Controllable Generation of Implied Volatility Surfaces with Variational Autoencoders
Controllable Generation of Implied Volatility Surfaces with Variational Autoencoders
This paper presents a novel deep generative framework for controllably synthesizing implied volatility surfaces (IVSs) using a variational autoencoder (VAE). The key innovation is the disentanglement of latent space into two components: controllable variables corresponding to financially interpretable shape features (volatility level, slope, curvature, and term-structure slope) and residual latent variables capturing additional variation. This design allows users to specify desired feature values while maintaining diversity in generated surfaces. The shape features are quantified via bivariate polynomial regression at the anchor point (short-maturity at-the-money), providing a principled way to extract and control these attributes.
Methodologically, the framework employs a conditional VAE where the decoder takes both the controllable features and sampled residual latents as inputs. Training maximizes an evidence lower bound (ELBO) with a β-weighting on the KL divergence to balance reconstruction and regularization. A ResNet architecture is used for both encoder and decoder to improve performance. To address potential no-arbitrage violations (butterfly and calendar spread arbitrage) that may arise when sampling residual latents from tail regions, the authors propose a post-generation latent-space repair algorithm. This algorithm adjusts only the residual latent variables to minimize arbitrage violations while keeping the specified features fixed, ensuring market validity without compromising controllability.
Numerical experiments demonstrate that the controllable VAE achieves high accuracy in generating IVSs with desired single or multiple features, and the repair algorithm effectively removes arbitrage violations. The framework combines interpretability, controllability, and flexibility, making it suitable for applications such as stress testing, risk management, and scenario generation in quantitative finance. Compared to black-box generative models, this approach offers explicit control over economically meaningful surface characteristics, bridging the gap between data-driven generation and financial interpretability.
Highlights
- 1Proposes a controllable VAE framework for generating implied volatility surfaces (IVSs) with explicit control over financially interpretable shape features (level, slope, curvature, term-structure).
- 2Introduces a disentangled latent representation: controllable variables for target features and residual latent variables for remaining diversity.
- 3Develops a post-generation latent-space repair algorithm to remove static no-arbitrage violations while preserving specified features.
- 4Demonstrates high accuracy in both single- and multi-feature control settings, enabling rapid generation of realistic IVSs.
Methods
- MVariational Autoencoder (VAE) with a conditional architecture that separates controllable features (y) from residual latent variables (z).
- MBivariate polynomial regression at the anchor point (τ→0+, m=0) to quantify shape features from observed IVSs.
- MReparameterization trick for training the VAE with Gaussian priors and posteriors.
- MPost-generation latent-space optimization to correct arbitrage violations (calendar spread and butterfly) while fixing controllable features.
Results
- RThe controllable VAE generates realistic IVSs that match specified feature values (level, slope, curvature, term-structure) with high accuracy.
- RThe framework preserves diversity through residual latent variables, producing varied surfaces even with identical feature specifications.
- RThe post-generation repair algorithm effectively removes arbitrage violations without altering the controlled features, maintaining market validity.
- RResNet architectures significantly improve generative performance compared to plain feedforward networks.
- RThe method outperforms black-box generators in terms of interpretability, controllability, and flexibility for scenario design.
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