本論文使用Longstaff, F.和 E. Schwartz在2001年所發展出來的Least- Square Monte-Carlo simulation approach 來估計美式利率交換選擇權。在此研究出來之前，蒙地卡羅模擬僅能運用於歐式選擇權的訂價，因無法判斷最佳提前履約點而不能解決美式選擇權的提前履約決策問題。而Longstaff和Schwartz 提出的LSM演算法正好可以有效預測出每個標的物路徑的最佳期望履約時點，並且不侷限於自變數個數的選取也不需路徑獨立的假設，而能透過價值最大化的過程有效計算出美式選擇權的價格。
以往美式利率交換選擇權的評價往往僅運用傳統美式選擇權的數值方法計算，忽略了過程中利率的變化對折現因子的影響效果，本論文將此影響考慮進入模型，以增加結果的正確性，並針對傳統方法和折現因子改良後的價值做出比較和因果分析。

We use the Least- Square Monte-Carlo simulation approach (Longstaff, F., E. Schwartz, 2001) to evaluate the American interest rate swaptions. Before this approach was developed, Monte-Carlo simulation could only be used in pricing the European options. Unable to help make optimal decisions for early exercise, it couldn’t apply to American options pricing. LSM could solve the problems above, provide a pathwise approximation to the optimal stopping rule that maximize the value of the American option regardless of if the underlying asset is path-dependent or not and how many stochastic variables are involved in the moving process.
In previous studies about American swaptions, people are used to estimate cash flows’ present value by fixed, predetermined discounting factor without concerning how the discounter will change during the contract period. In this thesis, we take this situation into account to help improve the precision of the estimated value, and then analyze the influence occurred by our discounting rate adjustment.

Contents 1
Chapter 1 2
Chapter 2 3
2.1 Valuation of interest rate swap 3
2.2 Valuation of European Swaptions 5
2.2.1 Swaption Introduction 5
2.2.2 Valuation by Black-Scholes Model 6
2.3 The LSM Approach 8
2.3.1 The practice of the LSM approach 8
2.3.2 The Framework 16
2.3.3 The Algorithm 18
2.3.4 The Convergence Result 20
Chapter 3 22
3.1 Models and Assumptions Used 22
3.1.1 Black-Scholes Model 22
3.1.2 Discounting Factor Assumption 23
3.2 Pricing American Swaption 24
3.3 Numerical Results 26
Chapter 4 37
4.1 Conclusions 37
4.2 Future Work 37
Bibliography 38

1.Cox, J., S. Ross, and M. Rubinstein, 1979, “Option Pricing: A Simplified Approach,” Journal of Financial Economics 7, 229-264.
2.Glasserman P, 2004,“Monte Carlo methods in financial engineering”, New York : Springer
3.Hull, J. C., 2000, “Options, Futures, and Other Derivatives”, N.J.: Prentice-Hall.
4.Hull, J., and A. While, 1993, “Efficient Procedures for Valuing European and American Path-Dependence Options,” Journal of Derivatives 1, 21-31
5.Kwok Y. K, 1998,“Mathematical models of financial derivatives”, Springer
6.L. Stentoft, 2001, “Assessing the Least Squares Monte-Carlo Approach to American Option Valuation,” Working Paper, Centre for Analytical Finance, University of Aarhus-Aarhus School of Business.
7.Longstaff, F., and E. Schwartz, 2001, “Valuing American Options by simulation: A Simple Least-Squares Approach,” The Review of financial Studies 14, 113-147.
8.Lyuu, 2002,“Financial engineering and computation”, Cambridge
9.Vasicek, O., 1977, “An Equilibrium Characterization of the Term Structure,” Journal of Financial Economics, 5, 177-188