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Operator Deep Smoothing for Implied Volatility
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Proposes operator deep smoothing: an interpolating graph neural operator trained on ten years of intraday S&P 500 options to map observed implied volatilities directly to arbitrage-constrained smoothed surfaces.
Problem
Implied volatility smoothing (nowcasting a smooth, arbitrage-free IV surface from irregularly sampled, time-varying option quotes) is conventionally solved by per-surface parametric calibration (e.g., SVI), which is slow and fragile online and does not fit classical fixed-size neural networks.
Approach
Treat smoothing as an operator learning problem with a discretization-invariant graph neural operator: a purely non-local first layer with Nystrom-subsampled in-neighborhoods lets one trained model map arbitrarily sized and placed option quotes to smooth surfaces, with no-arbitrage penalties added to the loss.
Key results
- Operator with monthly finetuning achieves MAPE around 0.5% on 2021 intraday S&P 500 data, while SVI fluctuates between 1% and 2%.
- A single GNO with about 102,529 trainable parameters smooths the full 2012-2021 history, versus 61,454 SVI slice instances.
- On end-of-day January 2021 data, the intraday-trained operator yields delta_abs of 0.00272 for SPX and 0.01629 for DJX, with near-zero arbitrage losses.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use OpDS instead of SVI for implied-volatility smoothing | S&P 500 options · S&P 500 options, 2021 test period | MAPE: around 0.5% vs. between 1% and 2% | Helped |
| Retain local transformations in subsequent GNO layers after the purely non-local first layer (ablation) | Interpolating GNO · Implied-volatility smoothing | performance | Helped |
| Subsample 50% of input points per surface | OpDS (GNO) · S&P 500 options, Jan-Apr 2018 backtest | smoothing accuracy | No clear effect |
Surprising findings
- A purely non-local first GNO layer substantially outperformed an alternative purely non-local architecture in experiments, showing local components matter.
- Operator trained only on intraday S&P 500 data generalizes to end-of-day data of NDX, DJX and RUT without retraining.
Limitations
- Loses the interpretability of parameters compared with SVI.
- Proprietary CBOE intraday data is expensive and not released, limiting reproduction.
- GNO outputs are not theoretically guaranteed smooth, requiring regularization and Nystrom subsampling that breaks guaranteed smoothness.
Experimental setup
Datasets: CBOE S&P 500 Index Option Quotes (20-minute intraday, 2012-2021) OptionMetrics Ivy DB US end-of-day options data (SPX, NDX, DJX, RUT)
Benchmarks: CBOE S&P 500 intraday test set 2021 OptionMetrics end-of-day SPX, NDX, DJX, RUT January 2021
Metrics: mean absolute percentage error (delta_abs) bid-ask spread-relative error (delta_spr) butterfly arbitrage loss calendar arbitrage loss
Baselines: SVI classical neural network smoothing (per-surface NN) SSVI
Builds on: graph neural operator Fourier neural operator SVI SSVI
Closest papers by meaning
- HyperIV: Real-time Implied Volatility Smoothing (ICML 2025 · similarity 0.89)
- From GARCH to Neural Network for Volatility Forecast (AAAI 2024 · similarity 0.83)
- Towards Financial World Modeling (NeurIPS 2026 · similarity 0.83)
- Market-GAN: Adding Control to Financial Market Data Generation with Semantic Context (AAAI 2024 · similarity 0.83)
- StockMixer: A Simple Yet Strong MLP-Based Architecture for Stock Price Forecasting (AAAI 2024 · similarity 0.82)
- MASTER: Market-Guided Stock Transformer for Stock Price Forecasting (AAAI 2024 · similarity 0.82)
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