Beyond Surrogate Modeling: Learning the Local Volatility Via Shape Constraints

TL;DR

This paper compares Gaussian process and neural network methods for no-arbitrage interpolation of option prices, focusing on local volatility surface learning with constraints, and demonstrates their respective advantages in accuracy and smoothness.

Contribution

It introduces two machine learning approaches for arbitrage-free local volatility surface estimation, highlighting their properties and performance relative to industry standards.

Findings

GP approach is arbitrage-free and has best calibration error

NN approach produces smoother local volatility surfaces

NN approach shows improved backtesting performance

Abstract

We explore the abilities of two machine learning approaches for no-arbitrage interpolation of European vanilla option prices, which jointly yield the corresponding local volatility surface: a finite dimensional Gaussian process (GP) regression approach under no-arbitrage constraints based on prices, and a neural net (NN) approach with penalization of arbitrages based on implied volatilities. We demonstrate the performance of these approaches relative to the SSVI industry standard. The GP approach is proven arbitrage-free, whereas arbitrages are only penalized under the SSVI and NN approaches. The GP approach obtains the best out-of-sample calibration error and provides uncertainty quantification.The NN approach yields a smoother local volatility and a better backtesting performance, as its training criterion incorporates a local volatility regularization term.