Deep Local Volatility

TL;DR

This paper introduces a deep learning method for interpolating European vanilla option prices that enforces no-arbitrage conditions to produce consistent local volatility surfaces, improving accuracy and reliability.

Contribution

It develops a novel deep learning framework that incorporates no-arbitrage constraints into local volatility surface estimation using modified loss functions and the Dupire formula.

Findings

Effective enforcement of no-arbitrage conditions during training

Accurate local volatility surface reconstruction on real datasets

Enhanced option pricing and Greeks computation

Abstract

Deep learning for option pricing has emerged as a novel methodology for fast computations with applications in calibration and computation of Greeks. However, many of these approaches do not enforce any no-arbitrage conditions, and the subsequent local volatility surface is never considered. In this article, we develop a deep learning approach for interpolation of European vanilla option prices which jointly yields the full surface of local volatilities. We demonstrate the modification of the loss function or the feed forward network architecture to enforce (hard constraints approach) or favor (soft constraints approach) the no-arbitrage conditions and we specify the experimental design parameters that are needed for adequate performance. A novel component is the use of the Dupire formula to enforce bounds on the local volatility associated with option prices, during the network fitting. Our methodology is benchmarked numerically on real datasets of DAX vanilla options.