臺灣博碩士論文加值系統

在評價選擇權時，若選擇權價格與標的資產的歷史路徑有關的話，我們則將之稱為路徑相依選擇權。路徑相依選擇權在實務上擁有低成本以及符合許多實務性的避險情境與需求。本文探討了過去評價路徑相依選擇權法的文獻，發現較少文獻在討論美式路徑相依選擇權，有深入探討的價值。
二元樹法與蒙地卡羅法最常被用來定價美式路徑相依選擇權，兩者在處理美式屬性以及路徑相依各有優劣。二元樹法由於事先生成了每個節點，每個節點的選擇權價值可藉由風險中立機率，針對其折現值求解期望值之方式計算，在處理提前履約與否較容易；蒙地卡羅法因為可藉由模擬來生成每條路徑，因此在處理路徑相依時方法也較簡單。
本文以二元樹模型為基礎，在節點間加入額外的股價路徑模擬，將之稱之為”布朗橋”，目的是在不增加提早履約時點的前提下，增加更多股價路徑的描繪。選擇權的時間價值可分為提早履約價值與價格波動價值，增加二元樹的期數可以視為增加提早履約的時間點，同時也增加股價變動的多樣性，但也大大增加選擇權價格的求解時間；而增加布朗橋期數可以視為增加價格變動所帶來的時間價值，但其增加的求解時間有限，而該機制也正好可以驗證選擇權中源於價格波動帶來的時間價值之效果。
最後在美式亞式選擇權的實證結果發現：在期數很多的二元樹中，節點間增加布朗橋路徑模擬，可以再提升其估計的精確度，表示布朗橋路徑可以有效捕捉選擇權的價格波動價值。另外，由於二元樹相鄰的二節點間股價差異很小，故布朗橋路徑波動程度有限，僅要少量的路徑數(約10條)即可使二元樹估計值有良好的收斂。

When it comes to pricing an option, if the option price is related to the historical path of the underlying asset, we refer to it as a path-dependent option. Path-dependent options have the low cost in practice and are suitable for a number of practical hedging scenarios and needs. This paper explores the literature of path-dependent options, and finds that less literature is discussing the American path-dependent option, which has the value of in-depth discussion.
The binary tree method and the Monte Carlo method are most often used to price the American path-dependent option. The two have advantages and disadvantages in dealing with American attributes and path dependence. The binary tree method is easier to handle early exercising because it generates each node in advance, and the value of each node can be calculated by discounting the expectation under risk-neutral probability. Monte Carlo method is easier to handle path-dependent issue because it can generate each path by simulation.
Based on the binary tree model, this paper adds an additional stock price path simulation between nodes, which is called “Brownian Bridge”. The purpose is to increase the depiction of stock price paths without increasing the nodes. The time value of the option can be divided into the early exercising value and the price fluctuation value. To increase the periods of binary tree can be regarded as increasing both the early exercising value and price fluctuation value, but it is time-consumption. However, to increase the nodes via the Brownian Bridge can be regarded as the increase of the time value of price changes and its time-consumption is limited. This “Brownian Bridge” mechanism can just verify the source of the time value of the option, which is from the price fluctuation.
Finally, the empirical results of the American Asian option show that in the binary tree with many nodes, the Brownian Bridge simulation can increase the accuracy of the estimation, indicating that the Brownian Bridge can effectively capture the volatility value of the option. In addition, since the stock price difference between the two nodes adjacent to the binary tree is small, the Brownian Bridge path has a limited degree of fluctuation, and only a small number of paths (about 10) can make the binary tree estimate converge.

第一章、 緒論
第一節、 研究動機與目的
第二章、 文獻探討
第一節、 二元樹法
第二節、 蒙地卡羅法
第三節、 解析公式解
第四節、 結論
第三章、 研究方法
第一節、 Hull and White (1993)樹狀模型
第二節、 Hull and White (1993)樹狀模型之強化機制－布朗橋
第三節、 Hansen and Jørgensen (2000)解析解
第四節、 數值驗證
第四章、 布朗橋在美式幾何平均執行價選擇權的效果
第一節、 布朗橋期數(n)及布朗橋路徑數(STPH)對二元樹評價效果之分析
第二節、 布朗橋期數(n)對選擇權價格之影響與二元樹期數(N)之抵換關係
第三節、 布朗橋路徑數(STPH)對選擇權價格之影響與二元樹期數(N)之抵換關係
第四節、 二元樹期數(N)與布朗橋期數(n)對執行時間的抵換關係
第五章、 布朗橋在其他亞式選擇權上的效果
第一節、 美式算術平均執行價選擇權
第二節、 美式幾何平均選擇權
第三節、 美式算術平均選擇權
第六章、 結論與建議
參考文獻
附錄A:美式幾何平均執行價選擇權檢定表
附錄B:美式算術平均選擇權檢定表
附錄C:美式幾何平均選擇權檢定表
附錄D:美式算術平均選擇權檢定表

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