The Avengers actor stars as the President of the United States, who finds himself stranded in the Finnish wilderness after Air Force One is attacked.
The British-Finnish film is directed by Rare Exports' Jalmari Helander, and also stars newcomer Onni Tommila.
The President has to team up with a 13-year-old boy, who is on his own mission to become a man, in order to stay alive and battle the terrorists.
Jim Broadbent, Felicity Huffman, Victor Garber and Ted Levine also have supporting roles in the film.
Big Game has yet to be given an official release date, but is expected later in 2015.