I don't think I would use either version. It's more natural to say any of these instead "There is nothing at home" - By itself, this means there is literally nothing at home. The house is empty. If the listener has more context, it could mean there nothing of a specific item. "There aren't any at home" - This assumes the listener knows what item you're talking about. "There aren't any [item name] at home" - This indicates what is not at home.