Actually, you're not exactly correct. Frank's player option is $10 million. The White Sox team option is $12 million. Frank would have to choose to stay for $10 million. The White Sox could choose to give him $12 million or buyout his contract for $3.5 million, regardless of what Frank chooses. So if Frank declines his $10 million player option, the Sox would have to decide if he is worth $8.5 million ($12 million team option - $3.5 million buyout).
http://sports.espn.go.com/mlb/news/story?id=1650383