  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,
respectively. 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 the 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.
