Latent-Space No-Arbitrage Geometry of Generative Models for Implied Volatility Surfaces

Jing Wang; Shuaiqiang Liu; Cornelis Vuik

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Latent-Space No-Arbitrage Geometry of Generative Models for Implied Volatility Surfaces

Jing Wang, Shuaiqiang Liu, Cornelis Vuik

arXiv:2609.00332v1Aug 31, 2026q-fin.CPcs.AIcs.LGmath.NA

TL;DR

Generative implied volatility models need latent outputs satisfying static no-arbitrage constraints, but admissibility can vary across latent space. The paper defines and analyzes a latent margin and its boundaries, then shows analytic boundary recovery and varied Heston-VAE admissibility, with local boundary correction moving many violating codes.

Problem
Generated implied volatility surfaces can fit distributions yet violate static no-arbitrage conditions, making them unusable consistently for pricing and hedging.
Method
A fixed deterministic generator is composed with a scalar no-arbitrage margin, whose nonnegative latent super-level set and zero-margin boundaries are analyzed using topology and level-set methods.
Results
The Heston VAE produced central admissible regions across five seeds; similar reconstruction errors coincided with different admissible areas and prior probabilities, and 33 of 40 violating codes crossed the computed boundary after correction.
Takeaways & Limitations
Latent-space boundaries can compare no-arbitrage properties beyond reconstruction accuracy and modify codes that generate violating surfaces.
Takeaways & Limitations
Guarantees are relative to the selected finite domain or grid, and positive discrete-grid margin does not ensure continuous-domain no-arbitrage without approximation and boundary or tail conditions.

Abstract

from arXiv · show

Generative models for implied volatility surfaces must produce outputs that satisfy static no-arbitrage constraints. We study these constraints in latent space. For a fixed generator, we assign each latent code a scalar margin determined by the no-arbitrage conditions of the generated surface. The codes with nonnegative margin form the admissible latent set. We establish conditions under which strictly admissible codes remain admissible under small perturbations and the boundary of the admissible set is characterized by zero margin. For regular boundary components, we formulate a level-set equation whose local dynamics are directed toward the zero-margin set. The analysis treats the generator as a map from latent variables to surfaces and is therefore not restricted to a particular architecture. It applies to variational autoencoders, generative adversarial networks, and other generative models with a deterministic realization map. Numerical tests recover known boundaries in analytic examples. Experiments with a variational autoencoder trained on Heston surfaces show that similar reconstruction errors can correspond to different admissible regions and that the latent prior may be concentrated inside such a region. The computed boundary can also be used to modify latent codes that generate violating surfaces.

1 Introduction

The paper studies static no-arbitrage constraints by partitioning a fixed generator’s latent space into codes producing admissible or violating implied volatility surfaces. It develops margin-based topology and boundary tools, then applies them to analytic examples and a Heston-trained VAE.

Distributional fit alone is insufficient because surfaces violating static no-arbitrage conditions cannot be used consistently for pricing and hedging.
The analysis keeps the trained generator fixed and studies latent codes whose generated surfaces satisfy prescribed no-arbitrage conditions.
A scalar margin partitions latent codes into strictly admissible, violating, and boundary regions, independently of the generator architecture.
Continuity, zero-margin boundary analysis, and a Hamilton–Jacobi level-set equation provide local stability and evolving-interface calculations.
The framework applies to VAE decoders, GAN generators, normalizing flows, and other fixed generators represented as latent-to-surface maps.

2 Generative Models and No-Arbitrage Constraints

The paper represents implied volatility surfaces through deterministic realization maps into total-variance surface spaces and evaluates calendar and butterfly no-arbitrage conditions on continuous domains or finite computational grids.

Surface representation: The computational domain is a compact region in log-moneyness and time to maturity, with the short-maturity endpoint excluded because total variance degenerates there.
Surface representation: Total implied variance is used because calendar-spread and butterfly-arbitrage conditions can be expressed directly through it and its derivatives.
Finite-domain formulation: On finite grids, derivatives are replaced by finite differences on interior points, while boundary and tail behavior requires separate accompanying conditions.
Generative maps: The generator maps each latent code to one positive total-variance surface, whether it outputs variance directly or converts implied volatility deterministically.
No-arbitrage conditions: Calendar and butterfly constraints become latent-dependent inequalities after composition with the generator, and their minimum defines a scalar latent margin field.

3 The No-Arbitrage Margin and the Admissible Set

The no-arbitrage margin is the minimum of encoded constraint values after mapping latent codes to generated surfaces. Its nonnegative super-level set defines admissibility, with topology and boundary determined by the induced margin rather than finite samples.

Margin construction: The combined margin uses the minimum of calendar and butterfly margins, with positive values denoting strict admissibility and negative values denoting violations.
Margin construction: A surface is admissible only when all encoded constraints hold throughout the checked continuous domain or at every checked grid point.
Latent admissible set: Composing the surface margin with the generative map transfers no-arbitrage conditions from generated surfaces into latent space.
Latent admissible set: The admissible set is the nonnegative super-level set of the latent margin, while strict admissible and violating regions are defined by positive and negative margins.
Topology: Continuity guarantees that the admissible set is closed and that its strict-admissible and violating regions are open; equality between the boundary and zero-margin set can fail at degenerate zeros.
Discrete formulation: Finite-difference implementations use interior grid points and may apply a tolerance ε to distinguish numerical error from computed violations.

4 Topology, Local Stability, and Boundary Regularity

Continuity makes strictly admissible latent codes locally stable, while non-degenerate zero-margin points form regular boundaries separating admissible and violating regions. These guarantees remain local rather than global.

Local stability: Under continuity, every positive-margin code has a neighborhood contained in the admissible set, but this does not imply global admissibility.
Regularity assumptions: If the generator and surface margin are continuous, the induced latent margin is continuous; local Lipschitz assumptions strengthen this to local Lipschitz regularity.
Boundary regularity: A non-degenerate interior zero-margin point lies on the admissible-set boundary, where the boundary and zero-margin set locally coincide as a C1 hypersurface.
Boundary regularity: The regular boundary’s normal direction is given by the normalized gradient of the latent margin.
Boundary limitations: Boundary codes are unstable under opposite normal perturbations, which enter admissible and violating regions respectively.
Global scope: For a standard Gaussian prior, high probability lies near the origin, but positive margin there provides local admissibility rather than a global no-arbitrage guarantee.

5 Computing the Admissible-Set Boundary

The admissible-set boundary is represented as a zero-margin level set and computed through a Hamilton–Jacobi evolution driven by the margin. The analysis establishes local attraction and viscosity well-posedness while emphasizing limitations of global coverage and computational cost.

5.1 Level-set representation and local attraction: The margin induces an implicit zero set Σ that locally equals the actual boundary Γ at non-degenerate zeros and accommodates multiple components, holes, corners, and topology changes.This representation avoids directly parameterizing the boundary and supports an evolution equation for numerical analysis.
5.1 Level-set representation and local attraction: The level-set equation uses the margin as front speed: positive margins move the front toward the boundary from the admissible side, negative margins move it oppositely, and zero margin stops motion.The convention is Φ < 0 on the admissible side and Φ > 0 on the violating side.
5.1 Level-set representation and local attraction: At a regular zero with nonzero gradient, the fixed-normal trajectory remains nearby, moves monotonically toward zero, and converges exponentially at rate ∥∇M(z∗)∥.This result concerns only the fixed normal cross-section, not general characteristics or the full evolving front.
5.1 Level-set representation and local attraction: Global boundary coverage is not guaranteed: initialization may miss components, and discretized-front convergence is not established.The stopping scale near a regular component is a margin tolerance of order ∥∇M∥h.
5.2 Well-posedness and numerical approximation: The level-set problem is well posed in the viscosity sense for bounded globally Lipschitz margins with suitable bounded uniformly continuous or periodic initial data.Viscosity solutions are used because Hamilton–Jacobi evolution can develop corners, shocks, or kinks even from smooth inputs.
5.2 Well-posedness and numerical approximation: A narrow-band method is not unconditionally cheaper than full-grid contouring; its relative cost depends on travel distance, caching, initialization, refinement, and the boundary components sought.For a d-dimensional grid, full-grid evaluation costs O(N^d) margin evaluations, while narrow-band updates can accumulate O(N^(d−1)N_t).

6 Toy Experiments

Toy experiments illustrate level-set motion, generator-induced calendar admissibility, and interactions among active constraints. Numerical fronts recover analytic boundaries with first-order refinement, while Gaussian mass and latent-tail analyses quantify violation regions.

Experimental roles: The toy experiments examine level-set fronts, generator-induced calendar margins, and competing calendar and butterfly constraints.Known boundaries are used for numerical reconstruction before analyzing admissible geometry induced by a generator.
Example 1: radial admissible set: The evolving zero level set converges to the no-arbitrage boundary Γ = {M = 0}.In the radial example, the explicit front approaches the stationary interface R = 1.
Example 2: generator-induced calendar margin: The decoder-induced calendar-admissible set contains all latent codes with z2 ≥ −2/3, while violations occur below its zero-level boundary.The central latent region remains strictly admissible, whereas a lower-tail region crosses the boundary.
Example 2: generator-induced calendar margin: The calendar boundary enters the latent box at (−6, −3.7178) and exits at (4.1278, −6), with a branch switch at z2 = −2/3.The analytic lower boundary matches the plotted zero contour in the generator-induced example.
Example 2: generator-induced calendar margin: Gaussian violation probabilities are negligible near the latent center but rise sharply as fixed z2 moves farther into the lower tail.For the decoder example, the regions ∥z∥≤2 and ∥z∥≤3 contain no violations and hold 86.5% and 98.9% of Gaussian mass; Monte Carlo estimated 6.0 × 10−7 violations.
Numerical reconstruction: Halving the grid spacing produces first-order refinement in both examples, with Example 1’s finest-grid radial residual equal to 1.2 × 10−2.The finest radial front gives Rnum(1) = 0.9129 versus Rana(1) = 0.9137; Example 2’s maximum vertical residual decreases to 4.8 × 10−2.
Example 3: competing constraints: When α ≥ 1, the butterfly constraint alone determines the boundary; when 0 < α < 1, calendar and butterfly constraints both contribute switching corners.The stylized margins are not derived from a generator-produced surface.

7 Admissible latent geometry of Generative Financial Models

The Heston VAE experiments show that reconstruction accuracy and no-arbitrage geometry capture different properties of a trained generator. The admissible region is central under the Gaussian prior, and local latent correction can move violating codes across the computed boundary.

Experimental setup: The experiment compares reconstruction error, held-out admissibility, admissible area, and Gaussian-prior probability across five VAE training seeds.The VAE is trained on synthetic Heston implied volatility surfaces and evaluated in a two-dimensional latent space.
Experimental setup: The discrete margin is the minimum of six checked no-arbitrage constraint values on a 28 × 28 surface grid.The checks include call-price bounds, strike-slope bounds, strike convexity, and calendar monotonicity.
Reconstruction and admissibility: 0.00646 mean test IV-RMSE was achieved, while selected-model seeds ranged from 0.00639 to 0.00653.The mean test IV-RMSE fell from 0.01017 for the baseline VAE to 0.00646.
Reconstruction and admissibility: 0.974 mean held-out admissibility coexisted with latent-box fractions from 0.334 to 0.421 and Gaussian-prior probabilities from 0.921 to 0.981.Held-out codes and prior samples concentrate near the central admissible region, whereas the uniform box measure weights outer regions more heavily.
Reconstruction and admissibility: Seed 3 combines 0.00645 test IV-RMSE with the smallest held-out admissibility, box-area fraction, and prior admissibility.Its corresponding values are 0.927, 0.334, and 0.921, respectively; strike convexity is active at 2,491 of 2,498 refined boundary points.
Scope and limitations: Positive margin gives local admissibility, not a global no-arbitrage guarantee; discrete-grid positivity does not certify the continuous strike–maturity domain.The scope is further limited for stochastic decoders, where conditional-mean admissibility need not imply admissibility of every draw.
Local correction: 33 of 40 violating latent codes reached the target margin through local ascent, with median displacement 0.872 and median final margin 6.98 × 10^-5.The initial codes had M(z) ≤ −10^-2; endpoint-active constraints included the lower call-price bound, calendar monotonicity, and strike convexity.
Scope and limitations: The computed geometry supports repair formulations but does not itself provide a projection algorithm, and numerical constraints are evaluated only to tolerance.Boundary calculations also depend on grids, derivative stencils, and tolerances, with computational cost increasing in latent dimension.

8 Conclusion

The paper concludes that scalar latent margins and zero-margin boundaries provide a geometry for analyzing deterministic generators of no-arbitrage surfaces. Heston VAE experiments show distinct admissible regions despite similar reconstruction errors, while correction and higher-dimensional extensions remain subject to computational and certification limits.

Main conclusions: A scalar margin defines the admissible latent set and its boundary, while continuity and non-degeneracy yield local stability and zero-margin boundary characterization.Regular boundary components can be computed with a Hamilton–Jacobi level-set equation whose local dynamics approach the zero-margin set.
Main conclusions: The Heston VAE has a central admissible region for every seed, and similar reconstruction errors accompany differences in admissible area and prior probability.Strike convexity determines almost all refined boundary points, and local correction moves 33 of 40 violating codes across the computed boundary.
Future scope: Higher-dimensional latent spaces require more efficient boundary calculations, while continuous-domain guarantees require error estimates relating discrete and continuous margins.Extensions to stochastic generators may require a probabilistic definition of admissibility.

B.1 Continuity of discrete margins

The discrete margin remains continuous when finite-difference operators and denominators satisfy the stated conditions, and a continuous decoder preserves this continuity after composition.

Continuity conditions: Finite-difference operators are linear, and the discrete butterfly operator is continuous where denominators such as w are bounded away from zero.The minimum over finitely many grid points preserves continuity.
Continuity conditions: A continuous decoder makes the latent discrete margin continuous through composition.This transfers surface-level continuity to the latent-space margin.

B.2 Closedness of tolerance-adjusted admissible sets

The tolerance-adjusted admissible set inherits closedness from continuity of the discrete margin, supporting local stability under the adjusted criterion.

Tolerance-adjusted admissibility: If the discrete margin satisfies Mh(z0) > −ε, then z0 is locally stable for the tolerance-adjusted admissibility criterion.The stated result concerns the closed tolerance-adjusted set.

C Viscosity Solutions and Hamilton-Jacobi Equations

The paper treats the latent-space level-set PDE as a first-order Hamilton-Jacobi equation and uses viscosity solutions to retain uniqueness, comparison, and stability when classical solutions fail. Numerical schemes and reinitialization support boundary computation while controlling interface errors and unnecessary margin evaluations.

The latent-space level-set PDE is a first-order Hamilton-Jacobi equation.
Viscosity solutions provide a weak formulation when gradients become discontinuous, preserving comparison and stability.Under suitable conditions, the viscosity solution is unique, exists, and remains stable under locally uniform Hamiltonian perturbations.
A monotone Godunov discretization uses the sign of M to select upwind differences and prevent spurious oscillations near the interface.In two dimensions, one-sided differences are combined according to the sign of Mij.
Reinitialization drives the level-set field toward |∇Φ| = 1, but aggressive implementation can slightly move the zero level set.
Narrow-band updates restrict margin evaluations to a neighborhood of the estimated boundary Γ, while adaptive refinement targets sign changes, large gradients, or near-zero margins.In two dimensions, marching-squares extraction can recover an approximate zero contour; higher dimensions can use local charts, sparse grids, active subspaces, or reduced-dimensional slices.