臺灣博碩士論文加值系統

過往的利率模型大多以固定的volatility來評價利率未來的走勢，然而依照目前世界財經狀況可以合理推估低利率的時代即將結束，未來利率必定調升的情況下，volatility必定有很明顯的波動。在明顯的波動情況下，我們所使用的利率模型並不符合市場狀況，因此使用過往的模型必定會產生顯著的誤差。為了更符合市場，在論文中我利用Heston模型引入了隨機volatility至利率模型當中，藉由Vasicek模型來預估未來的利率，最後使用樹狀圖的方法來預估出可能的利率走勢。

In the past, most of the short rate models were pricing with constant volatility. However, constant volatility does not fit the real situation of our financial condition now because Janet L. Yellen implied that the interest rate will continuous go up soon in the future. When the interest rate goes up, the volatility of interest rate has significant fluctuation. Consequently, the models we used to simulate the interest rate with constant volatility are out-of-date. From the above, stochastic volatility should apply into interest rate model to get more precise interest rate. I, therefore, apply stochastic volatility into vasicek model with tree based method to elaborate my work in the thesis.

Acknowledgement i
中文摘要 i
Abstract ii
Contents 1
List of Figures 3
List of Tables 4
1. Introduction 5
2. Literature Review 6
2.1 Heston Model and Pricing Methods 6
2.2 Short Rate Model 13
I. The Evolution of the Forward Rate Approach 14
II. The Evolution of the Short-Term Interest Rate 15
2.3 Vasicek Model 16
3. Methodology 18
3.1 Setup and Notation 19
3.2 Construct the Model 20
3.2.1 Binomial Variance Tree Approximation 22
3.2.2 Trinomial Stock Price Tree Approximation 27
3.2.3 Combine Stock and Variance Tree, and Match Correlation 29
4. My Model with Stochastic Volatility 37
4.1 Vasicek model with Stochastic Volatility in Heston Model 37
4.1.1 1st-order Taylor Expansion of the Moment 37
4.1.2 Transition Probability of the Interest Rate Process 43
5. Result 53
5.1 Variance Tree 53
5.2 Interest Rate Tree 55
5.3 Combine Interest Rate and Variance Tree, and Match Correlation 57
6. Conclusion 60
Appendix 61
Derive Equations 61
Transition Probability of the Interest Rate Process 64
Reference 67

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