This study presents a comparative analysis of supervised and unsupervised neural network approaches for pricing multi-asset options, with a particular focus on exchange options involving two underlying assets. The supervised methodology utilises data-driven neural networks trained on simulated option prices. In contrast, the unsupervised approach employs Physics-Informed Neural Networks that incorporate the governing partial differential equation along with boundary and initial conditions directly into the loss function. Addressing a notable gap in the literature, this work evaluates the relative efficiency of PINNs and supervised neural networks in pricing multi-asset derivatives. Additionally, we propose a novel grid search framework to systematically identify optimal hyperparameters for both approaches, enabling a fair comparison in terms of accuracy, computational speed, and overall efficiency. Empirical results indicate that the supervised neural network outperforms the PINNs approach, achieving superior accuracy and significantly reduced execution times, which underscores its suitability for real-time financial applications. However, the findings also highlight that PINNs remain a valuable alternative in scenarios where data availability is limited, offering a flexible model-free solution for complex option pricing problems.