What are the challenges of administrating an International stock trading database?
Because it's a system, you will have a LOT of interacting subsystems, including power, environmental controls, communications, hardware, and then on top of that the data management and everything else.

There are facets which cut across all these layers, like security, redundancy, business continuity, etc.

Just from the database side, you would need to consider the transactional load, the architecture for balancing the load, the storage requirements, the response time requirements.

Everything depends upon the requirements - of which there will be many from regulatory bodies, clients, users, business partners.

As to whether cloud-based databases, I don't think any of those can be viable if your organization has to be responsible for the system.

Any time you outsource anything, you limit your ability to be responsive to requirements.  Some things you delegate are easier to mitigate than others.  For instance, you decide to use a software library as a component in your system - you can always modify the library if you have the source, or rewrite it yourself or swap it out.  Or picking a RDBMS like Oracle or SQL Server where you can get a vendor to do a proof of concept and have a high level of support because the sale is large.

That's not to say by relaxing your requirements, you couldn't use Google Apps BigTable or Amazon or Azure or whatever, but the simple fact is that even with a contract, you have a limited amount of control over another organization's system and you are relying on them.

All systems are built by balancing tradeoffs.