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研究生： 柯坤義
Kun-Yi Ko
論文名稱： 高斯數值積分在選擇權評價上的應用研究
Fast Accurate Option Valuation UsingGaussian Quadrature
指導教授： 張森林
San-Lin Chang
張傳章
Chuang-Chang Chang
口試委員:
學位類別： 碩士
Master
系所名稱： 管理學院 - 財務金融學系
Department of Finance
畢業學年度： 91
語文別： 英文
論文頁數： 55
中文關鍵詞： 數值積分 、新奇選擇權 、GARCH 模型 、選擇權評價
外文關鍵詞： option pricing, GARCH model, exotic option, numerical quadrature
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    • I
    本論文應用數值積分方法來迅速且正確地評價選擇權的價值。吾人所建議的數
    值積分方法為高斯數值積分，因為其能達到數值積分的最高階次，所以可以非
    常逼近真實的選擇權價格。高斯數值積分的理念在於它不僅能夠選擇積分點的
    權重同時也能自由地決定積分點的位置，因此在同樣的積分點數之下，高斯數
    值積分的收斂階次將會是辛普森法的兩倍。數值結果顯示，本方法可以應用在
    非常廣泛的選擇權類型上同時也能應用在不同的標的資產演化過程上。利用本
    方法，我們將能進一步萃取市場上美式選擇權或其他新奇選擇權的隱含波動度以從事更進一步的研究。

    This paper develops an efficient and accurate method for numerical evaluation of the
    integral equations in option pricing problems. We suggest using the Gaussian
    quadratures, the highest order method in numerical integration, to approximate the
    option values. The idea of Gaussian quadratures is to give ourselves the freedom to
    choose not only the weight coefficients, but also the location of the abscissas at
    which the function is evaluated. It turns out that we can achieve Gaussian quadrature
    formulas whose convergence order is, essentially, twice that of Newton-Cotes
    formula (such as the Simpson''s rule) with the same number of points. The numerical
    results are extremely well for a broa d range of options and underlying asset price
    processes. With this powerful tool, it would be possible to extract information such
    as implied volatility from the market prices of American options and other exotic
    options.

    II Abstract ...........................................................................................................................I List of Tables................................................................................................................III List of Graph ................................................................................................................III 1. Introduction:................................................................................................................I 2. Literature Review:...................................................................................................... 4 3.Gauss-Legendre formula ：....................................................................................... 9 4. Compare of Gaussian quadrature Method and AWDN’s Simpson Method............. 11 4.1 Illustration of Gaussian quadratue Method .................................................... 11 4.2 Single observation ：European call case......................................................12 4.3 Multiply observations： ................................................................................16 4.3.1 Bermudan put case ： ........................................................................17 4.3.2 Other exotic options in AWDN’s paper ..............................................19 Case 1: Discrete barrier option.............................................................20 Case 2: Moving barrier option .............................................................21 Case 3: Compound call option .............................................................22 Case 4: American call option with changing strike price ....................23 Case 5: American option with dividends .............................................25 5.1 Reset option....................................................................................................35 5.2 knock-in option ..............................................................................................41 5.3 Pricing option with two underlying assets:....................................................42 5.4 GARCH Model: .............................................................................................44 References ....................................................................................................................54

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