# 臺灣博碩士論文加值系統

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 在本文中，我們介紹如何使用模擬的方法來估算可及早履約選擇權的價值。首先，我們說明如何在任意一個有限並且是離散時間的馬可夫過程(Markovian process)中去計算它的最佳停止時間。接著，去利用條件期望值來估計最佳停止時間。而在馬可夫過程的假設下，迴歸模型可以用來幫助我們估計條件期望值。在這裡，我們介紹了兩種方法：局部線性迴歸(Local linear regression)和簡單線性迴歸。最後，則是運用拔靴法(Bootstrap method)和局部線性迴歸在未知波動的情況下來調整我們的估計。
 In this thesis, we show how to value the early exercise options with simulation. Above all, we present how to value the optimal stopping time for any Markovian process in finite discrete time and the estimation of decision rule to early exercise by conditional expectation. For Markovian process, the conditional expectation can be estimated with regression models. Local polynomial kernel estimators and simple linear regression are used in our experiments. After that, we apply bootstrap method and local polynomial kernel method to adjust our estimate without knowing the real volatility .
 Contents1.Introduction...........................12.Description of Theory..................32.1 The Intrinsic Value of an Option....32.2 The Optimal Stopping Time...........42.3 Valuation of European Options.......62.4 Valuation of American Call Option...72.5 The Approximation Algorithm of the Early Exercise Options.........................83.Regression Model......................103.1 Local Polynomial Kernel Estimators.103.2 Simple Linear Regression...........163.3 Approximating the Value of an American Put Option..................................18 3.4 Properties of the Estimator....194.Improve Estimation by Bootstrap Method234.1 Taylor Series......................234.2 Bootstrap Method...................244.3 Approximating the Value of American Put Option by Bootstrapping.................254.4 Comparison of Estimators...........264.5 Properties of Estimators...........275.Conclusion............................32Reference...............................33
 Reference1.Andre, I. Khuri, 1993, “Advanced Calculus with Applications inStatistics,” 108-112.2.Carriere, J., 1996, “Valuation of Early-Exercise Price of options UsingSimulations and Nonparametric Regression,” Insurance: Mathematics andEconomics, 19, 19-30.3.Chow, Y.S., H. Robbins and D. Siegmund, 1971, Great Expectations: The Theoryof Optimal stopping. Houghton Mifflin, New York, NY.4.Davison, A. C. and Hinkley, D. V., 1997, “Bootstrap Methods and TheirApplication,” Cambridge University Press.5.Efron, B., 1979a, “Bootstrap Methods: Another Look at the Jackknife,Insurance: Annals of Statistics, 7, 1-26.6.Longstaff, F. and Schwartz, E., 2001, “Valuing American Options bySimulation: A Simple Least-Square Approach,” Insurance: The Review ofFinancial Studies Spring 2001 Vol. 14, No. 1, 113-147.7.Schimek, Michael G.,“Smoothing and Regression,” 229-276.8.Ross, Sheldon M. “Introduction to Probability Models,” Eighth edition, 350.9.Ross, Sheldon M “Simulations,” Third Edition, 118-124.10.Tiley, J.A., 1993, “Valuing American options in a path simulation model,”Insurance: Transaction, Vol. XLV. Society of Actuaries, Schaumburg, 499-549.
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