The Value Of Fixed-Rate Mortgages

In recent years, the mortgage developed fastly in our country. Most of them are Floating-Rate Mortgage. In order to adjust to the developing market,Studying the Fixed-Rate Mortgage is very meaningful. In the mortgages, there are two main risks:the prepayment risk and the default risk. In option, they are prepayment option and default option. In recent papers,some scholars got some result which effect the mortgage value via the single factor. Other scholars studied the value which influence the prepayment option and the default option via the two factors.Firstly, we introduced the prepayment option which mainly related to the short instantaneous interest rate and default option which mainly related to the house value in this paper. We considered the instantaneous interest rate follow the Vasicek model and the house value follow the Brown motion. We got the Partial Differential Equation of the value of mortgage by using the theory of the hedge and Ito’s Lemma.Secondly, the random process of the interest rate and house value exist correlation coefficient and heteroscedasticity, the partial differential equation which we had got ex-ist mixed partial derivative term. Through our analysis, we got two new variable which correlation coefficients was equal to zero and variance were same by using the variable transformation. Through the theory of the hedge and Ito’s Lemma, we deduced the par-tial differential equation of the loan contract value under the mew variables. Compared with the partial differential equation transformation before,there was no mixed partial derivative term. We discreted the transformed equation by alternative direction implicit difference algorithm and got the corresponding equations which we had transformed.In numerical analysis, we analyzed the boundary conditions and program the e-quations across MATLAB software, geting the option values and mortgage value and analysing the effects of various factors to contract value.Finally, the house value effected by the policy maybe exist jumping phenomenons, we got the the partial differential equation under the double factors by using the Ito’s lemma of semi-martingale.