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Volume 13, Issue 6
A Semi Discrete Model for Mortgage Valuation and Its Computation by an Adaptive Finite Element Method

D.-J. Xie & S.-Y. Zhang

Int. J. Numer. Anal. Mod., 13 (2016), pp. 831-851.

Published online: 2016-11

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  • Abstract

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 for guiding mortgage practice.

  • Keywords

Finite element method, parabolic equation, free boundary problem, mortgage valuation.

  • AMS Subject Headings

91B28, 91B66, 35K15, 65M06

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{IJNAM-13-831, author = {D.-J. Xie and S.-Y. Zhang}, title = {A Semi Discrete Model for Mortgage Valuation and Its Computation by an Adaptive Finite Element Method}, journal = {International Journal of Numerical Analysis and Modeling}, year = {2016}, volume = {13}, number = {6}, pages = {831--851}, abstract = {

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 for guiding mortgage practice.

}, issn = {2617-8710}, doi = {https://doi.org/}, url = {http://global-sci.org/intro/article_detail/ijnam/468.html} }
TY - JOUR T1 - A Semi Discrete Model for Mortgage Valuation and Its Computation by an Adaptive Finite Element Method AU - D.-J. Xie & S.-Y. Zhang JO - International Journal of Numerical Analysis and Modeling VL - 6 SP - 831 EP - 851 PY - 2016 DA - 2016/11 SN - 13 DO - http://doi.org/ UR - https://global-sci.org/intro/article_detail/ijnam/468.html KW - Finite element method, parabolic equation, free boundary problem, mortgage valuation. AB -

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 for guiding mortgage practice.

D.-J. Xie and S.-Y. Zhang. (2016). A Semi Discrete Model for Mortgage Valuation and Its Computation by an Adaptive Finite Element Method. International Journal of Numerical Analysis and Modeling. 13 (6). 831-851. doi:
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