房貸的提前清償一直是個重要的議題，且它會對未來住宅抵押貸款證券的評價或一般貸款的價值有著重要的影響。提前清償即借款人會因為某些因素而在每個月償還部分或全部貸款本金。
本篇論文主要是運用隨機過程中的停止時間與提前清償的相似處進一步對住宅抵押貸款證券做評價或是計算一般貸款的價值。本篇我們考慮當利率低於某個點時或是房價高於某價值時，借款人會提前清償即把剩下的餘額全部繳清，而且也考慮當房價低於某價值時，借款人會進行違約，進而做進一步的評價。隨機過程的停止時間是指我們設定一個想要到達的點，並且找出這隨機過程何時到達該點，一旦到達該點就不考慮該點後的隨機過程，此與房貸提前清償有雷同的想法。

This paper uses the stopping time of a stochastic process to value loans and mortgage-backed securities. We consider the implications of a borrower being faced with low interest rates or high home prices and how that scenario may influence a borrower to prepay the loan. Conversely, when the house price is below a certain value, the borrower has an option to default. By using the stopping time of a stochastic process as a proxy for loan termination, we can set a point a borrower desires to reach and determine when the stochastic process will arrive at pre-determined point. Once the point is reached, we will not consider that the stochastic process further. Thus, we use the pre-determined point (stopping time) of the stochastic process as a proxy for loan termination.

Sheldon Lin, X., 1998. Double barrier hitting time distributions with applications to exotic options. Insurance: Mathematics and Economics 23, 45-58

Gerber, H.U., Shiu, E.S., 1994. Martingale approach to pricing perpetual American options. Astin Bulletin 24, 195-220

Gerber, H.U., Shiu, H.S., 1996. Martingale approach to pricing perpetual American options on two stocks. Mathematical finance 6, 303-322

Lo, C.F., Hui, C.-H., 2006. Computing the first passage time density of a time-dependent Ornstein–Uhlenbeck process to a moving boundary. Applied mathematics letters 19, 1399-1405

KARATZAS, I., WANG, H., 2005. A barrier option of American type. Applied Mathematics and Optimization 23, 15-31

Jacka, S., 1991. Optimal stopping and the American put. Mathematical Finance 1, 1-14

Lo, C.F., Chung, T.-K., Hui, C.-H., 2007. Double Barrier Hitting Time Distribution of a Mean-reverting Lognormal Process and Its Application to Pricing Exotic Options. Available at SSRN 1085983

Buist, Henry and T. T. Yang. 1998. Pricing the Competing Risks of Mortgage Default and Prepayment in Metropolitan Economies. Managerial Finance, 28(9/10):110-128

Stephen, W., Y. Li, V. Lekkas, J. Abraham, C. Calhoun and T. Kimner. 1995. Conventional Mortgage Home Price Index. Journal of Housing Research, 6:389-418

Yang, T. T., C. C. Lin. and Man Cho. 2011. Collateral Risk in Residential Mortgage Defaults. Journal of Real Estate Finance and Economics, 42:115-142