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Controllable Generation of Implied Volatility Surfaces with Variational Autoencoders

T0 review · 4 major / 6 minor · reviewed 2026-08-05 · deepseek-v4-flash

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Pith's one-line read A controllable VAE generates implied volatility surfaces whose level, slope, curvature, and term-structure slope match user targets within fractions of a percent, while staying free of static arbitrage for in-distribution samples.

desk verdict Useful applied step for controllable IVS generation, but the control evidence is partly a closed loop around extrapolated polynomial features; deserves refereeing, not desk rejection. read the letter →

arxiv 2509.01743 v1 pith:PKUCUQUP submitted 2025-09-01 q-fin.CP

Jing Wang (1) , Shuaiqiang Liu (1 , 2) , Cornelis Vuik (1) ((1) Numerical Analysis , Delft University of Technology , Delft , the Netherlands , (2) ING Bank
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Amsterdam the Netherlands)
This is my paper · ORCID
classification q-fin.CP MSC 68T0791G20
keywords impliedvolatilitysurfacevariationalautoencodercontrollablegenerationshapefeaturesno-arbitrageconditionslatentspacedisentanglementsyntheticmarketdatastresstesting
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

The paper sets out to build a generative model of implied volatility surfaces that takes explicit orders: give it a target volatility level, skew slope, smile curvature, or term-structure slope, and it produces a synthetic surface with those characteristics, rather than just any plausible-looking surface. The mechanism is a variational autoencoder whose latent space is split into a controllable part — the four shape features, quantified by polynomial regression at a short-maturity at-the-money anchor point — and a residual part that preserves diversity. The reported results claim control errors below 0.1% for the level feature in most single-feature trials, comparable accuracy when three or four features are controlled together, and zero static-arbitrage violations among all 60,000 in-distribution generated surfaces, with a latent-space repair step correcting about half of the violations that appear under extreme sampling. If the claims hold, the payoff is a practical tool for stress testing and market simulation, where the user specifies the kind of market state the synthetic data should represent.

What carries the argument

The load-bearing object is the split latent representation (y, z) of a conditional VAE: y holds the commanded shape features, z holds residual variation. The features are the coefficients of a bivariate Taylor expansion of the surface at the anchor point (τ→0+, m=0) — level Σ0,0, slope Σ0,1, curvature Σ0,2, term-structure slope Σ1,0 — estimated by polynomial regression on the observable grid. The decoder maps (y, z) to the full surface, and a β-weighted KL term keeps z near a standard Gaussian prior. The repair step minimizes LCalendar + LButterfly + LMSE over z alone with L-BFGS, which lets the pipeline remove arbitrage violations without undoing the requested features.

What would settle it

Fit the paper's bivariate polynomial to surfaces generated by a model with known short-time asymptotics — for instance a rough-volatility model whose ATM slope diverges as τ→0 — while only observing the grid τ ∈ [0.1, 0.6]. If the extrapolated coefficients systematically miss the known limits (which finite-order polynomials must, at a singularity), then the 'slope' and 'term-structure' control variables are artifacts of the fitting basis, and a user commanding those values is not commanding the market feature the paper claims to control. A direct check of the same kind is available on real dat

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Extended reading notes

Core claim

Central discovery: quantitative control of IVS shape and financial validity can live in one VAE. Four shape features (level, slope, curvature, term structure) are anchor-point derivatives estimated by bivariate polynomial regression and fed to the decoder as controllable latent variables. A post-generation step re-optimizes only the residual latent variables against calendar-spread and butterfly penalties, leaving commanded features untouched. Results: level-generation error below 0.1% in most trials; four-feature control keeps errors mostly below 10^-2 and renders residual latents nearly inert; all 60,000 in-distribution surfaces satisfy no-arbitrage conditions; about 45.5% of rare tail vio

Load-bearing premise

The four controlled 'shape features' are coefficients of a polynomial fitted to the surface on a grid whose shortest maturity is τ = 0.1 and then extrapolated to the anchor point τ → 0+, m = 0; if that extrapolation does not capture the economically meaningful limiting shape, hitting the coefficients precisely does not control the market-relevant surface.

Editorial extensions

If this is right

  • Stress-test scenario design becomes a matter of dialing in feature values: surfaces with elevated volatility levels, steepened skews, or inverted term structures can be generated in a single decoder pass, without re-calibrating a pricing model.
  • Surfaces sampled within the convex hull of the training feature space are, according to Section 5.3, observed to be free of calendar-spread and butterfly arbitrage in all 60,000 draws, so downstream pricing or hedging tasks can consume them directly.
  • When fewer than four features are controlled, the residual latent variables still carry meaningful structure (z2 correlates with slope at about -0.991 in the single-feature experiment), so the model doubles as an exploratory tool for finding stylized facts the user did not specify.
  • When all four stylized factors are supervised, the residual latent variables become nearly inert (feature fluctuations on the order of 10^-3 across an eight-standard-deviation sweep), meaning the four-feature model gives essentially complete control over the dominant modes of IVS variation.
  • Extreme latent draws can produce slope values beyond the training range, so the generator can extrapolate to market states — steep skews or inverted term structures — that are rare or absent in the training data.

Reading between the lines

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

  • The same split-latent recipe should transfer wherever a high-dimensional market object is summarized by a handful of interpretable coefficients — for instance implied correlation surfaces, variance swap term structures, or CDS surfaces — since nothing in the architecture depends on equities except the choice of the four features.
  • A natural upgrade the authors flag but do not pursue: replace the polynomial basis with functions that can represent the exploding ATM slope of short-expiry surfaces, as in rough-volatility models. Until then, 'slope' means the polynomial's slope on the τ ≥ 0.1 grid extrapolated to τ = 0, and that extrapolation deserves validation against true short-time asymptotics before the control variable is
  • The repair algorithm keeps y fixed by construction but can push z away from the Gaussian prior; checking the posterior density of repaired z's and re-testing arbitrage after a second repair cycle would clarify whether the 45.5% correction rate is a property of the decoder or an artifact of L-BFGS initialization, which the paper itself flags as a local-optimization procedure.
  • For deployment, the convex-hull result suggests a cheap safety wrapper: constrain each sampled y to the training feature hull (a few linear inequality checks) and z to the central region, and arbitrage violations should effectively never occur — trading a little diversity for a certificate of validity.
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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

4 major / 6 minor

Summary. The paper proposes a controllable VAE for generating implied volatility surfaces (IVSs) with user-specified shape features. Four features (level, slope, curvature, term-structure slope) are quantified as coefficients of a bivariate polynomial regression evaluated at an anchor point (τ→0, m=0). These features serve as controllable variables y, while residual latent variables z capture remaining variability; training uses a conditional VAE objective with a β-weighted KL term. Experiments on 60,000 synthetic Heston/SABR surfaces report low generation errors in single- and multi-feature control, latent-traversal diversity, and no-arbitrage checks with a post-generation latent-space repair algorithm.

Significance. If the central claims hold, the framework would be a practically useful tool for stress testing and scenario generation, combining interpretable financial features with deep generative modeling. The paper contains a clean derivation of the conditional VAE ELBO (Appendix C), tests on two stochastic-volatility model families, and explicit no-arbitrage validation on generated surfaces. However, the evidence for the main claim of 'precise controllability' is weakened by two intertwined issues: the anchor-point features are extrapolated far outside the data support, and the evaluation applies the same regression operator that defines the conditioning labels to the generated surfaces. These issues make the reported sub-0.1% errors partly self-referential and not yet established as control of market-relevant surface shape.

major comments (4)
  1. [§3.3, Eq. (33), and Appendix B] The four controlled features are regression coefficients fitted on τ∈[0.1,0.6] and m∈[−0.27,0.27] but evaluated at the anchor point (τ=0, m=0). In particular, Σ1,0 is the τ-derivative at τ=0 estimated from a polynomial fit far from the origin, and the Remark after Eq. (12) acknowledges that the regression cannot capture singular short-end slopes. Since the data grid does not include τ near 0, the features are extrapolations, not interpolations. The sub-0.1% errors in §5.1.1 therefore establish control over polynomial coefficients, not necessarily over economically meaningful surface shape. The paper should provide a stability analysis with respect to the polynomial orders J,L(J) (which are never stated) and validate control errors using features measured on the observed grid, e.g., at the shortest maturity τ=0.1.
  2. [§5.1.1 and Eq. (10)] The generation error is computed by applying the same differential operator P (Eqs. 9–12) that defines the conditioning labels y(given) to the generated surface to obtain y(generated). The decoder is trained with labels extracted via this same operator (Section 4.3). Low |e| thus partly reflects how well the decoder inverts P on the training manifold, rather than how well the generated IVS matches an independently defined shape target. This closed loop does not invalidate the method, but it means the headline 'control accuracy' is not yet evidence of control over the underlying surface geometry. Please add an independent evaluation, e.g., fit SVI or a different parametric form to generated surfaces and compare those parameters, or measure features by finite differences at the shortest available maturity.
  3. [§5.3, Table 3] The post-generation repair algorithm fixes only 45.5% of the violating surfaces, leaving 2,950 of 60,000 generated samples (4.9%) in violation of static no-arbitrage conditions. The text concludes that the method 'can be corrected effectively' and that violations 'remain rare and can be corrected effectively'; given that over half of the violations are unrepaired, this is an overstatement. Moreover, Algorithm 2 optimizes z while keeping y fixed, but the decoded repaired surface may still have a different measured y, and LMSE only constrains the surface deviation, not the feature deviation. Please report the distribution of y errors before and after repair, and discuss the residual 4.9% violation rate.
  4. [§5 (all experiments)] The paper repeatedly contrasts the framework with 'black-box generators' and claims that it 'combines interpretability, controllability, and flexibility', but no baseline comparisons are provided. Without comparisons to, e.g., a conditional VAE that conditions on raw grid points or on PCA scores, a linear decoder-plus-regression model, or an existing IVS GAN, the experimental section cannot support the relative advantages asserted in the introduction and conclusion. At minimum, report control error and arbitrage-violation rates for a simple conditional VAE or a regression-based generator on the same dataset.
minor comments (6)
  1. [§3.3 / Appendix B] The polynomial orders J and L(J) in Eq. (33) are never specified, and no sensitivity study is reported. Since these orders define the feature values, the reader cannot reproduce the experiments or assess the stability of the anchor-point extrapolation.
  2. [§5.2 / Table 1] The KL weight β is listed as 'problem-specific' but only the three-feature experiment states a value (β=5×10^-8). Please report β for the single- and four-feature experiments, and describe how it was selected.
  3. [§4.4 / Algorithm 2] The thresholds ε1, ε2, maximum iterations T, and the learning rate η in Algorithm 2 are not given numerically, and no convergence statistics are reported. Please provide the implementation values or a reference to released code.
  4. [§5.1.3] The latent traversal uses z2=±8, which is 8 standard deviations under the standard normal prior and has negligible prior probability. The claim that the model 'can extrapolate meaningfully' would be stronger if these out-of-distribution surfaces were also checked for no-arbitrage violations and compared with in-distribution latent samples.
  5. [Figures 11 and 16] The legend entries in Figures 11 and 16 appear as 'uni000000/...' encoding artifacts rather than readable feature names; these should be fixed.
  6. [§1 and §5.1.1] Typos and minor wording: 'A serial of studies' should be 'A series of studies'; in Section 5.1.1, the sentence beginning 'whered y(given)L' has a stray 'd'.

Circularity Check

1 steps flagged · score 4.0 of 10

Control metric is closed-loop: generation error is measured with the same regression operator P that defines the controllable labels, so sub-0.1% accuracy partly restates decoder inversion of P.

  1. self definitional [Section 5.1.1 (Overall Performance), referencing Section 3.3 Eqs. (9)-(12)]
    "For each case, we define the generation error eL = y(generated)L − y(given)L , where y(given)L stands for the desired level which is given to the input of the generator (i.e., the already trained decoder), and y(generated)L stands for the volatility level of a generated IVS, which is computed using the regression method in Section 3.3."

    The controllable labels y are extracted by the regression operator P (Eq. 10: yi = Pg(τ,m|xi)), whose coefficients define the very 'shape features' that the paper claims to control. The success metric then computes y(generated) by applying the same operator P to the decoded surface. Consequently, the reported sub-0.1% generation error measures how well the decoder inverts P on user-supplied labels (P∘Decoder ≈ Id) rather than providing an independent check that economically meaningful surface shape has been controlled. Moreover, because y is a deterministic function of the training surface x and the decoder is trained to reconstruct x under an MSE loss (Eq. 45), matching y is largely inherited from the reconstruction objective, making the 'precise controllability' claim partly a restatemen

full rationale

The paper does not rely on load-bearing self-citations: the anchor-point regression method is attributed to external work ([27],[34]) and the VAE machinery is standard. The central architecture—explicit controllable y, residual latent z, and post-generation arbitrage repair—is independently implemented and tested against no-arbitrage conditions that are not derived from the feature operator. The main circularity concern is the evaluation protocol: the features that define the control targets are the same regression coefficients used to measure generation success. Thus the quantitative controllability results are self-referential: they demonstrate that the decoder can invert the feature-extraction map, not that the underlying polynomial coefficients correspond to robust, economically meaningful shape attributes. The paper itself acknowledges a limitation in the Remark after Eq. (12) that the regression cannot capture singular short-end slopes, which is exactly the regime where the anchor-point extrapolation (from τ∈[0.1,0.6] to τ→0+) is most fragile. That extrapolation instability is a correctness risk rather than circularity. Overall, the circularity is partial: the framework has independent content, but the headline 'precise control' metric is closed-loop and should be interpreted as control over the regression-defined features, not as an externally validated control of surface shape.

Assumptions & free parameters 4 free parameters · 4 assumptions · 0 invented entities

The central result does not introduce new physical or financial entities. It depends on a set of modeling choices (feature definition via extrapolated polynomial regression, independence factorization, Gaussian latent assumptions) and on several hand-tuned hyperparameters (beta, latent dimension, regression order, repair thresholds) that are not fully specified.

free parameters (4)
  • beta (KL weight) = 5e-8 (three-feature); problem-specific in Table 1
    Trade-off between reconstruction accuracy and latent regularization; authors report that smaller beta helps residual feature capture.
  • latent dimension = 5
    Fixed after experimenting with different dimensionalities; claim that higher dims give no improvement.
  • polynomial regression orders J, L(J) = not reported
    The Taylor expansion order in Eq. (33) is not specified, but it determines the feature values controlling generation.
  • Algorithm 2 thresholds and max iterations = not reported
    L-BFGS termination parameters (epsilon1, epsilon2, T) affect repair success rate but are not given.
assumptions (4)
  • standard math Implied volatility is uniquely defined by inverting the Black-Scholes formula.
    Needed to construct IVS from option prices; Section 2.1.
  • domain assumption The no-arbitrage conditions, expressed via total variance w = sigma^2 tau, are those of Gatheral-Jacquier (calendar and butterfly).
    Used to validate and repair generated surfaces; Section 2.3.
  • domain assumption The IVS can be represented by a bivariate polynomial around the anchor point, and the limiting Taylor coefficients exist and are finite.
    This makes feature quantification via regression possible; Eq. (33) and Appendix B. The paper acknowledges the slope may blow up near the anchor, which already violates this assumption.
  • ad hoc to paper The joint distribution factorizes as p(x,y,z)=p(z)p(y)p(x|y,z) with independent y and z.
    Equation (13) assumes disentanglement between controllable features and residual latents; on real data this may not hold.

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Cite this review

Pith. "Pith review of Controllable Generation of Implied Volatility Surfaces with Variational Autoencoders." pith.science (2026). https://pith.science/paper/PKUCUQUP

@misc{pith2026250901743,
  author       = {Pith},
  title        = {Pith review of: Controllable Generation of Implied Volatility Surfaces with Variational Autoencoders},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/PKUCUQUP}},
  note         = {Machine review of arXiv:2509.01743}
}
read the original abstract

This paper presents a deep generative modeling framework for controllably synthesizing implied volatility surfaces (IVSs) using a variational autoencoder (VAE). Unlike conventional data-driven models, our approach provides explicit control over meaningful shape features (e.g., volatility level, slope, curvature, term-structure) to generate IVSs with desired characteristics. In our framework, financially interpretable shape features are disentangled from residual latent factors. The target features are embedded into the VAE architecture as controllable latent variables, while the residual latent variables capture additional structure to preserve IVS shape diversity. To enable this control, IVS feature values are quantified via regression at an anchor point and incorporated into the decoder to steer generation. Numerical experiments demonstrate that the generative model enables rapid generation of realistic IVSs with desired features rather than arbitrary patterns, and achieves high accuracy across both single- and multi-feature control settings. For market validity, an optional post-generation latent-space repair algorithm adjusts only the residual latent variables to remove occasional violations of static no-arbitrage conditions without altering the specified features. Compared with black-box generators, the framework combines interpretability, controllability, and flexibility for synthetic IVS generation and scenario design.

Figures

Figures reproduced from arXiv: 2509.01743 by the authors.

Figure 1
Figure 1. Flowchart of the controllable VAE framework for IVS generation. [PITH_FULL_IMAGE:figures/full_fig_p009_1.png] view at source ↗
Figure 2
Figure 2. Neural network architecture with shortcut connections used in the controllable VAE. [PITH_FULL_IMAGE:figures/full_fig_p009_2.png] view at source ↗
Figure 3
Figure 3. Convergence of the log-transformed MSE loss during training. [PITH_FULL_IMAGE:figures/full_fig_p013_3.png] view at source ↗
Figures from the paper (32 more)
Figure 4
Figure 4. Figure 4: Distribution of the log absolute errors log [PITH_FULL_IMAGE:figures/full_fig_p014_4.png]
Figure 5
Figure 5. Figure 5: From left to right, given/generated yL = 9.83/9.86, 34.30/34.39, 50.00/50.12(%), while z = (0, 0, 0, 0, 0). 14 [PITH_FULL_IMAGE:figures/full_fig_p014_5.png]
Figure 6
Figure 6. Figure 6: From left to right, given/generated yL = 9.83/9.83, 34.30/34.31, 50.00/50.12(%), while z = (0, −6, 0, 0, 0) . Importantly, to satisfy arbitrage-free conditions, the generator does not simply shift the IVS. For example, in [PITH_FULL_IMAGE:figures/full_fig_p015_6.png]
Figure 7
Figure 7. Figure 7: Given y = (yL) = (0.3875), the figure shows the effect of varying z2 = −8, −4, 0, 4, 8 from left to right, while keeping all other components of z fixed at zero. The corresponding slope values from left to right are 0.5792, 0.4078, −0.1166, −0.5211, and −0.8095. 0.1 0.…
Figure 8
Figure 8. Figure 8: Given y = (yL) = (0.3875), the figure shows the effect of varying z2 = −8, −4, 0, 4, 8 from left to right, while keeping all other components of z fixed at zero. The corresponding term structure values from left to right are 0.1245, 0.0341, 0.0097, −0.1351, and −0.3368…
Figure 9
Figure 9. Figure 9: Given y = (yL) = (0.3875), the figure shows the effect of varying z4 = −8, −4, 0, 4, 8 from left to right, while keeping all other components of z fixed at zero. The corresponding curvature values from left to right are 1.1707, 0.7628, 0.1623, −0.0487, and 0.1603. Howe…
Figure 10
Figure 10. Figure 10: Given y = (yL) = (0.3875), the figure shows the effect of varying z4 = −8, −4, 0, 4, 8 from left to right, while keeping all other components of z fixed at zero. The corresponding term structure values from left to right are −0.1557, −0.1221, 0.0097, 0.1355, and 0.171…
Figure 11
Figure 11. Figure 11: Distribution of log absolute errors for controlled level, slope, and term-structure features under [PITH_FULL_IMAGE:figures/full_fig_p018_11.png]
Figure 12
Figure 12. Figure 12: From left to right, given/generated yL = 9.83/9.83, 34.30/34.31, 50.00/50.01(%) with fixed z. 18 [PITH_FULL_IMAGE:figures/full_fig_p018_12.png]
Figure 13
Figure 13. Figure 13: From left to right, given/generated yS = −0.5642/ − 0.5641, −0.1097/ − 0.1096, 0.3881/0.3882 with fixed z. 0.1 0.2 0.3 0.4 0.5 0.6 Time to Maturity ( ) 10 20 30 40 50 60 Implied Volatility (%) Given Term Structure: -0.1841 Generated Term Structure: -0.1843 log(K/S0) =…
Figure 14
Figure 14. Figure 14: From left to right, given/generated yT = −0.1841/ − 0.1843, 0.0069/0.0071, 0.2497/0.2500 with fixed z. 0.2 0.1 0.0 0.1 0.2 Log-Moneyness (log(K/S0)) 30 35 40 45 50 55 Implied Volatility (%) z2 = 8.00 = 0.10 = 0.35 = 0.60 (m=0, =0.1) 0.2 0.1 0.0 0.1 0.2 Log-Moneyness (…
Figure 15
Figure 15. Figure 15: Given y = (yL, yS, yT ) = (0.3875, −0.2139, −0.0236), the figure shows the effect of varying z4 = −8, −5, −2, 1, 4 from left to right, while keeping all other components of z fixed at zero. The corresponding curvature values from left to right are 2.5750, 1.5353, 0.57…
Figure 16
Figure 16. Figure 16: Histogram of log absolute errors for simultaneous control of level, slope, curvature, and term [PITH_FULL_IMAGE:figures/full_fig_p020_16.png]
Figure 17
Figure 17. Figure 17: From left to right, given/generated yL = 9.83/9.82, 34.30/34.28, 50.00/49.96(%) with fixed z. 0.2 0.1 0.0 0.1 0.2 Log-Moneyness (log(K/S0)) 10 20 30 40 50 60 Implied Volatility (%) Given Slope: -0.5642 Generated Slope: -0.5669 = 0.10 = 0.35 = 0.60 (m=0, =0.1) 0.2 0.1 …
Figure 18
Figure 18. Figure 18: From left to right, given/generated yS = −0.5642/ − 0.5669, −0.1097/ − 0.1095, 0.3881/0.3863 with fixed z. At the same time, the latent space z no longer causes substantial additional shape variations. First, we examine the impact of varying z1 on the generated IVS sh…
Figure 19
Figure 19. Figure 19: From left to right, given/generated yC = −0.5757/ − 0.6093, 0.1850/0.1850, 1.3751/1.3754 with fixed z. 0.1 0.2 0.3 0.4 0.5 0.6 Time to Maturity ( ) 10 20 30 40 50 60 Implied Volatility (%) Given Term Structure: -0.1841 Generated Term Structure: -0.1842 log(K/S0) = -0.…
Figure 20
Figure 20. Figure 20: From left to right, given/generated yT = −0.1841/ − 0.1842, 0.0069/0.0080, 0.2497/0.2503 with fixed z [PITH_FULL_IMAGE:figures/full_fig_p022_20.png]
Figure 21
Figure 21. Figure 21: Visualization of the dataset feature domain (orange) and its convex hull (blue). [PITH_FULL_IMAGE:figures/full_fig_p023_21.png]
Figure 22
Figure 22. Figure 22: Distribution of features extracted from the training IVSs: [PITH_FULL_IMAGE:figures/full_fig_p028_22.png]
Figure 23
Figure 23. Figure 23: 3D IVS varying level feature (yL) while latent variables z fixed. 34 [PITH_FULL_IMAGE:figures/full_fig_p034_23.png]
Figure 24
Figure 24. Figure 24: 3D example: generated IVS with a pronounced “smile” while controlling the [PITH_FULL_IMAGE:figures/full_fig_p035_24.png]
Figure 25
Figure 25. Figure 25: Effect of varying latent dimension z2 on the generated IVS with other latent variables set to zero. 0.3 0.2 0.1 0.0 0.1 0.2 0.3 Log-Moneyness (log(K/S0)) 0.1 0.2 0.3 0.4 0.5 0.6 Time to Maturity ( ) 10 20 30 40 50 60 Implied Volatility (%) z4 = 8.00 0.3 0.2 0.1 0.0 0.…
Figure 26
Figure 26. Figure 26: Effect of varying latent dimension z4 on the generated IVS with other latent variables set to zero. 35 [PITH_FULL_IMAGE:figures/full_fig_p035_26.png]
Figure 27
Figure 27. Figure 27: 3D visualization of controlled level variations under three-feature control with fixed latent vari￾ables. 0.3 0.2 0.1 0.0 0.1 0.2 0.3 Log-Moneyness (log(K/S0)) 0.1 0.2 0.3 0.4 0.5 0.6 Time to Maturity ( ) 10 20 30 40 50 60 Implied Volatility (%) Given Slope: -0.5642 G…
Figure 28
Figure 28. Figure 28: 3D visualization of controlled slope variations under three-feature control with fixed latent vari￾ables. 36 [PITH_FULL_IMAGE:figures/full_fig_p036_28.png]
Figure 29
Figure 29. Figure 29: 3D visualization of the effect of latent variable [PITH_FULL_IMAGE:figures/full_fig_p037_29.png]
Figure 30
Figure 30. Figure 30: Three-dimensional illustration of controlled variation in the level feature. [PITH_FULL_IMAGE:figures/full_fig_p037_30.png]
Figure 31
Figure 31. Figure 31: Three-dimensional illustration of controlled variation in the slope feature. [PITH_FULL_IMAGE:figures/full_fig_p037_31.png]
Figure 32
Figure 32. Figure 32: Three-dimensional illustration of controlled variation in the curvature feature. [PITH_FULL_IMAGE:figures/full_fig_p038_32.png]
Figure 33
Figure 33. Figure 33: Three-dimensional illustration of controlled variation in the term structure feature. [PITH_FULL_IMAGE:figures/full_fig_p038_33.png]
Figure 34
Figure 34. Figure 34: Impact of varying latent variable z1 on IVSs under complete four-feature control scenario. 38 [PITH_FULL_IMAGE:figures/full_fig_p038_34.png]
Figure 35
Figure 35. Figure 35: 3D visualization of controlled “term-structure” variations under three-feature control with fixed [PITH_FULL_IMAGE:figures/full_fig_p039_35.png]

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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. Latent Flow Matching for Arbitrage-Aware Implied Volatility Surface Generation

    q-fin.CP 2026-08 conditional novelty 6.0 of 10

    Latent flow matching with an arbitrage-regularized VAE generates implied volatility surfaces that match the empirical distribution and pass static no-arbitrage tests at a higher rate than GAN and diffusion baselines.

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Pith tools

Reviewed August 5, 2026 · model on record in the stance chip above.