Nathan is a narcissistic personality who also has delusions of grandeur. He is also very paranoid and has an explosive temper. He lies about who he is in order to affect others' opinion of him. He is with Sophie in part because she is beautiful and desirable, which reflects positively on him. But he also stays with her because she is vulnerable and grateful to him for protecting her and rescuing her, and this gives him control over her. It seems much more likely that Nathan lies about his occupation to feed his own delusion about his power and place in society, which is also a form of denial about his illness and emotional problems.