Most philosophers believe that objective morality exists. It is only each person, religion and culture's interpretation of morality that varies hence giving an impression that morality is relative. There has been a long and on-going philosophical debate over this and the arguments for moral objectivism (realism) are winning against moral relativism which used to be popular during the 18th century until the mid-20th century among social darwinists.

If a morality is objective then why does it vary among cultures? Isn't that a contradiction itself?

The moral discrepancies between cultures represent only a difference in their perception and understanding of morality since morality gets distorted by economics, politics, culture, religion and other factors, not that it proves that morality is not objective.