過去的三十年間由於評價美式選擇權所產生的自由邊界問題已經有相當的研究成果。本論文將證明自由邊界問題的解為遞增函數。更進一步提出自由邊界凹性的嚴謹証明。利用我們的結論可以得知美式選擇權的最佳履約邊界對時間而言為嚴格遞減的凹函數。這個結果對可用來求導最佳履約邊界的漸近解。 對於美式交換選擇權，我們將其自由邊界問題轉換成單變數的積分方程，同時提供一個永續型美式交換選擇權的評價公式。對於有限時間的美式交換選擇權的最佳履約邊界，我們將提供一個接近到期日的漸近解並發展一個數值方法求其數值解。數值計算的結果顯示漸近解在接近到期日時與數值解非常接近。 對於評價美式選擇權，我們提出使用混合整數非線性規劃(MINLP)的模型，這個模型的最佳解同時提供賣方的完全避險策略、買方的最佳交易策略與美式選擇權的公平價格。因為求算MINLP模型的解需耗用大量的計算時間，我們證明此模型和其非線性規劃的寬鬆問題有相同的最佳解，所以只需求算寬鬆問題即可。觀察數值結果亦顯示非線性規劃的寬鬆問題可以大幅的降低計算的時間。此外，當市場的價格低於公平價格時，我們提出一個最小化賣方期望損失的數學規劃模型，此模型的解提供賣方最小化其期望損失的避險策略。 / In the past three decades, a great deal of effort has been made on solving the free boundary problem (FBP) arising from American option valuation problems. In this dissertation, we show that the solutions, the price and the free boundary, of this FBP are increasing functions. Furthermore, we provide a rigorous verification that the free boundary of this problem is concave. Our results imply that the optimal exercise boundary of an American call is a strictly decreasing concave function of time. These results will provide a useful information to obtain an asymptotic formula for the optimal exercise boundary. For pricing of American exchange options (AEO), we convert the associated FBP into a single variable integral equation (IE) and provide a formula for valuating the perpetual AEO. For the finite horizon AEO, we propose an asymptotic solution as time is near to expiration and develop a numerical method for its optimal exercise boundary. Compared with the computational results, the values of our asymptotic solution are close to the computational results as time is near to expiration. For valuating American options, we develop a mixed integer nonlinear programming (MINLP) model. The solution of the MINLP model provides a hedging portfolio for writers, the optimal trading strategy for buyers, and the fair price for American options at the same time. We show that it can be solved by its nonlinear programming (NLP) relaxation. The numerical results reveal that the use of NLP relaxation reduces the computation time rapidly. Moreover, when the market price is less than the fair price, we propose a minimum expected loss model. The solution of this model provides a hedging strategy that minimizes the expected loss for the writer.