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姓名 柯坤義(Kun-Yi Ko)  查詢紙本館藏   畢業系所 財務金融學系
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)
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摘要(中) 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。
摘要(英) 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.
關鍵字(中) ★ 數值積分
★ 新奇選擇權
★ GARCH 模型
★ 選擇權評價
關鍵字(英) ★ option pricing
★ GARCH model
★ exotic option
★ numerical quadrature
論文目次 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
參考文獻 [1] Abramowitz, M., and I.A. Stegun (ed.), 1964 Handbook of Mathematical
Functions, National Bureau of Standa rds Applied Mathematics Series 53, USGPO,
Washington, D.C.
[2] Andricopoulos Ari D., Widdicks Martin, Duck Peter W., Newton David P.,
Universal Option Valuation Using Quadrature Methods.
[3] Black, F., and M. Scholes, 1973, The Pricing of Options and Cor porate Liabilities,
Journal of Political Economy, 81, 637-654.
[4] Broadie Mark, Detemple Jerome, 1996, American Option Valuation: New Bounds,
Approximations, and a Comparison of Existing Methods, Review of Financial Studies,
Vol.9, No.4, pp.1211-1250.
[5] Duan Jin-Chuan, Simonato Jean-Guy, 2001, American option pricing under
GARCH by a Markov chain approximation, Journal of Economic Dynamics &
Control 25, 1689-1718.
[6] Figlewski Stephen, Gao Bin, 1999, The adaptive mesh model: a new approach to
efficient option pricing, Journal of Financial Economics 53, 313-351.
[7] Heston Steve, Zhou Guofu, 2000, On The Rate Of Convergence Of Discrete-Time
Contingent Claims, Mathematical Finance, Vol. 10, No. 1, 53-75.
[8] Hull, J., 2000, Options, futures, and other derivatives, fourth edition, Prentice Hall.
[9] Longstaff Francis A., Schwartz Eduardo S., 2001,Valuing American Options by
Simulations: A Simple Least-Squares Approach, Review of Financial Studies Vol.14,
No. 1,113-147.
[10] Press,W., S.Teukolsky, W.Vetterling, and B. Flannery, 1992, Numerical Recipes
in C 2nd Editon, Cambridge University Press, New York.
[11] Sullivan Michael A., Pricing Discretely Monitored Barrier Options.
[12] Sullivan Michael A., 2000, Valuing American Put Options Using Gaussian
Quadrature,, Review of Financial Studies Vol. 13, No.1, 75-94.
[13] Pearson Neil D., 1995, An Efficient Approach For Pricing Spread Options, The
Journal Of Derivatives, 76-91.
指導教授 張森林、張傳章
(San-Lin Chang、Chuang-Chang Chang)
審核日期 2003-6-19
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