Deep self-consistent learning of local volatility

TL;DR

This paper introduces a deep learning-based algorithm for calibrating local volatility surfaces from market option prices, leveraging neural networks to solve Dupire's PDE and enforce no-arbitrage conditions, resulting in more accurate and smooth volatility estimates.

Contribution

The paper proposes a novel deep self-consistent learning approach that jointly approximates option prices and local volatility, improving calibration accuracy and smoothness over existing methods.

Findings

Reduced interpolation errors

Enhanced smoothness of local volatility surfaces

Improved market option price re-pricing accuracy

Abstract

We present an algorithm for the calibration of local volatility from market option prices through deep self-consistent learning, by approximating both market option prices and local volatility using deep neural networks. Our method uses the initial-boundary value problem of the underlying Dupire's partial differential equation solved by the parameterized option prices to bring corrections to the parameterization in a self-consistent way. By exploiting the differentiability of neural networks, we can evaluate Dupire's equation locally at each strike-maturity pair; while by exploiting their continuity, we sample strike-maturity pairs uniformly from a given domain, going beyond the discrete points where the options are quoted. Moreover, the absence of arbitrage opportunities are imposed by penalizing an associated loss function as a soft constraint. For comparison with existing approaches, the proposed method is tested on both synthetic and market option prices, which shows an improved performance in terms of reduced interpolation and reprice errors, as well as the smoothness of the calibrated local volatility. An ablation study has been performed, asserting the robustness and significance of the proposed method.