This paper introduces a novel approach to option pricing by combining neural stochastic differential equations (SDEs) with classical martingale pricing theory. The key idea is to treat neural SDEs as universal approximators for Itô processes, allowing the underlying asset price dynamics to be learned directly from data without parametric assumptions. The authors propose training the neural SDE using the Wasserstein-1 distance as a loss function, which measures the discrepancy between the empirical distributions of the model and target processes. This method, a variation of SDE-GAN, avoids the need for adversarial training and provides a direct gradient-based optimization. A central conjecture is that the error in the computed option price is bounded by the Wasserstein-1 distance achieved during training, leveraging the Lipschitz property of option payoffs.

Experiments on synthetic data from geometric Brownian motion demonstrate that the neural SDE can accurately learn the underlying dynamics and produce European call option prices that closely match the Black-Scholes formula. The results also show a strong correlation between the training loss and option pricing error, providing empirical support for the conjectured bound. While the current implementation is limited to one dimension due to computational constraints of the Wasserstein distance, the framework offers a flexible, unsupervised alternative to traditional option pricing models. Future work includes extending to higher dimensions via dual formulations or GAN-based approaches, and proving the conjectured bound in full generality.