Killing Little Grandpa (idea)

Not a euphemism for male masturbation, but rather a popular exposition of time-travel paradox:
You enter a time machine, and set the date for, say, 50 years in the past, after your grandfather was born but before your father was born. You find Grandpa and kill him. Since there is no Grandpa, there can be no Dad, and no "you": you have altered history to make yourself an anomaly.
So what happens? There are several theories to cope with this paradox:

• You create a loop across a sort of meta-time, where you kill Grandpa, cause yourself to not-exist and therefore don't kill Grandpa, come to exist, kill Grandpa... This is the assumption of the Back to the Future films.
• Time "forks", and two possible parallel worlds each start progressing in their exclusive timelines: one where you exist because Grandpa was unkilled, and one where you don't exist because an extradimensional assassin (you) killed him.
• "Backward" time travel is not possible, and you cannot affect causal conditions of the past. Forward time travel does not create a gaping paradox like this, so it may be possible, but such asymmetry is ugly.
  
  A Blather of Paradoxes