"Words can't describe how we feel right now and how much we miss Morgan.
"It's hard to summarise his life and all his good qualities. He's going to leave a huge hole in our family."
A New Zealand Defence Force spokeswoman said Fraser was in the First Royal New Zealand Infantry Regiment.
A Facebook post from chief of army Major General Peter Kelly said Fraser was one of their own.
"Morgan was a keen outdoorsman, and over the past few days I've been privileged to hear stories about this young soldier - a good southern lad from Gore who was a determined and capable soldier with a passion for mountaineering.
"Morgan enlisted into the Army in April 2013, and was posted to Victor Company, 1 RNZIR, and then Signals Platoon, before most recently moving to Reconnaissance and Surveillance Platoon in March this year.
"Morgan was Aumangea Programme qualified, and had a promising career ahead of him.
Morgan's loss will be keenly felt by his family, comrades and mates. My thoughts and prayers are with them as they come to terms with their loss. His service to New Zealand will not be forgotten."
Fraser was reported missing on Wednesday after failing to return from a climbing trip. Search and Rescue teams found his body just after midnight.