Have you ever looked at GARCH? It is really good in predicting future volatility/variance, and the packages in R and python are really easy to use.
https://en.wikipedia.org/wiki/Autoregressive_conditional_heteroskedasticity
I experienced the same. I looked at marginal distributions, correlations between OHLC and correlations across time. They were all similar to real data. In addition to that, I looked at characteristic that are present in specific markets such as fat tails, volatility clustering, and leverage...
I recently completed a research project focused on creating synthetic data using Generative Adversarial Networks (GANs). This innovative approach has shown significant promise in enhancing trading strategy development by addressing overfitting challenges and expanding datasets.
Why Synthetic...