1973年，Black-Scholes以熱傳導原理發表了著名的選擇權定價模型，在往後的30多年的時間為近代財務開啟了一條康莊大道。但是，Black-Scholes的基本假設過於簡化現實生活當中可能會面臨到的問題，因此許多新模型皆陸續的誕生，如隨機波動模型、隨機利率模型、跳躍-擴散模型等等。而這些新模型也都是秉持同樣一個理念，也就是進一步修改Black-Scholes模型使其新模型能夠更準確的去計算出選擇權價格。
    自從1987年發生了全球股災後，許多學者便針對所謂波動度微笑現象來做研究並加以改善此現象的發生。因此本文主要探討Heston的隨機波動度模型是否在台指選擇權方面也能比Black-Scholes模型之定價誤差來的小；並且加入另外兩種也是以改善Black-Scholes波動度假設的模型，包含了Merton的跳躍-擴散模型以及固定彈性波動度模型(CEV)。實證研究主要去衡量及比較模型價格和市場價格之誤差，並進行顯著性分析及誤差分析。本文的實證結果指出：
1. 樣本內的買權，Merton、Heston隨機波動度或CEV模型皆優於Black-Scholes模型。樣本內的賣權，整體部分以Heston隨機波動度模型為較佳模型。
2. 樣本外買權部分，以CEV模型之預測效果為較佳。另外，Merton模型未有明顯優於Black-Scholes模型。樣本外賣權部分CEV和Heston模型較好。
3. 樣本外誤差分析，主要的影響變數為價性程度以及到期期間，利率在買權部分有較明顯的影響。

This study is mainly to correct one of the Black-Scholes model assumptions. To assume volatility is a constant value which is not appropriate and to correct it. Since the Black-Scholes model occurred the famous volatility smile effect in the 1987’s market crash. My study is to use Merton’s jump-diffusion model, Heston’s stochastic volatility model and CEV model. These three models improved B-S model and to reduce volatility smile; moreover, to predict precisely the option price. Through these three models against BS model, to see whether there is a better improvement in using the TAIEX Options. The methodology is to use Fast Fourier Transform, Fast Fourier Transform no need to assume the underlying follow some distributions. It only needs the characteristic function of the model to calculate the model price and to do error analysis with the market price. The conclusion is that these three models improve the BS model indeed.

目錄
第一章 緒論	1
   第一節 研究背景與動機	1
   第二節 研究問題與目的	5
   第三節 研究架構	6
第二章 文獻回顧	8
   第一節 隨機波動度模型之演進	8
   第二節 文獻探討	20
第三章 研究方法	24
   第一節 快速傅立葉轉換之介紹	24
   第二節 選擇權評價模型	26
   第三節 參數估計方法	31
   第四節 資料來源	32
   第五節 定價模型績效衡量與誤差分析	33
第四章 實證結果與分析	35
   第一節 台股指數選擇權之參數估計與定價誤差	35
   第二節 模型間顯著性分析	47
   第三節 定價誤差之結構分析	58
第四節 波動度微笑之降低效果	62
第五章 結論與建議	65
   第一節 研究結論	65
   第二節 後續研究之建議與發展	66
參考文獻	67
附錄	70
圖目錄
圖1-1指數期貨與選擇權五年日均量比較圖	1
圖1-2根據Black & Scholes模型假設逐一放寬所延伸的模型	3
圖1-3研究流程圖	5
圖4-1波動度微笑	62
圖4-2 Heston波動度微笑曲面圖	63
圖4-3 BS模型和Merton模型之波動度微笑	63
圖4-4 BS模型和Heston模型之波動度微笑	64
圖4-5 BS模型和CEV模型之波動度微笑	64
表目錄
表2-5-1隨機波動度所服從之隨機過程	18
表4-1-1模型的參數估計值	34
表4-1-2樣本內買權、賣權整體之績效比較	38
表4-1-3樣本內近月買權、賣權之績效比較	39
表4-1-4樣本內遠月買權、賣權之績效比較	40
表4-1-5樣本外買權、賣權整體之績效比較	43
表4-1-6樣本外近月買權、賣權之績效比較	44
表4-1-7樣本外遠月買權、賣權之績效比較	45
表4-2-1買權樣本內近月四種模型之Wilcoxon符號等級檢定統計	48
表4-2-2賣權樣本內近月四種模型之Wilcoxon符號等級檢定統計	49
表4-2-3買權樣本內遠月四種模型之Wilcoxon符號等級檢定統計	50
表4-2-4賣權樣本內遠月四種模型之Wilcoxon符號等級檢定統計	51
表4-2-5買權樣本外近月四種模型之Wilcoxon符號等級檢定統計	53
表4-2-6賣權樣本外近月四種模型之Wilcoxon符號等級檢定統計	54
表4-2-7買權樣本外遠月四種模型之Wilcoxon符號等級檢定統計	55
表4-2-8賣權樣本外遠月四種模型之Wilcoxon符號等級檢定統計	56
表4-3-1買權及賣權樣本外整體之結構誤差分析	69
表4-3-2買權及賣權樣本外近月之結構誤差分析	60
表4-3-3買權及賣權樣本外遠月之結構誤差分析	61

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