Pricing of mortgages with prepayment and default options: numerical methods for the case with adjustable (floating) rate

Abstract

In this paper we consider the valuation of adjustable-rate mortgages with prepayment and default options where the underlying stochastic variables are the house price and the interest rate. Additionally, we obtain the value of the insurance associated to the contract. The mortgage value is understood as the discounted value of the future monthly payments. The mathematical model to obtain the mortgage price is posed in terms of a sequence of complementarity problems associated with partial differential equations (PDEs). One of the main innovative aspects is the consideration of a floating interest rate, which is linked to an index and fixed for each year of the loan life. The model for the value of the insurance is also posed. In this setting, we propose appropriate numerical methods to solve the models, which are mainly based on a Lagrange–Galerkin discretization of the PDE in time and space, jointly with an augmented Lagrangian active set method to deal with the complementarity formulation. Finally, some numerical results are presented in order to illustrate the behaviour of the numerical schemes, the properties of the mortgage value and the comparison with alternative fixed rate mortgages.