Authenticity is a bit of a buzz word. Everybody speaks about authenticity. Everybody wants to be authentic. In truth, vulnerability and authenticity go together because being authentic means to be real, genuine, and true to yourself.
As a leader, when you are under pressure, the first thing which tends to disappear is your sense of humour. If you think about it, using humour in a stressful situation shows that you are in control of yourself and the situation.