# 臺灣博碩士論文加值系統

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 一開始我們先公式化Black-Scholes方程並且表示出它的收斂性. 然後我們找到一個基底函數使得我們得以公式化成Petrov-Galerkin有限元素法. 我們證明這樣的雙線性模式是coercive與連續的. 我們也說這樣的離散解有order h的誤差. 最後我們也用數值模擬了整個二維模型並且與一維模型做比較. 我們也得到這樣的二維模型是滿足M矩陣.
 In this dissertation we first formulate the Black-Scholes equation with a tensor (or matrix) diffusion coefficient into a conservative form and present a convergence analysis for the two-dimensional Black-Scholes equation arising in the Hull-White model for pricing European options with stochastic volatility. We formulate a non-conforming Petrov-Galerkin finite element method with each basis function of the trial space being determined by a set of two-point boundary value problems defined on element edges. We show that the bilinear form of the finite element method is coercive and continuous and establish an upper bound of order O(h) on the discretization error of method, where h denotes the mesh parameter of the discretization. We then present a finite volume method for the resulting equation, based on a fitting technique proposed for a one-dimensional Black-Scholes equation. We show that the method is monotone by proving that the system matrix of the discretized equation is an M-matrix. Numerical experiments, performed to demonstrate the usefulness of the method, will be presentd.
 1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 The Continuous Problem and its Solvability . . .. . 42.1 The Continuous Problem . . . . . . . . . . . . . . . . .. . 42.2 The Existence and Uniqueness . . . . . . . . . . . . . . . 62.2.1 Coercivity of the Bilinear Form B(·, ·; t) . . . . .. . . . 72.2.2 Continuity of the Bilinear Form B(·, ·; t) . . . . . . . . 83 The Finite Element Method . . . .113.1 The Finite Element Formulation of the Discretization Scheme . . . . . 113.2 Stability and Error Analysis of the Method . . . . . . 153.2.1 Lower Bound for the Bilinear Form B(·, ·) . . . . . 163.2.2 Error Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 The Finite Volume Method. . . . . . . . . 254.1 The Formulation for the Flux κ(f) on Different Intervals . . . . . . . . 264.2 The Finite Volume System . . . . . . . . . .. . . . . . . . 294.3 Boundary Conditions and Payoff Conditions . . . 325 Numerical Experiments . . . . . . . . 345.1 Ramp Payoff Final Condition . . . . . . . . . . . . . . .. . . 345.2 Cash-or-Nothing Final Condition . . . . . . . . . . . . . 375.3 Portfolio of Options . . . . . . . . . . . . . . .. . . . . . . 395.4 2-D and 1-D Simulations . . . . . . . . . . .. . . . . . . . . 406 Conclusions . . . . 45Bibliography. . . . 46
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 1 多維度選擇權價值於Black-Scholes方程之有限體積法

 1 馮丁樹、王永明。1987。循環式乾燥機乾燥榖粒之模擬與研究。農業工程學報 2 馮丁樹。1996。循環式稻穀乾燥模式之建立及應用。農業機械學刊5:1-16。

 1 關於趨近平坦之徑向基底函數求解橢圓算子的特徵模組問題 2 關於趨近平坦之多類型徑向基底函數的內插問題 3 以二邊逐次修正平行演算法加速物流派車問題最佳解的搜尋 4 模糊數排程問題的CUDA改良基因演算法 5 電視購物產業消費滿意度研究 6 混合資料探勘與改良型支撐向量機應用於短期負載預測 7 國中課程綱要代數分年細目詮釋的等號類型之內容分析 8 Levy過程之下的一般化Sharpe指標 9 股價操縱因素對於交叉上市股價差異的影響-以兩岸三地股票市場為例 10 跨功能團隊運作效能之研究 11 服飾零售業在國際合資的挑戰；以台灣瑪莎百貨為例 12 虛擬社群發展策略之研究–以愛情公寓為例 13 精品名牌包消費模式之消費者知覺風險探討 14 多維度選擇權價值於Black-Scholes方程之有限體積法 15 加速區域分解法之實作

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