Penalty-Based No-Arbitrage Enforcement for Neural Volatility Surfaces Under Sparse Strike Data

Neural network approaches to implied volatility surface fitting tend to violate no-arbitrage conditions when trained on sparse or noisy option price data. This paper introduces a penalty-based loss formulation that encodes static no-arbitrage constraints — including calendar spread monotonicity and butterfly spread positivity — directly into the training objective of a neural network surface model. The approach requires no explicit constraint projection or post-processing step: the constraints are soft-enforced through differentiable penalty terms added to the empirical loss. We evaluate the method on option price panels with varying degrees of sparsity and demonstrate that the penalty formulation substantially reduces arbitrage violations compared to unconstrained baselines while retaining competitive interpolation accuracy. The implementation is fully reproducible via the accompanying open-source repository and a live interactive demo.