In this paper we adopt the adaptive mesh approach to solve the model for pricing the fixed-rate mortgage, which is a convection-dominated problem when the volatility or the asset price is small. It is well known that when using the standard inite difference method to solve those convection-dominated problems, numerical difficulty can be caused. Our numerical  method combines  a central difference spatial discretization on a piecewise uniform mesh with an implicit time stepping technique. The matrix associated with discrete operator is an $M$-matrix, which ensures that the scheme is stable for arbitrary volatility and
arbitrary interest rate without any extra conditions. We apply the maximum principle to the discrete linear complementarity problem in two mesh sets and derive the error estimates. It is proved that the scheme is second order convergent with respect to the spatial variable. Numerical results indicate that the numerical solutions by our method are non-oscillatory and the scheme is second order convergent with respect to the spatial variable.