# Arbitrage-Free Implied Volatility Surface Generation with Variational Autoencoders

arXiv: https://arxiv.org/abs/2108.04941 · HTML: https://mlsanity.com/paper/2108.04941/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 5 of its 38 references that are in the map.
- First posted on arXiv 10 Aug 2021.

## Builds on (most cited first, 5 of 5)

- [Scalable Gradients for Stochastic Differential Equations](https://mlsanity.com/paper/2001.01328/) (2020)
- [Deep Smoothing of the Implied Volatility Surface](https://mlsanity.com/paper/1906.05065/) (2019)
- [Arbitrage-free neural-SDE market models](https://mlsanity.com/paper/2105.11053/) (2021)
- [Online Adaptive Machine Learning Based Algorithm for Implied Volatility Surface Modeling](https://mlsanity.com/paper/1706.01833/) (2017)
- [Deeply Learning Derivatives](https://mlsanity.com/paper/1809.02233/) (2018)

## Built on by (most cited first, 9 of 9)

- [Reinforcement Learning with Dynamic Convex Risk Measures](https://mlsanity.com/paper/2112.13414/) (2021)
- [Multi-Asset Spot and Option Market Simulation](https://mlsanity.com/paper/2112.06823/) (2021)
- [FuNVol: A Multi-Asset Implied Volatility Market Simulator using Functional Principal Components and Neural SDEs](https://mlsanity.com/paper/2303.00859/) (2023)
- [DYSANOS Generative Dynamic Smooth Arbitrage-free Non-parametric Option Surfaces](https://mlsanity.com/paper/2608.12587/) (2026)
- [A Risk-Neutral Neural Operator for Arbitrage-Free SPX-VIX Term Structures](https://mlsanity.com/paper/2511.06451/) (2025)
- [ProbRes: Volatility Learning for Probabilistic Time-Series Forecasting](https://mlsanity.com/paper/2606.02117/) (2026)
- [PIVOT: Bridging Black-Scholes Implied-Volatility and Price Objectives via Differentiable J"ackel Operator](https://mlsanity.com/paper/2606.17065/) (2026)
- [Arbitrage-Aware Multi-Step Forecasting of Implied Volatility Surfaces: Modelling Surface Trajectories Using Latent Diffusion](https://mlsanity.com/paper/2608.22478/) (2026)
- [Latent-Space No-Arbitrage Geometry of Generative Models for Implied Volatility Surfaces](https://mlsanity.com/paper/2609.00332/) (2026)

## Abstract (arXiv)

We propose a hybrid method for generating arbitrage-free implied volatility (IV) surfaces consistent with historical data by combining model-free Variational Autoencoders (VAEs) with continuous time stochastic differential equation (SDE) driven models. We focus on two classes of SDE models: regime switching models and L'evy additive processes. By projecting historical surfaces onto the space of SDE model parameters, we obtain a distribution on the parameter subspace faithful to the data on which we then train a VAE. Arbitrage-free IV surfaces are then generated by sampling from the posterior distribution on the latent space, decoding to obtain SDE model parameters, and finally mapping those parameters to IV surfaces. We further refine the VAE model by including conditional features and demonstrate its superior generative out-of-sample performance.