Deep self-consistent learning of local volatility

Zhe Wang, Ameir Shaa, Nicolas Privault and Claude Guet

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 partial differential equation solved by the parameterized option prices to provide 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 is 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, and it 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 is performed, asserting the robustness and significance of the proposed method.