During a game a player picked up a chance card which told him to go to a utility. He however owned both utilities and it was therefore his property. He argued that the bank therefore owes him the 10× his roll. I said that he would simply owe himself the 10× dice roll. What is the rule here?
Nothing happens (or if you want to say he owes himself money you can). Monopoly spells out what happens in the rules fairly well. The Chance and Community Chest cards also do a good job of explaining what should happen. Nothing in that Chance card says anything about the Bank paying in the event the property is owned by the player. It simply says
If OWNED, throw dice and pay owner a total ten times the amount thrown.
"Pay" is clearly an instruction to the player who drew the card, not the bank.