This excellent but also somewhat intriguing tale is told using the third person limited point of view. This means that we have a narrator that is external to the action of the story, but who follows it from the perspective of one of the character's alone. In this case, the narrator zooms in on the banker and his thoughts and ideas of what has happened and how he reacts to it. The reason why Chekhov made this choice in terms of point of view is that we as readers are privy to his reactions to the letter of the young lawyer he defaults on the terms of the bet so dramatically:
When the banker had read this, he laid the page on the table, kissed the strange man on the head, and went out of the lodge, weeping. At no other time, even when he had lost heavily on the Stock Exchange, had he felt so great a contempt for himself. When he got home, he lay on his bed, but his tears and emotion kept him for hours from sleeping.
Clearly the banker is shown through the point of view to be greatly affected by the letter, and feels himself to be contemptible. Thus it is that we are surprised that in the next paragraph, he keeps the letter of the lawyer, presumably so that he can have proof that the lawyer relinquished the money in case he changes his mind later on. Thus the point of view clearly establishes that, although he was impacted by the letter, he has clearly not evolved beyond his own greed and self-interest. The choice of point of view thus highlights the banker's failings as a character and can be linked in to the way that the author establishes the theme of the story, which could be argued to relate to the lack of integrity that people have.