臺灣博碩士論文加值系統

選擇權評價模型發展至今已是非常成熟的工具，實務上也廣泛地運用，然而台指選擇權仍然是相當新穎的衍生性金融商品。相關的實證研究並不多，因此本論文探討最適合台指選擇權價格評價之模型。以 Black-Scholes 模型為基礎進行蒙地卡羅模擬，因股價波動的隨機性質，我們必須使用蒙地卡羅模擬，然而未經改善的模擬其變異數相當地大，並不能讓人放心的相信，在研究中，我們藉由數種降低模擬變異數的方法，期望能得到更貼近於市場價格且能有效地將模擬的變異數降低，讓模擬所得到的數值更有說服力。在我們的實證中，也確實發現我們提出的方法能有效地降低模擬的變異數且貼近市場價格，所以本研究嘗試以台指選擇權為研究對象，找出最適合進行台指選擇權模擬之模型。

1 前言
2 數學知識 … 2
2.1 布朗運動與平賭. . . . . . . . . . . . . . . . . . . . . . . . 2
2.2 The Ito Calculus . . . . . . . . . . . . . . . . . . . . . . 7
2.2.1 The Itˆo Calculus of simple function . . . . . . . . 8
2.2.2 The general Itˆo Integral . . . . . . . . . . . . . . 12
2.2.3 Itˆo Formula . . . . . . . . . . . . . . . . . . . . . 15
2.3 測度轉換 - Cameron-Martin-Girsanov 定理 . . . . . . . . . 16
3 蒙地卡羅運用於歐式選擇權評價
3.1 Black-Scholes 模型 . . . . . . . . . . . . . . . . . . . . . . 23
3.2 蒙地卡羅模擬介紹 . . . . . . . . . . . . . . . . . . . . . . . 27
3.2.1 蒙地卡羅積分. . . . . . . . . . . . . . . . . . . . . 27
3.2.2 製造隨機亂數 . . . . . . . . . . . . . . . . . . . . . 28
3.2.3蒙地卡羅模擬運用於歐式選擇權評價. . . . . . . . . . 30
3.3 降低蒙地卡羅模擬變異數方法介紹. . . . . . . . . . . . . . . 31
3.3.1 動差配適模擬法 (Moment Matching Simulation) . . 31
3.3.2 相法亂數法 (Antithetic Random Numbers) . . . . . 32
3.3.3 控制變數法 (Control Variates) . . . . . . . . . . . . 33
4 實證分析
4.1 資料來源與說明 . . . . . . . . . . . . . . . . . . . . . . . . 37
4.2 蒙地卡羅模擬評價台股選擇權之結果 . . . . . . . . . . . . . . 39
5 結論與建議 54
參考文獻. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

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