I would like to remove an old Facebook fanpage but I cannot find the option
It is possible, although this option is well-hidden :) go to your fanpage and click "Edit Page>Edit Settings" (at the top, below the blue bar with Facebook logo). Scroll down and here you can find an option for removing your fanpage: Remove Page - Delete your Page. Choose it to remove the fanpage. Click on "Permanently remove - name of your page" and confirm your choice.