I suggest you do some research on this page:
http://www.lufthansa.com/online/port...en&cid=1000390

If you choose to "search by price +/- 3 days from travel dates" you will get a nice matrix of departure and return dates with prices you can compare. For departure in December fares start going up a little around 10th but do not seem to vary much except at the weekend before Christmas when they become more expensive. What really seems to make a difference is the return date in January. The price seems to be lower for a return on or after 14th.

With airline tickets you have to pay for flexibility. I don't know if there's such a thing as a cheap changeable ticket. However some tickets are changeable for a fee. lufthansa.com should tell you which fare buckets are changeable.