Lets take a look at all the different ways you can play your PS2 games through HDMI depending on your budget:

Amazon Affiliate links:
Composite to HDMI converter: http://amzn.to/2GjISlW
Component to HDMI converter: http://amzn.to/2IMTs3d
Monoprice Component Cable: http://amzn.to/2II5LxU