本篇論文包括兩個主題，在第一個主題，本論文提出一個前進式的蒙地卡羅法，用以對美式選擇權進行評價。這種方法的主要優點在於不用傳統倒退式的方法，而採用了一個較簡單的方法來判斷所模擬股價是否觸及提早履約的區域。本論文針對vanilla的例子提出了數學上的證明以支持其方法的有效性，同時也將此方法的應用擴展到其它的美式選擇權，如美式後定選擇權（chooser options）及美式互換選擇權（exchange options），並利用一系列的數值例子來展示此方法的評價效率。由試驗的例子中顯示出，前進式的方法不管在準確率或是計算的時間上均比傳統使用最小平方法（迴歸）的蒙地卡羅有效率。
論文的第二個主題則討論美式冪次買權（power call options）永不提早履約的充分條件及當存在提早履約機會時它的解析上界。在vanilla的例子中，當股利率為零為其永不提早履約的唯一條件，但在美式冪次買權，則存在一個範圍可導致選擇權永遠不會提早履約。本論文導出當冪次大於1或小於0時，滿足美式冪次買權在BS模型、二個跳躍擴散模型及Variance Gamma模型下永不提早履約的一般充分條件。當冪次介於1與0之間時，除非存在負的股利率，否則該買權將永遠具有提早履約的機會。此外，當股利率落在永不履約的區域外時，我們同時計算其價格之解析上界，文中同時提供數值例以展示解析上界的有效性。

This thesis contains two essays on American options. The first essay proposes a forward Monte Carlo method for the pricing of American options. The main advantage of this method is that it does not use backward induction as required by other methods. Instead, the proposed approach relies on a wise determination about whether a simulated stock price has entered the exercise region. The validity of the proposed method is supported by the mathematical proofs for the vanilla cases. With some adaption it is shown that this forward method can be extended to price other American style options such as chooser and exchange options. This study demonstrates the effectiveness of the proposed approach using a series of numerical examples, revealing significant improvements in numerical efficiency and accuracy in contrast with the standard regression-based method of Longstaff and Schwartz (2001).
The second essay discusses the sufficient condition under which the American power call options should never be early exercised. Unlike in the vanilla case where the dividend yield q = 0 is the only condition, there actually exists a range of q such that it is never optimal to exercise the American power call option early. We first derive the general (model free) sufficient condition of q for American power call option with n>1 or n<0 (n is the power coefficient). We then provide sufficient conditions for specific models where a wider range of q and n can be found. Moreover, when q falls outside the never-early-exercise region, we also give the analytical upper bounds for the American power call prices. These analytical formulae are first derived for the fundamental Black-Scholes model, and are then extended to two jump-diffusion models and the variance gamma model. Numerical examples are provided to demonstrate the validity of the derived analytical formulae.

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