基于提前支付強度過程的抵押貸款定價與均衡

摘 要：基于提前支付強度過程考察了固定利率抵押貸款合同的定價和市場均衡問題。將均衡問題描述成代表性抵押人與市場之間的博弈。均衡由市場決定的內生抵押貸款利率和抵押人的最優再融資策略描述。在時齊Markov鏈利率及正線性比例再融資成本假設下，抵押人的規劃問題可以簡化成一個僅包含三個離散狀態變量的Markov決策鏈，且一定存在唯一解。從而，均衡可由一個抵押利率決定函數和抵押人的最優再融資策略組成。一個簡單的數值例子說明了計算均衡的迭代算法。結果表明，抵押人選擇再融資往往是不明智的短視行為。

關鍵詞：抵押貸款；提前支付強度過程；Markov決策鏈；內生抵押貸款利率

中圖分類號：F830.9 文獻標識碼：A 文章編號：1003-5192（2008）05-0069-06

Valuation and Equilibrium of Mortgage: An Intensity-based Approach

QIAN Yan-xiang

(School of Finance， Shanghai University of Finance and Economics， Shanghai 200433， China)

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Abstract:Based on prepayment intensity process， the valuation of fixed-rate mortgage contracts is studied and an equilibrium model is built with the specification of a game between the representative mortgagor and the market whose optimal strategy are about the endogenous mortgage rates and whether to refinance a new contract or continue with the current one， respectively. Under the assumption of the rate process of time-homogeneous Markov chain and positive linear refinancing cost， the dynamic programming problem of the mortgagor is simplified to a Markov decision chain with only three discrete state variables which is proved， by standard dynamic programming theory， to have a unique solution. Thus， the equilibrium can be given by a pair of the endogenous rate and the optimal refinancing strategy of the mortgagor. A simple numerical example with an iteration algorithm is finally provided to show how to compute the equilibrium. The results show that the mortgagor is usually too hasty to refinance， thus refinancing is typically a myopic behavior.

Key words:mortgage; prepayment intensity process; Markov decision chain; endogenous mortgage rate

1 引言

抵押貸款（mortgage）的定價是抵押貸款證券化的核心問題。一般認為，抵押貸款合同的價值依賴于提前支付風險（prepayment risk）和違約風險（default risk）。前者指貸款人（如銀行）允許抵押人（mortgagor）在到期日之前就償還當前的貸款余額。后者指抵押人選擇放棄抵押財產而不繼續還款。如果不存在違約風險，那么僅僅包含提前支付風險的抵押貸款合同的價值等于未來現金流（抵押人的付款）的期望現值。抵押貸款可以分為固定利率抵押貸款（fixed-rate mortgage）和浮動利率抵押貸款（floating-rate mortgage）。這里的固定和浮動是針對合同約定的貸款利率而言的。對于固定利率抵押貸款，抵押人每期支付一個固定的金額（coupon）。浮動利率抵押貸款的估值可以非常復雜，所以關于抵押貸款合同估值的文獻大多是考察固定利率的情況。……