Controllable Generation of Implied Volatility Surfaces with Variational Autoencoders

Paper: arXiv 2509.01743

Authors: Jing Wang, Shuaiqiang Liu, Cornelis Vuik

Abstract

This paper presents a deep generative modeling framework for controllably synthesizing implied volatility surfaces (IVSs) using a variational autoencoder (VAE). Unlike conventional data-driven models, our approach provides explicit control over meaningful shape features (e.g., volatility level, slope, curvature, term-structure) to generate IVSs with desired characteristics. In our framework, financially interpretable shape features are disentangled from residual latent factors. The target features are embedded into the VAE architecture as controllable latent variables, while the residual latent variables capture additional structure to preserve IVS shape diversity. To enable this control, IVS feature values are quantified via regression at an anchor point and incorporated into the decoder to steer generation. Numerical experiments demonstrate that the generative model enables rapid generation of realistic IVSs with desired features rather than arbitrary patterns, and achieves high accuracy across both single- and multi-feature control settings. For market validity, an optional post-generation latent-space repair algorithm adjusts only the residual latent variables to remove occasional violations of static no-arbitrage conditions without altering the specified features. Compared with black-box generators, the framework combines interpretability, controllability, and flexibility for synthetic IVS generation and scenario design.

Complexity vs Empirical Score

• Math Complexity: 8.0/10
• Empirical Rigor: 6.5/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced deep learning architectures (VAEs, conditional VAEs) and incorporates theoretical financial constraints (no-arbitrage conditions) which require complex mathematical handling. It presents numerical experiments on synthetic data (Heston/SABR models) with quantitative metrics like regression error and arbitrage violation rates, demonstrating a strong empirical component for a methodological paper.

Research Flowchart

  flowchart TD
  A["Research Goal: Controllable IVS Generation<br>via VAEs with Interpretable Features"] --> B["Methodology: Disentanglement & Control"]
  B --> C["Data: Implied Volatility Surface Dataset"]
  C --> D["VAE Framework: Encoder & Decoder"]
  D --> E["Control Mechanism: Target Features<br>Embedded via Anchor Regression"]
  D --> F["Residual Latent Variables<br>Preserve Shape Diversity"]
  E --> G["Output: Synthetic IVSs<br>with Specific Features<br>Slope, Curvature, Vol-Level"]
  F --> G
  G --> H["Outcomes: High Accuracy & Market Validity"]
  H --> I["Post-Generation: Latent-Space Repair<br>for No-Arbitrage Conditions"]
  I --> J["Key Findings: Flexible, Interpretable<br>Controllable Generation for Scenarios"]

Cited References (68)

1. The moment formula for implied volatility at extreme strikes (2004)
2. Dynamics of implied volatility surfaces (2002)
3. The Volatility Surface: A Practitioner’s Guide (2006)
4. Does net buying pressure affect the shape of implied volatility functions? (2004)
5. The moment formula for implied volatility at extreme strikes (2002)
6. The term structure of variance swap rates and optimal variance swap investments (2010)
7. Pricing event risk: Evidence from concave implied volatility curves (2025)
8. How important is the term structure in implied volatility surface modeling? Evidence from foreign exchange options (2011) — link

2 of 68 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.