A Steady State Solution to a Mortgage Pricing Problem

This paper considers a mortgage contract where the borrower pays a fixed mortgage rate and has the choice of making prepayment. Assume the market interest follows the CIR model, the problem is formulated as a free boundary problem where the free boundary denotes the level of market interest rate at which it is optimal for the borrower to make prepayment. Here we focus on the infinite horizon problem. Using variational method, we obtain an analytical solution to the problem, where the free boundary is implicitly given by a transcendental algebraic equation. 1. Formulation of the problem We consider a mortgage contract where the borrower pays a fixed rate of c (year−1) to the lender. In reality this mortgage rate is implicitly represented by a continuous payment of m $/year. At each time t when the contract is effect, the borrower has two choices: to continue the mortgage by paying mdt for the next dt period or to close the mortgage by paying off all the loan balance M(t), where the loan balance M(t) is determined by dM(t) dt = −m+ cM(t). (1.1) When the contract duration T is given and M(T ) = 0 specified, the above ODE has a unique soultion M(t) = m c (1− e). (1.2) Here we assume the borrower always has sufficient amount of capital. The borrower chooses not to pay M(t) even though he is financially capable to do so if the expected future market return from an equal amount of investment is higher enough. On the other hand, if the expected future market return from an equal amount of investment is lower enough, he should choose to settle M(t). From the lender’s point of view, the value of the contract V , as a function of time t and market interest rate x, is determined by the market interest rate x. The higher the market interest rate x is, the lower the contract value V , but V shall never be lower than 0. The lower the market interest rate x is, the higher the contract value V , but V shall never be exceed M(t) since the borrower has the choice to settle the loan once V reaches M(t). From standard mathematical finance theory, one can find the value of the contract V (x, t) and the optimal level of market interest x = h(t) at which the borrower should make prepayment of M(t) by solving 1 2 DEJUN XIE the following free boundary problem: