Arbitrage-Aware Probabilistic Multi-Step Forecasting of Implied Volatility Surfaces: Joint Generation of Surface-Return Trajectories Using Latent Diffusion

The gist: A conditional latent diffusion framework for generating joint 30-step trajectories of implied volatility surfaces and underlying returns is proposed to model option market evolution while preserving economic admissibility.

How it works

The proposed method employs a twostage latent diffusion model where an autoencoder compresses daily surfaces into low-dimensional codes, and a conditional diffusion model generates the future joint evolution The autoencoder learns a low-dimensional surface representation, while the diffusion model captures the conditional joint evolution. This separation assigns representation learning and economic regularisation to the autoencoder, and the conditional law over future paths to the diffusion model.

Data Construction and Representation

The data protocol involves constructing daily implied volatility surfaces from OptionMetrics data on WRDS, covering January 2000 to August 2025 A pretrained 3 graph neural operator maps the irregular (BSM) IV quote set to a smooth, essentially static–arbitrage-free surface on prescribed coordinates. Each daily t–IVS is transformed into a normalized log surface, and this is then compressed into an eight-dimensional latent state using a deterministic, RAE-inspired autoencoder. The training objective includes reconstruction terms and latent regularisation terms to guide the model towards admissible surfaces.

Latent Trajectory Generation

The conditional diffusion model generates the next 30 days of surfaces jointly with daily equity returns in a single non-autoregressive pass, conditional on recent surface history. The latent representation uses anchored displacements, where persistence of the latest observed state corresponds to future anchored displacements remaining at their value at the forecast origin. A mean-scaling gate is fitted to correct the predictive mean by adding a term related to persistence, such as zˆ scaled h = zˆ persistence + λbh (zˆ mean h − zˆ persistence).

Evaluation and Results

The framework is evaluated on SPX surfaces, generating realistic probabilistic multi-step scenarios while also outperforming the persistence benchmark in point forecasting. The generated trajectories preserve the dominant smile and term-structure geometry, recover the realised level, skew, and curvature factors of surface movements. While underdispersion is a limitation—predictive intervals widen with the horizon but remain too narrow—the predictive mean also contains information not available from persistence. The model overtakes persistence from approximately weekly horizons onward, with gains being strongest at short maturities and in the wings.

Economic Admissibility and Distributional Quality

Arbitrage restrictions are evaluated using finite differences on the grid, showing virtually no calendar violations and only sparse butterfly violations. The continuous ranked probability score (CRPS) shows that diffusion outperforms bootstrap persistence in eight (of nine) surface descriptors by horizon 10, indicating better distributional improvements at medium and longer horizons. However, the predictive mean's gains are driven by horizondependent shape adjustments rather than uniform level shifts. The return channel matches the zero-return baseline, producing informative distributions without implying return predictability.

Conclusion

The work introduces a conditional latent diffusion framework for generating 30-day joint trajectories of implied volatility surfaces and returns, which produces realistic, coherent, low-dimensional, and essentially arbitrage-free scenarios. Its main limitations are underdispersion and weak one-day-ahead performance, reinforcing that the framework is best viewed as a probabilistic trajectory model rather than a one-step point forecaster. This work provides a foundation for a reproducible IVS-forecasting benchmark. More fundamentally, future work should identify which metrics are most appropriate for evaluating functional trajectories. Applications to hedging, risk measurement, and stress testing should then assess whether statistical gains translate into practical value.

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Arbitrage-Aware Multi-Step Forecasting of Implied Volatility Surfaces

Surfaces

Modelling Surface Trajectories Using Latent Diffusion

Dominik Manuel Buchegger

University of St.Gallen

St.Gallen, Switzerland

Lukas Gonon

University of St.Gallen

Abstract

Implied volatility surfaces summarise the option market and are central to many financial applications. Forecasting their future evolution requires modelling two-dimensional geometry, temporal dependence, and predictive uncertainty while preserving economic admissibility. We propose a conditional latent diffusion framework for generating joint 30-step trajectories of implied volatility surfaces and underlying returns. An arbitrage-aware autoencoder learns a low-dimensional surface representation, while the diffusion model captures the conditional joint evolution. Evaluated on SPX surfaces, the framework generates realistic probabilistic multi-step scenarios while also outperforming the persistence benchmark in point forecasting.

Keywords

Implied volatility surfaces, probabilistic forecasting, latent diffusion models, trajectory generation, static arbitrage, option markets

1 Introduction

The implied volatility surface (IVS) summarises option-implied uncertainty across moneyness and maturity and is a core input to pricing, hedging, and risk management. Although the shape of the IVS varies across assets and over time, three properties frame the modelling problem. First, an admissible IVS must satisfy shape restrictions that rule out static arbitrage. Second, a forecast of the IVS trajectory, the surface’s evolution over multiple days, must reproduce the realised dependence structure. Third, the IVS is generally persistent, so its current state already provides a strong prediction of its future evolution. A useful model must demonstrate value beyond carrying the most recent surface forward.

Together, these properties make point-forecast accuracy insufficient as the sole modelling target. A model that merely copies the current IVS can achieve low errors, especially at short horizons, while learning little about the conditional law of future surface movements. A useful forecasting model must instead respect the economic structure of option markets, generate informative trajectories with realistic uncertainty, and demonstrate predictive value relative to persistence rather than merely visual plausibility or realistic unconditional sampling.

Existing work does not jointly address these requirements. In particular, probabilistic forecasting of complete IVS trajectories has received almost no attention. We address this gap with a twostage latent diffusion model. An autoencoder compresses daily surfaces into latent codes, while its decoder is regularised towards no-static–arbitrage consistency, making admissibility a property of the reconstruction map rather than of a parametric family or a post-processing step. On these latent codes, a conditional diffusion model generates the next 30 days of surfaces jointly with daily equity returns in a single non-autoregressive pass, conditional on the recent surface history. This separation assigns representation learning and economic regularisation to the autoencoder, and the conditional law over future paths to the diffusion model.

Our contribution is twofold. First, we provide a transparent, reproducible protocol for constructing IVS forecasting datasets from OptionMetrics data on WRDS, allowing licensed users to recreate a common evaluation setting. Second, we develop a latent diffusion model for conditional, multi-step generation of joint IVS–return trajectories and evaluate it systematically against persistence. Code is available on GitHub.

The results support the model primarily as a probabilistic scenario generator and multi-step trajectory model, rather than as a one-day point forecaster. Generated surfaces preserve the dominant smile and term-structure geometry, recover the realised level, skew, and curvature factors of surface movements. These trajectories improve substantially upon persistence, while the generated surfaces remain essentially free of static arbitrage. The principal limitation is underdispersion: predictive intervals widen with the horizon but remain too narrow, indicating that the model captures the directions and dependence of future movements more accurately than their amplitude. Beyond this distributional performance, the predictive mean also contains information not available from persistence. Persistence remains superior at the one-day horizon, but the model overtakes it from approximately weekly horizons onward. Gains are strongest at short maturities and in the wings, whereas the stable long-maturity centre remains difficult to improve upon.

2 Related Work

Representation and admissibility. Implied volatility surfaces are empirically low-dimensional, with level, skew, and curvature explaining most daily variation [13]. Useful representations must also respect (static) no-arbitrage restrictions [31]. Learnt models address this either by operating within an admissible parametric family [27] or by combining a flexible representation with soft arbitrage penalties [1, 37]. SANOS provides a recent nonparametric alternative that is arbitrage-free by construction [6].

Dynamic surface forecasting. Most prior machine-learning approaches produce deterministic single-step forecasts. Zhang et al. [39] compress the surface, forecast its low-dimensional features, and reconstruct. Cao et al. [7] regress expected IVS changes on returns, moneyness, maturity, and VIX. Neither yields predictive uncertainty. Generative work began without dynamics.

1. Bold surface geometry reproduction: The model closely reproduces the dominant smile and term-structure geometry, allowing it to generate realistic surfaces that preserve the dominant smile and term-structure geometry in multi-step trajectories.

2. Conditional trajectory generation: The system can generate joint 30-step trajectories of implied volatility surfaces and underlying returns conditional on recent history, moving beyond deterministic single-step forecasts.

3. Economic admissibility enforcement: The framework incorporates constraints to ensure admissibility by construction, as the autoencoder is regularised towards no-static–arbitrage consistency, making this a property of the reconstruction map rather than a post-processing step.

4. Low-dimensional representation learning: The autoencoder compresses daily surfaces into an eight-dimensional latent state while simultaneously enforcing no-static–arbitrage conditions during training, optimizing the trade-off between information retention and economic consistency.

5. Probabilistic scenario generation: The model functions as a probabilistic scenario generator and multi-step trajectory model, generating trajectories where the predictive intervals are widening with the horizon but remain too narrow, indicating that the model captures the directions and dependence of future movements more accurately than their amplitude.

6. Horizon-dependent predictive value: The system demonstrates gains over persistence from approximately weekly horizons onward, specifically showing that the model overtakes it from approximately weekly horizons onward in point forecasting.

Implied volatility surfaces summarise the option market and are central to many financial applications. Probabilistic forecasting of their future evolution requires modelling two-dimensional geometry, temporal dependence, and predictive uncertainty while preserving economic admissibility. We propose a conditional latent diffusion framework for generating joint 30-step trajectories of implied volatility surfaces and underlying returns. An arbitrage-aware autoencoder learns a surface representation, while diffusion captures the evolution. Evaluated on SPX surfaces, the framework generates realistic probabilistic multi-step scenarios. Its predictive mean also outperforms persistence beyond the shortest horizons.