What's the gender? Males tend to get oily from marking things.
If it is a male, is he neutered? They get less oily when neutered.
How often do you clean the cage? If you are cleaning it regularly he might be on hiper drive trying to remark it causing him to be extremely oily.
What is the diet he is on?
How long has he been on the diet?
How old is he?
What are his cage mates like? Are they oily too?
What does it smell like?
Has it been getting worse or has it been the same for a few months?
Have you tried taking him to a vet?
I'm honestly not really sure why it's so oily. I'm just asking questions to see if maybe someone or you could figure it out. If no one knows then you should go to the vet.