Operator Deep Smoothing for Implied Volatility

Paper: arXiv 2406.11520

Abstract

We devise a novel method for nowcasting implied volatility based on neural operators. Better known as implied volatility smoothing in the financial industry, nowcasting of implied volatility means constructing a smooth surface that is consistent with the prices presently observed on a given option market. Option price data arises highly dynamically in ever-changing spatial configurations, which poses a major limitation to foundational machine learning approaches using classical neural networks. While large models in language and image processing deliver breakthrough results on vast corpora of raw data, in financial engineering the generalization from big historical datasets has been hindered by the need for considerable data pre-processing. In particular, implied volatility smoothing has remained an instance-by-instance, hands-on process both for neural network-based and traditional parametric strategies. Our general operator deep smoothing approach, instead, directly maps observed data to smoothed surfaces. We adapt the graph neural operator architecture to do so with high accuracy on ten years of raw intraday S&P 500 options data, using a single model instance. The trained operator adheres to critical no-arbitrage constraints and is robust with respect to subsampling of inputs (occurring in practice in the context of outlier removal). We provide extensive historical benchmarks and showcase the generalization capability of our approach in a comparison with classical neural networks and SVI, an industry standard parametrization for implied volatility. The operator deep smoothing approach thus opens up the use of neural networks on large historical datasets in financial engineering.

Complexity vs Empirical Score

• Math Complexity: 8.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced mathematics including neural operators, graph neural operators, and PDE-related theory, reflecting high complexity. It demonstrates strong empirical rigor by backtesting on 10 years of raw intraday S&P 500 options data, comparing against industry standards (SVI) and showcasing robustness, though it lacks explicit code or live trading implementation details.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Implied Volatility Smoothing"] --> B["Data Input<br>Raw S&P 500 Options Data"]
  B --> C["Methodology<br>Graph Neural Operator"]
  C --> D["Computational Process<br>Operator Mapping"]
  D --> E["Key Outcome 1<br>Single Model Instance"]
  D --> F["Key Outcome 2<br>Adheres to No-Arbitrage Constraints"]
  D --> G["Key Outcome 3<br>Robust to Subsampling"]
  E --> H["Final Result<br>Operator Deep Smoothing"]
  F --> H
  G --> H

Cited References (39)

1. Deep smoothing of the implied volatility surface (2020)
2. Deep calibration with random grids — → 2306.11061
3. Variational autoencoders: A hands-off approach to volatility (2021) — link
4. Cboe GLOBAL MARKETS REPORTS TRADING VOLUME FOR DECEMBER AND FULL YEAR 2023 (2024)
5. Deep local volatility (2020)
6. Detecting and Repairing Arbitrage in Traded Option Prices (2020) — link
7. Robust calibration and arbitrage-free interpolation of SSVI slices (2019)
8. The normalizing transformation of the implied volatility smile (2012)

2 of 39 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.