# Deep Smoothing of the Implied Volatility Surface

arXiv: https://arxiv.org/abs/1906.05065 · HTML: https://mlsanity.com/paper/1906.05065/

## Key facts

- Cited by 19 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 12 Jun 2019.

## Built on by (most cited first, 19 of 19)

- [Arbitrage-Free Implied Volatility Surface Generation with Variational Autoencoders](https://mlsanity.com/paper/2108.04941/) (2021)
- [FuNVol: A Multi-Asset Implied Volatility Market Simulator using Functional Principal Components and Neural SDEs](https://mlsanity.com/paper/2303.00859/) (2023)
- [Multi-Domain Causal Discovery in Bijective Causal Models](https://mlsanity.com/paper/2504.21261/) (2025)
- [Data-Efficient Realized Volatility Forecasting with Vision Transformers](https://mlsanity.com/paper/2511.03046/) (2025)
- [Recurrent Neural Networks for Dynamic VWAP Execution: Adaptive Trading Strategies with Temporal Kolmogorov-Arnold Networks](https://mlsanity.com/paper/2502.18177/) (2025)
- [Interpolation of Missing Swaption Volatility Data using Gibbs Sampling on Variational Autoencoders](https://mlsanity.com/paper/2204.10400/) (2022)
- [VWAP Execution with Signature-Enhanced Transformers: A Multi-Asset Learning Approach](https://mlsanity.com/paper/2503.02680/) (2025)
- [A Risk-Neutral Neural Operator for Arbitrage-Free SPX-VIX Term Structures](https://mlsanity.com/paper/2511.06451/) (2025)
- [Shallow Representation of Option Implied Information](https://mlsanity.com/paper/2603.17151/) (2026)
- [ProbRes: Volatility Learning for Probabilistic Time-Series Forecasting](https://mlsanity.com/paper/2606.02117/) (2026)
- [Neural Slack Variables for Shape Constraints](https://mlsanity.com/paper/2606.13803/) (2026)
- [Beyond the Smile: A Hybrid Convolutional VAE for Crypto Volatility Surfaces](https://mlsanity.com/paper/2606.16961/) (2026)
- [PIVOT: Bridging Black-Scholes Implied-Volatility and Price Objectives via Differentiable J"ackel Operator](https://mlsanity.com/paper/2606.17065/) (2026)
- [Inverse Learning of Latent Risk-Neutral Densities from Irregular Option Quotes](https://mlsanity.com/paper/2607.27188/) (2026)
- [Amortizing the Calibration Triple: A Projection-Consistent Neural Operator for Local-Stochastic Volatility](https://mlsanity.com/paper/2608.01217/) (2026)
- [Arbitrage-Aware Multi-Step Forecasting of Implied Volatility Surfaces: Modelling Surface Trajectories Using Latent Diffusion](https://mlsanity.com/paper/2608.22478/) (2026)
- [Graph-Based Modeling of Financial Volatility Dynamics](https://mlsanity.com/paper/2608.26127/) (2026)
- [Latent-Space No-Arbitrage Geometry of Generative Models for Implied Volatility Surfaces](https://mlsanity.com/paper/2609.00332/) (2026)
- [Asymptotically-informed neural networks for Black-Scholes implied volatility computation](https://mlsanity.com/paper/2609.05491/) (2026)

## Abstract (arXiv)

We present a neural network (NN) approach to fit and predict implied volatility surfaces (IVSs). Atypically to standard NN applications, financial industry practitioners use such models equally to replicate market prices and to value other financial instruments. In other words, low training losses are as important as generalization capabilities. Importantly, IVS models need to generate realistic arbitrage-free option prices, meaning that no portfolio can lead to risk-free profits. We propose an approach guaranteeing the absence of arbitrage opportunities by penalizing the loss using soft constraints. Furthermore, our method can be combined with standard IVS models in quantitative finance, thus providing a NN-based correction when such models fail at replicating observed market prices. This lets practitioners use our approach as a plug-in on top of classical methods. Empirical results show that this approach is particularly useful when only sparse or erroneous data are available. We also quantify the uncertainty of the model predictions in regions with few or no observations. We further explore how deeper NNs improve over shallower ones, as well as other properties of the network architecture. We benchmark our method against standard IVS models. By evaluating our method on both training sets, and testing sets, namely, we highlight both their capacity to reproduce observed prices and predict new ones.