Abstract: The authors consider the mortgages with the constant market enviroment and the constant contract rate.Suppose that the asset price moved as a geometric Brownian motion, an one-dimensional parabolic variational inequality which arises from the fixed mortgages is studied, the existence and uniquencess of the solution are obtained. Moreover, the monotonicity and the regularity of the free boundary are proved as well.

Key words: Option pricing, Fixed mortgages, Obstacle problem, Free boundary