Actually, Christopher only holds the right to the Silmarillion and beyond. The Saul Zaentz Company holds the right to the Lord of the Rings and the Hobbit. To quote their website:
"The Saul Zaentz Company also owns exclusive rights to certain copyrightable elements of The Lord of the Rings and The Hobbit, including, among other things, film and legitimate stage rights."
http://www.tolkien-ent.com/