coverage close to or modestly above nominal 95%.Tables 17–19 report empirical coverage for each retained specification; contract-level variation remains within the family-level summaries.Source: §C.3 · p. 52, Table 17 · p. 58, Table 18 · p. 59

C.4. Predictive-variance attribution.

• Realized-variance claims are predominantly posterior-driven, while the conditional surrogate component is often comparable to or larger than the posterior component for barriers.Forward-start claims are generally posterior-driven away from the most in-the-money specifications, though surrogate uncertainty can remain material; the decomposition is descriptive, not causal.Source: §C.4 · pp. 52–53, Table 20 · p. 60, Table 21 · p. 61

Appendix D. Extended XAI analysis

• Point-estimator maps explain MSE- and MAE-based inverse-network outputs, whereas Hellinger-SHAP and KL-SHAP explain posterior contraction relative to the chosen prior.Qualitative agreement across maps indicates convergent feature importance across distinct estimands, not equality; differences need not be inconsistencies.Source: Appendix D · p. 62
• Figure 10 reports mean absolute KL Kernel-SHAP attributions for each of the six rough Heston parameters, averaged over 500 held-out IV surfaces.The figure provides the posterior-information attribution comparison referenced by the appendix discussion.Source: Appendix D · p. 62, Figure 10 · p. 63
• Figures 11 and 12 report mean absolute Kernel-SHAP attributions for MSE- and MAE-trained point estimators across all six parameters, averaged over 500 held-out IV surfaces.Figure 11 covers κ, θ, and ν; Figure 12 covers ρ, Vo, and H, complementing the Hellinger-SHAP attributions in Figure 10.Source: Figure 11 · p. 64, Figure 12 · p. 65

Appendix E. Methodological considerations on the training of the neural estimators

• Section E.1 describes preprocessing and standardization of neural-network inputs.Source: Appendix E · p. 66
• Section E.2 details architectures and loss functions for point estimators, ratio estimators, and exotic-option surrogate pricers, plus TRE post-training calibration and SBC checks.Source: Appendix E · p. 66
• Section E.3 reports the hyperparameter grids used during neural-network training.Source: Appendix E · p. 66

E.1. Data preprocessing and standardization procedures.

• The 63-dimensional IV surfaces are projected onto 20 leading principal directions, retaining over 99.9999% of training-set variability; whitening is computed in float64 for numerical stability.The PCA fit uses the training split only and is then applied to validation and test data.Source: §E.1 · p. 66
• Each parameter is standardized to [−1, 1], and local XAI backgrounds use the 100 nearest surfaces in the unwhitened retained-score representation.Nearest neighbours are selected using Euclidean distance in 20 dimensions.Source: §E.1 · pp. 66–67

E.2. Neural network multi-head architecture and loss functions.

• A shared MLP encoder extracts features for all outputs, while independently parameterized heads fine-tune each output; networks are amortized for reuse across inputs.Preprocessing and the multi-head design are used to reduce the amortization gap and multi-output optimization issues.Source: §E.2 · p. 67

E.2.1. Neural point estimators.

• Scaled MSE and MAE point-estimator objectives target the posterior mean and median, respectively, while scaling puts componentwise losses on a common scale.The neural point-estimator architecture and validation losses are shown in Figures 13–15.Source: §E.2.1 · pp. 67–68, Figure 13 · p. 68, Figure 14 · p. 68

E.2.2. Neural surrogate pricers.

• Surrogate pricers use one network per exotic family and one output head per contract; the retained realized-variance set contains 10 active contracts.The RV contracts are filtered to exclude some higher-strike contracts with many zero or economically irrelevant Monte Carlo prices.Source: §E.2.2 · p. 69
• The lognormal price-predictive assumption allows probability mass on [−ε, 0], a limitation that could in principle be removed with a more complex predictive distribution.The RV price model uses ε = 10^-5 for numerical stability because some retained Monte Carlo prices are zero or extremely small.Source: §E.2.2 · p. 69
• TRE estimates sequential conditional posterior-to-prior ratios from the IV surface and parameter prefixes, then combines the head outputs to recover the full ratio.β-calibration is applied per head; unlike Platt scaling and isotonic regression, which can degrade performance, it is selected for this setting.Source: §E.2.3 · pp. 69, 70, 73
• Marginal NRE directly estimates each parameter’s posterior marginal, which is needed for XAI because marginalizing posterior samples does not provide marginal posterior densities.The architecture has separate heads for the individual parameter marginals.Source: §E.2.3 · p. 71, Figure 19 · p. 72
• β-calibrated TRE conditional posterior approximations achieve near-perfect empirical HPD coverage, while uncalibrated classifiers show under-confidence.The figure caption also reports that β-calibration qualitatively improves coverage.Source: §E.2.3 · p. 74, Figure 21 · p. 74

E.3.1. Neural point and ratio estimators.

• The inverse-estimator search evaluates 192 configurations for each of three estimators, ranking point estimators by scaled MSE or MAE and ratio estimators by validation BCE.Hyperparameters are selected using validation-set loss.Source: §E.3.1 · p. 75, Table 23 · p. 75

E.3.2. Surrogate pricers.

• Each exotic-family surrogate-pricer search evaluates 128 configurations and ranks models by validation Gaussian negative log-likelihood on log-prices.The forward-estimator search uses a separate grid for each family.Source: §E.3.2 · p. 75, Table 24 · p. 75