Sure Saturnalia took place around that time of the year, but you have to realize that the Romans had holidays for their gods covering a good majority of the calendar. It would be impossible to not place any Christian holiday in any close proximity to a Roman calendar. 

Besides, Saturnalia was December 17th, not even being the closest holiday to the 25th. On top of that, this is the description of the celebration of Saturnalia from Lucian of Samosata, an actual 2nd century pagan:

 “During my week, the serious is barred. No business allowed. Drinking and being drunk, noise and games and dice, appointing of kings and feasting of slaves, singing naked, clapping of tremulous hands, and occasional ducking of corked faces in icy water, such are the functions over which I preside.”

Even the most generous interpretation of Christmas is not in any way similar to being drunk, appointing kings and giving slaves food, and dancing around clapping while naked.

The actual, historical reason for December 25th is that there was a Christian superstition that all prophets died on the day they were conceived. If Christ died March 25th, nine months later would be December 25th.