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Arbitrage-Aware Multi-Step Forecasting of Implied Volatility Surfaces: Modelling Surface Trajectories Using Latent Diffusion
Dominik Manuel Buchegger, Lukas Gonon
TL;DR
Forecasting implied volatility surfaces requires capturing geometry, temporal dependence, uncertainty, and economic admissibility beyond what persistence provides. The paper uses an arbitrage-aware autoencoder with conditional latent diffusion to generate joint 30-day surface–return trajectories, producing realistic scenarios and frequently outperforming persistence in point forecasting while remaining limited by underdispersion and weak one-day performance.
Problem
Forecasting implied volatility surface trajectories requires realistic uncertainty and economic admissibility, while persistence is a demanding benchmark because surfaces are highly persistent.
Method
An arbitrage-aware autoencoder learns low-dimensional surface codes, and a conditional diffusion model generates joint 30-day surface–return trajectories from recent surface history.
Results
The model generates realistic, coherent, essentially arbitrage-free scenarios and frequently outperforms functional persistence in point forecasting.
Takeaways & Limitations
The framework is best viewed as a probabilistic multi-step trajectory model and scenario generator rather than a one-step point forecaster.
Takeaways & Limitations
Generated trajectories are underdispersed, and one-day-ahead point forecasting is weak relative to persistence.
Abstract
from arXiv · showhide
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.
1 Introduction
The paper addresses probabilistic multi-step forecasting of implied volatility surfaces while preserving no-static-arbitrage structure and testing value beyond persistence. It introduces a reproducible latent diffusion framework for joint surface–return trajectories and evaluates its forecasting performance.
- Motivation: Implied volatility surfaces support pricing, hedging, and risk management but require forecasts that preserve shape restrictions, temporal dependence, and predictive value beyond persistence.The surface is persistent, making a carry-forward forecast a demanding benchmark.
- Research gap: Existing work rarely provides probabilistic forecasts of complete implied volatility surface trajectories satisfying these combined requirements.The paper identifies a gap in joint trajectory forecasting with uncertainty and economic admissibility.
- Method: The framework uses an autoencoder to compress daily surfaces into latent codes while regularising the decoder toward no-static-arbitrage consistency.Admissibility is imposed through the reconstruction map rather than a parametric family or post-processing step.
- Method: A conditional diffusion model generates the next 30 days of surfaces jointly with daily equity returns in one non-autoregressive pass, conditioned on recent surface history.The autoencoder handles representation and economic regularisation, while diffusion models the conditional future-path law.
- Contributions: The study contributes a reproducible OptionMetrics-on-WRDS dataset protocol, a joint latent diffusion model, and systematic evaluation against persistence.Together these components form a first step toward a reproducible benchmark.
- Results: The model produces realistic multi-step scenarios and improves substantially upon persistence, while remaining essentially free of static arbitrage.Its principal weakness is underdispersion, with predictive intervals narrower than the realised distribution.
2 Related Work
Prior work has explored low-dimensional representations, deterministic dynamics, neural SDEs, and diffusion models for implied volatility surfaces, but generally does not combine conditional multi-step uncertainty with thorough persistence benchmarking.
- Representation and admissibility: Implied volatility surfaces are commonly represented through low-dimensional factors while models address no-arbitrage using admissible parametric families, soft penalties, or arbitrage-free constructions.Level, skew, and curvature explain most daily variation.
- Dynamic surface forecasting: Most prior machine-learning approaches produce deterministic single-step forecasts and therefore do not provide predictive uncertainty.Earlier generative approaches learn latent spaces or SDE parameterisations but do not jointly solve the complete forecasting problem.
- Dynamic surface forecasting: Existing multi-step approaches include ConvLSTMs, SVI coefficient dynamics, functional autoregression, and neural-SDE models, with differing treatment of admissibility.The closest comparator generates surfaces jointly with price paths and assesses admissibility empirically.
- Dynamic surface forecasting: DYSANOS models arbitrage-free low-dimensional states for dynamic generation, but targets generative market simulation rather than conditional forecasting.It provides a related alternative for combining learned dynamics with admissible representations.
- Latent diffusion: Existing diffusion applications to implied volatility surfaces focus on completion, static generation, exogenous shocks, or one-day-ahead forecasting rather than multi-step conditional trajectories.These applications operate directly on surfaces and remain static or single-step.
- Persistence benchmark: Persistence is rarely confronted even though daily surfaces are highly persistent, so low errors and visual plausibility may mainly reflect copying the latest surface.Few studies evaluate thoroughly against persistence, and fewer beat it convincingly.
3 Theoretical Background
The theoretical background defines implied volatility surfaces from option prices and explains their construction and static-arbitrage constraints. Because market quotes are finite and noisy, surface construction requires interpolation, smoothing, and extrapolation under shape restrictions.
- Options and the implied volatility surface: An option gives its holder the right, but not the obligation, to buy or sell an underlying at a fixed strike and remaining maturity.Options are traded for purposes including speculation and hedging against future uncertainty.
- Options and the implied volatility surface: Implied volatility is the unobservable volatility value that reproduces an observed market option price under the Black–Scholes–Merton model.The supplied passage defines it through matching the model price to the market price.
- Options and the implied volatility surface: The implied volatility surface is the function mapping strike and maturity to implied volatility, whose nonflatness reflects the failure of constant volatility to explain the full option-price cross-section.Black–Scholes–Merton implied volatility remains the standard convention for quoting and comparing option prices.
- Surface construction and static arbitrage: Constructing a continuous implied volatility surface from finite irregular quotes requires interpolation, smoothing, and extrapolation across unobserved strike–maturity combinations.Quotes may also contain bid–ask spreads, nonsynchronous recording, and occasional errors.
- Surface construction and static arbitrage: Static-arbitrage absence provides a model-independent internal-consistency requirement that translates into shape constraints on the time-scaled implied volatility surface.It can be checked from the contemporaneous option-price cross-section without specifying a stochastic model for the underlying.
- Surface construction and static arbitrage: Within the bounded empirical domain, conditions C1–C4 apply, while C5 and C6 concern behaviour outside the domain.C3 excludes butterfly arbitrage, C4 excludes calendar-spread arbitrage, and C5 concerns asymptotic large-moneyness behaviour.
4 Methodology
The methodology constructs standardized, arbitrage-aware implied-volatility surfaces, compresses them into an eight-dimensional latent representation, and models 30-day joint surface–return continuations with conditional diffusion. Anchored displacements, Transformer denoising, and a horizon-specific mean-scaling gate support coherent multi-step forecasting while preserving sampled variability.
- 4.1 Data: The preprocessing pipeline transforms each daily surface into a standardized log representation and augments conditioning information with SPX returns and rolling return-level and volatility summaries.The common observation mask is retained, while transformations and standardization use training statistics only.
- 4.2 Latent Surface Representation: The deterministic RAE-inspired autoencoder compresses each normalized surface into an eight-dimensional latent state and uses regularization so decoded latent changes produce admissible surfaces.The encoder receives the normalized log surface and observation mask, while the architecture combines explicit and implicit regularization.
- 4.2 Latent Surface Representation: A coordinate-based decoder reconstructs surfaces from m–𝜏 coordinates, while smoothness, positivity, butterfly, and calendar-spread constraints guide static-arbitrage control.Butterfly and calendar-spread conditions are evaluated on a denser coordinate set, and violations are penalized in the training objective.
- 4.3 Conditional Latent Trajectory Diffusion: The diffusion model observes 21 trading days and jointly generates 30 future latent surface states and daily returns as a one-sided continuation problem.Anchored displacements place observed and future trajectories in a common coordinate system and focus the target on departures from persistence.
- 4.3 Conditional Latent Trajectory Diffusion: A Transformer denoiser jointly predicts denoising targets for all future steps, using historical anchored-displacement tokens and future tokens containing latent displacements and returns.The forward process uses 500 diffusion steps with a linear variance schedule and v-parameterization.
- 4.3 Conditional Latent Trajectory Diffusion: A frozen diffusion model is followed by a horizon-specific mean-scaling gate that corrects forecast magnitude while preserving sampled residual variability.The average fitted gate value is approximately 0.80, and the gate is trained for 15 epochs against realized latent codes.
5 Results
The model generates realistic, economically admissible multi-step surface and return scenarios, capturing dominant surface geometry, low-dimensional movement factors, and dependence structure. Distributional and point-forecast improvements over persistence emerge mainly at weekly and longer horizons, although generated uncertainty and movement amplitudes are understated.
- Generative realism: The model closely reproduces the dominant smile and term-structure geometry, while smoothing some local movements and mildly attenuating the left wing.Global IV level is reproduced closely, but residual discrepancies are structured and concentrated in local shape.
- Low-dimensional dynamics: The first three principal components explain 91.76% of realised increment variance, and generated loadings align with the realised level, skew, and curvature directions.PC1 captures level movements, PC2 skew, and PC3 curvature.
- Calibration and limitations: Generated factor dispersion becomes increasingly understated at longer horizons, with PC1 standard deviations of 0.769 versus 1.021 at h=30.The model captures dominant movement modes more accurately than their longer-horizon magnitude.
- Joint dependence: The model captures broader joint dependence from h=5 onward and reproduces negative return–surface dependence more closely at weekly-to-monthly horizons than on day one.The immediate leverage relation is too weak, although agreement improves at multi-day horizons.
- Economic admissibility: Generated surfaces are essentially free of economically material static arbitrage, with typical repairs of 1.4 IV bps below local half-spreads.Disabling the no-arbitrage mechanism reduces the arbitrage-free share to about 24% and increases corrections two- to fivefold.
- Distributional forecast quality: Diffusion beats bootstrap persistence in eight of nine surface descriptors and eight of nine grid buckets at h=30, while trajectory distributions improve from weekly horizons onward.At one day, persistence remains better for several descriptors and the multivariate energy score.
- Calibration and limitations: Predictive intervals are undercovered: nominal 50% coverage falls from 0.400 to 0.340 and nominal 95% coverage from 0.856 to 0.811 between h=1 and h=30.Intervals widen over time but not sufficiently; return intervals are closer to the diagonal, with slight undercoverage at their widest levels.
- Point forecasting: Compared with persistence, RMSE improves by 11.69% at short maturities, 6.46% at medium maturities, and 4.99% at long maturities, with gains turning positive around h=4.The central region improves by only 2.45%, and the model is initially worse because it introduces too much movement.
6 Conclusion
The framework provides a foundation for reproducible IVS forecasting benchmarks by generating coherent, essentially arbitrage-free 30-day joint trajectories and often outperforming functional persistence.
- The model generates 30-day joint trajectories of implied volatility surfaces and returns that are realistic, coherent, low-dimensional, and essentially arbitrage-free.
- The framework frequently outperforms functional persistence in point forecasting, while its main limitations are underdispersion and weak one-day-ahead performance.
- The work provides a foundation for a reproducible IVS-forecasting benchmark.
- Future evaluation should compare alternative diffusion models under common data construction and assess metrics and practical value for hedging, risk measurement, and stress testing.