Arbitrage-Aware Multi-Step Forecasting of Implied Volatility Surfaces: Modelling Surface Trajectories Using Latent Diffusion
Dominik Manuel Buchegger, Lukas Gonon
q-fin.MF, cs.LG
2026-08-24
A St. Gallen latent-diffusion model generates 30-day SPX implied-vol and return paths; RMSE beats persistence by 6.1% overall, but loses by 15.7% at one day.
An implied volatility surface (IVS) spreads option-implied uncertainty across moneyness and maturity. Desks use it for pricing, hedging, and risk scenarios. An admissible surface also has to obey static no-arbitrage shape constraints: the same-day option cross-section cannot hide a free lunch.
Most machine-learning work still does one-step point forecasts, or samples a surface that merely looks plausible. Daily IVS paths are highly persistent, so copying yesterday already produces low error. Papers that skip that baseline look better than they are. This one asks for a joint 30-trading-day trajectory of surfaces and underlying returns, with no-arbitrage checks, scored against persistence.
Daily SPX surfaces come from OptionMetrics end-of-day quotes, January 2000 to August 2025, 6,429 days. Irregular quotes are mapped onto a 170-point log-forward-moneyness × maturity grid by a pretrained graph neural operator (operator deep smoothing). Splits are chronological: train through 2021 (5,511 days), validate on 2022 (251), test from 2023 (667).
Two stages. First, a deterministic autoencoder inspired by regularised autoencoders. A residual convolutional encoder compresses the log time-scaled surface into an 8-dimensional code. The decoder is a coordinate neural field: any (moneyness, maturity) query returns one volatility value, with the latent code FiLM-modulating each layer. The loss is masked reconstruction MSE plus latent L2, a butterfly-arbitrage penalty, and a calendar-spread penalty. Smooth ops and a log parametrisation take care of differentiability and positivity. Training runs 300 epochs.
Second, conditional diffusion in that latent space. At each origin the model sees 21 days and generates the next 30 latent displacements and daily returns in one non-autoregressive pass. Displacements are anchored at the start of the window, so copying the latest surface means future displacements stay put; the training target is the departure from persistence. The denoiser is a 4-block prenorm Transformer over a 52-token sequence (current state, 21 history tokens, 30 noisy future tokens). The forward process uses 500 steps with a linear variance schedule and v-parameterisation, trained for 100 epochs.
On validation, the diffusion mean overstated movement. The authors freeze the diffuser and fit a horizon-specific scaling gate that pulls the mean toward persistence; the average fitted factor is about 0.80. At sampling they shift only the mean and keep residuals, so path dispersion is not pinched. Each test origin draws 1,000 30-day trajectories with DDPM.
The model mean reproduces the smile and term structure. Residuals are small and structured: a slightly damped left wing and a lifted long-maturity centre, already present in autoencoder reconstructions.
Surface motion is low-dimensional. The first three principal components of realised increments explain 91.76% of variance; generated loadings align at 0.996 / 0.962 / 0.831. Amplitude lags. At horizon 21 the PC1 standard deviation is 0.669 generated versus 0.923 realised.
| Metric | Model | Baseline |
| RMSE over all 30 steps | 0.01262 | persistence 0.01343 (+6.09%) |
| 1-day RMSE | 0.00619 | persistence 0.00535 (-15.72%) |
| 30-day RMSE | 0.01460 | persistence 0.01567 (+6.83%) |
| Short-maturity RMSE gain | 11.69% | centre of the surface only 2.45% |
On admissibility, 88.1% of individual samples are fully clean, 98.4% of ensemble means. Turning the arbitrage penalties off drops the clean share to about 24%. Typical repair is 1.4 IV basis points, below half the bid–ask. The negative return–vol link is too weak at one day (realised -0.79, generated -0.46) and close from weekly horizons on.
Distributional skill uses CRPS. Versus deterministic persistence, average IV improves 25.5% / 31.7% / 33.2% at horizons 1 / 10 / 30. Versus a bootstrap of historical change paths, the one-day gain shrinks to 4.4% and the 10-day gain to 10.7%. Nominal 50% interval coverage falls from 0.400 to 0.340; 95% coverage from 0.856 to 0.811. Intervals widen with horizon, just not enough.
The return channel matches a zero-return baseline. It belongs to the joint scenario generator; it does not claim return predictability.
SPX options already bake visible information into today's surface, so beating persistence at one day is hard. Recasting the job as a multi-step scenario generator is the honest framing: directions, factor structure, and return–vol dependence are learned; amplitude is not. That is closer to how risk, stress, and hedge desks would use such a model than a one-day RMSE contest.
Code and the WRDS reconstruction protocol are public. This forecasting problem still lacks a shared benchmark; pinning persistence as the mandatory baseline is the useful part.
This is not a trading signal. One-day point forecasts lose, intervals are tight, and only SPX was tested.
The authors name underdispersion and weak one-day skill as the main failures. Autoencoder reconstruction RMSE is already 0.0039, a large share of short-horizon error. A richer autoencoder makes the diffusion harder; they did not find a better split.
Comparisons are almost only versus persistence. DYSANOS, SANOS diffusion, neural SDEs, and ConvLSTMs are discussed, not evaluated under the same protocol. The test window is a single post-2022 stretch. The scaling gate is a post-hoc correction, which means the raw diffusion mean systematically overshoots. The return channel has no predictive content, and no hedge or risk backtest shows that the joint paths are worth money.