Odell will take up the role on October 1 and will be responsible for Volvo's global operations out of its headquarters in Gothenburg, Sweden.
Odell will report to Lewis Booth, executive vice president of Ford Motor Company and chairman of Volvo Car Corporation, who will continue to oversee the strategic direction of Volvo.
Booth thanked Arp for his leadership of Volvo over the past three years.
Booth said: "Fredrik Arp has steered the Volvo team through some difficult times since joining the company three years ago.
"Fredrik has decided that now is the right time to hand over to a new president and CEO who will lead the Volvo team through the next stage of its recovery."