This thesis investigates how well two generative models can produce realistic synthetic financial time series. Both models are trained on daily S&P 500 returns and evaluated using the same dataset and the same criteria. The comparison focuses on whether the models reproduce five key statistical properties of financial markets, known as stylised facts. The results show that the MMD-Signature model reproduces all five stylised facts and generates return series that closely match real market data. In contrast, the diffusion model does not reproduce these properties, despite training successfully and remaining stable after several implementation improvements. The findings indicate that the difference in performance is caused by how each model represents the data rather than by the training process. Overall, the thesis shows that a generative model must be designed for the statistical structure of the data it is intended to model in order to generate realistic financial time series.