A semi discrete model for mortgage valuation and its computation by an adaptive finite element method

In traditional models for valuation of mortgages with a stochastic interest rate, one parabolic equation starting from the maturity is assumed to govern the whole life of a mortgage. Following the valuation of zero-coupon bond, a new model is proposed, where an initial value problem is restarted after a mortgage payment each month. In addition, the low and high limits on the interest rate are incorporated into the initial-boundary value problems, so that the partial differential equation remains regular and the solution better approximates the real value. We show the existence and uniqueness of the solution and the free boundary (which determines early prepayment). A finite element method is introduced with a convergence analysis. Numerical tests are presented and the results are interpreted guiding mortgage practice.