What are the present-day names for cathay and Persia?

Cathay is what today we call China. Persia is what we now call Iran. One thing you must keep in mind; and it is the fact that former Cathay is today's China doesn't mean that the territory is exactly the same as today. Cathay was a tad smaller than today's China. About Iran, today it is smaller than Persia was. Hope this helps.