臺灣博碩士論文加值系統

中文摘要
本篇論文是使用Longstaff和Schwartz(2001)所發展之蒙地卡羅最小平方法(Least-Square Monte-Carlo simulation approach, LSM)來估計美移動平均選擇權之價值。蒙地卡羅模擬法以往無法解決美式選擇權提前履約的問題，因不知模擬出來的股價路徑最佳履約時點，而Longstaff和Schwartz提出的LSM演算法剛好可以有效選出每股價路徑的最佳履約點，解決蒙地卡羅評價美式選擇權的問題。
以美式賣權為例，LSM是利用蒙地卡羅法模擬一段期間每日(j)股票價格(Xj)之可能路徑,再依據選擇權契約所訂定的履約價格(K),算出每個股票價格路徑上每日之收益(Payoff，Max(Xj－K, 0))。然後從到期日開始，以到期日之全部股價路徑之收益诙因變項值，以到期昨日之股價為自變項值，做最小平方法簡單迴歸，求出β0、β1之(ｘj-1, E(Yj｜X=ｘj-1))簡單迴歸線的截距和斜率。再求出因變項平均值作為與當日收益價值作比較，若收益大於因變項平均值，則履約；反之，則不履約。再推往至前一日，重覆上述步驟，若每一條股價路徑有提前履約，則提前履約，直到契約成立日。再將每一條股價路徑中的最佳約日的收益，折現到契約成立日再進行算術平均，即可估算出美式賣權的價值
由於現今金融商品不斷創新，且日益複雜。而蒙地卡羅模擬法可價移動平均選擇權。但要評價美式移動平圴選擇權，須取前期股價，和移動平均珼做自變項變數，且為了讓消弭相關性，用Laguerre Polynomial模型使變數間彼此成正交。為了求證LSM演算法估計的準確性，用Dai(1999)的AuxiliaryState Variables和Ritchken & Trevor(1999)改良CRR Tree模丑來評價美式移動平均選擇權，結果可發現LSM和CRR所估計出來的值，差異都非常小。可觀察到LSM演算法實在非常強，可以評價如此複雜的金融商品。
本篇論文亦在以美式移動平均選擇權評價為例，進行探討如仃使LSM演算法估計更為準確。分為兩個部分，一個是迴歸自變項變數個數選取，另一是如何選擇較佳的自變數模型。在自變項變數個數選取上，至K = 2時，就非常準確，與K = 3所估計的值幾乎無差異。在選擇自變數模型上，比較了Monomials和Laguerre Polynomials，發現沒有做正交處理的Monomials模型，其估計出來的值和CRR與用Laguerre Polynomials做正交處理的LSM估計值，幾乎也無差異，這是一個較特別和驚訝的發現。若這發現是正確的，則用LSM演算法評價複雜式金融商品，都不用再對自變項變數間做正交處理了。

Content
1 Introduction 3
2 Preliminaries on Options Pricing 5
2.1 Simulation and option pricing………………………………………………5
2.1.1 Simulation from a Geometric Brownian Motion………………………5
2.1.2 Pricing European Options using simulation…………………………...6
2.1.3 Pricing American options using simulation……………………………6
2.2 Tree Models and Auxiliary State Variables…………………………………7
2.2.1 The CRR model………………………………………………………..8
2.2.2 Auxiliary State Variables………………………………………………9
3 The LSM Valuation Algorithm 10
3.1 The LSM valuation framework……………………………………………10
3.2 The LSM algorithm…………………………………………………….....11
3.3 The LSM algorithm to pricing American option………………………….12
3.3.1 The presentation of pricing American call options…………………...12
3.4 Convergence results ………………………………………………………14
4 Pricing Moving-Average-Lookback Options 16
4.1 Definition the AMVALs…………………………………………………..16
4.2 Pricing American-Style AMVALs………………………………………...17
4.2.1 The LSM methods……………………………………………………17
4.2.2 The CRR models……………………………………………………..18
4.3 Numerical Results…………………………………………………………19
4.3.1 Case1：Stock Price v.s. Volatility……………………………………19
4.3.2 Case2：Stock Price v.s. LB…………………………………………..20
4.3.3 Case3：Dividend Rate v.s. Reset Date……………………………….21
4.3.4 Case4：Reset Rate v.s. Different Reset Condition…………………...22
4.3.5 Case5：Moving –Average Number v.s. Volatility……………………23
4.3.6 Summary……………………………………………………………...24
5 What to Choose the Robustness of LSM? 25
5.1 Altering the number of regressors………………………………………...25
5.2 Using alternative polynomial families……………………………………26
6 Conclusions 28
Bibliography 29

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