對於利率模型的建構，主要有兩個理論方法。一個是一般均衡理論，另一個則是無套利理論。2001年Jin及Glasserman發表的論文中對於這兩種不同的模型提供了一完整的連結。藉著在連續時間中建構了一個pricing kernel model以作為這兩個理論之間的橋樑。透過這樣的連結，給定兩者中某一個的模型參數，即可以轉換至另一個模型上。除此之外，也經由這個連結，能夠對無套利模型提供一個支持它的生產經濟的均衡模型。
本論文試圖在離散時間的架構下追求一個相似的目標，也就是建立離散時間下兩種理論的連結。並得到以下四個主要貢獻：一、建構一個離散時間的pricing kernel model 以作為利率期間結構模型。二、建立pricing kernel model 與Heath, Jarrow, and Morton’s model離散時間下的連結，並提供兩者之間轉換的方式。三、建立一個生產經濟的均衡模型來支持所建立之pricing kernel model。四、在此模型下探討利率之特性，諸如：利率為正值、債券價格隨到期時間遞減的性質。

There are two main approaches in constructing models of interest rates. One is the general equilibrium approach. The other is the arbitrage-free approach. Jin and Glasserman (2001) developed a precise connection among the two approaches by using a pricing kernel model as the “bridge” in a continuous time economy. Given the primitive data of either perspective, they can obtain primitive data of the other, i.e. each of the two models can be transformed into the other. Through these connections they also find a production economy with a representative-consumer to support arbitrage-free models.
This thesis attempts to accomplish a similar goal in a discrete time economy. There are four main contributions in this thesis. First, to construct a discrete time pricing kernel model, we incorporate the discounted marginal utility of optimal consumption and fit the model to the term structure of interest rate. Second, we build a connection between the pricing kernel model and Heath, Jarrow, and Morton’s discrete time framework, and gives transformation formulas between the primitive parameters of these two models. Third, a production economy is offered to support the pricing kernel model. Fourth, we attempt to provide foundations for the properties of the interest rates including that the interest rates are positive and that the bond price vanishes to zero as its expiration time approaches infinity, which require adding more assumptions in the pricing kernel model.

1 Introduction 1
1.1 Motivation 1
1.2 Objectives 3
1.3 Framework 4
2 Review of the Literature 5
2.1 An Overview of Interest Rate Models 5
2.2 Heath, Jarrow, and Morton’s Continuous Time Model 7
2.3 Heath, Jarrow, and Morton’s Discrete Time Model 9
2.4 Cox, Ingersoll, and Ross’s Continuous Time General
librium Model 11
2.5 Sun (1992): Discrete Time Approximation of CIR 13
2.6 Jin and Glasserman (2001): The Pricing Kernel Approach 15
3 A Pricing Kernel Model 17
3.1 Terminology 17
3.2 Pricing Kernel 19
3.3 Interest Rates and Bank Account’s Derivation 21
4 The Connection with HJM’s Model (Arbitrage-Free
Approach) 24
4.1 Assumptions 24
4.2 Connection to HJM''s Model 29
5 The Connection with CIR’s Model (Equilibrium Approach)32
5.1 Supporting Equilibrium 32
5.2 Properties 34
6 Conclusion and Thoughts for Further Study 37
6.1 Conclusion 37
6.2 Thoughts for Further Study 38
Reference 40

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