"Period around now" means "starting recently and finishing soon". What "soon" and "recently" mean is a matter of context.
I play tennis.
(I can play tennis and I play frequently. I might not be on the tennis court now. Tennis is my sport.)
I am playing tennis.
(I am on the court right now. I started earlier and I expect to finish soon. For tennis "soon" means in a few hours or less)
Sometimes the present continuous can emphasise the idea that the activity is in progress, and taking some time, it may indicate repetition:
I'm jumping on the trampoline
(repeatedly jumping, the activity started recently and is taking some time.)
I jump off the box.
(Single action, started and finished as you were speaking, quite a rare situation)
I jump to school everyday.
(Statement of fact, not an activity that is ongoing right now).
Sometimes the continuous can emphasise the short period of time
I live in London.
(statement of fact, not expected to end soon(
I'm living with my parents.
(a temporary situation, expected to finish soon. In this case "soon" means in a few months or less.)