這篇文章首先由一個純投資的觀點來觀察抵押契約，並且整合了一些新方向的研究工作。其中
包含了固定抵押利率下，契約的價值決定以及平衡問題。這之中的工作內容有，由抵押契約的價值
表示中去尋找合適的” 還款動機過程”(intensity process) 來表示借款人的還款意願，以及由求得的
價值表示式中去解釋，價值可以分成本金部分加上實際利率與抵押利率的” 交互差項”(swap)。在平
衡問題中，我們考慮抵押利率不只受到實際利率的影響也會因為借款人的行為而改變(行為可以由”
還款動機過程”(intensity process) 來認識)。也就是說，在馬可夫鏈中(Markov chain)，我們考慮使
用” 內生的”(endogenous) 抵押利率去重整(經由使用” 動態規劃”(dynamic))” 外生的”(exogenous)
抵押利率。一些工作可以被推廣到連續型的情況，更進一步的，我們能用這種連續型的價值表示式
對” 抵押擔保債券”(collateralized mortgage obligations) 定價。

This thesis starts with a pure-investment perspective to observe the mortgage contracts, and
then summarizes some researches on a new approach, namely, the valuation of fixed-mortgage-rate
contracts and the equilibrium problems (in discrete-time). Working among the representation of
valuation makes us seek appropriate intensity process to represent the prepayment incentive. And
the obtained valuation equals to the outstanding principal plus the value of the certain swap between
short rates and mortgage rates. In the equilibrium problem, we consider that the mortgage rates
not only affected by short rates but also the mortgagor’s behavior (via the intensity process of the
prepayment time). That is, we consider using endogenous mortgage rates to refinance (basing on
dynamic) exogenous mortgage rates in a Markov decision chain. Some works can be generalized to
the continuous-time case. Furthermore, we use the valuation representation in continuous-time to
price the collateralized mortgage obligations (CMOs).

1 Introduction 1
2 From American option to prepayment option 1
3 Introduction to the discrete-time mortgage contract
with endogenous mortgage rates 6
4 (Optimal) refinancing and (optimal) equilibrium
solutions of discrete-time mortgage contract 11
5 Mortgage contracts in continuous time and an introduction
to the Collateralized Mortgage Obligations (CMOs) 16
6 Appendix A 23
7 Appendix B 26
8 Appendix C 33

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