An option-theoretic valuation model for residential mortgages with stochastic conditions and discount factors
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Standard mathematical mortgage valuation models consist of three components: the future promised payments, the financial option to default, and the financial option to prepay. In this thesis we propose and analyze new concepts introduced into the standard models. The new concepts include discount factors, coherent boundary conditions, and stochastic terms. In this framework, the value of a mortgage satisfies a Black-Scholes type stochastic PDE. The approximate solution to our model involves a numerical method based on the Wiener-Ito chaos expansion, which breaks the stochastic PDE into a sequence of deterministic PDEs. These PDEs involve a free boundary, are discretized by finite differences, and solved through the PSOR method. Finally, extensions to MBS valuation are discussed. This work represents a timely study of mortgage valuation in the wake of the recent MBS/financial crisis.
This thesis is broadly organized as follows: In chapter 1, we briefly introduce some concepts that are part of the foundations of the standard mortgage models. In chapter 2, we review the standard mortgage valuation PDE models. In chapter 3, we discuss the discount factors, the coherent boundary conditions, and the stochastic terms. In chapter 4 we give a quick overview of the Wiener-Ito chaos expansion. In chapter 5 we analyze the simulation of our model and present some numerical results. Finally, in chapter 6 we make some remarks regarding the valuation of MBS.