# Deep Smoothing of the Implied Volatility Surface

## TL;DR

This paper introduces a neural network approach to fit and predict implied volatility surfaces that guarantees arbitrage-free prices, improves with sparse or erroneous data, and can be integrated with classical models for better accuracy.

## Contribution

The paper proposes a neural network method with arbitrage constraints for improved IVS modeling, compatible with traditional models, and capable of quantifying uncertainty in sparse data regions.

## Key findings

- Effective in sparse or noisy data scenarios
- Guarantees arbitrage-free implied volatility surfaces
- Enhances classical models with neural network corrections

## Abstract

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.