Optimal prepayment and default rules for mortgage-backed securities

We study the optimal stopping problems embedded in a typical mortgage. Despite a possible non-rational behaviour of the typical borrower of a mortgage, the problem is worth to be solved for the lender to hedge against the prepayment risk, and because many mortgage-backed securities pricing model incorporate this suboptimality via a so-called prepayment function which can depend, at time t, on the fact that the prepayment is optimal or not. We state the prepayment problem in the context of the optimal stopping theory and present an algorithm to solve the problem via weak convergence. Numerical results in the case of the Vasicek model and of the CIR model are also presented. The work is extended to the case when both the prepayment as well as the default are possible.