The Old Man could argue that he killed his son in self-defense. After all, the young man tried to steal some money from him and threatened to kill him. Under the circumstances, the Old Man might argue that he had no choice. It was either him or the boy.
But the real reason for the Old Man's killing his son is altogether different and more disturbing. As the Old Man tells his dead mother he felt he had to put an end to the seemingly endless cycle of greed and dissolution that has destroyed this family for many years.
The Old Man's father was a notorious drunk and all-round ne'er-do-well, whose riotous spending habits brought the family to ruin. And the Old Man is worried that history is repeating itself in the shape of his son. All the indications seem to be that the young man will grow up to be a dissolute rake just like his grandfather. There's simply no way that the Old Man will every tolerate going through that again.
So he metes out the same punishment to his son as he did to his father amidst the burning of the old family pile. In doing so, he finally puts an end to this tragic cycle of events.