臺灣博碩士論文加值系統

Black and Scholes (1973)對於報酬率提出以B-S模型配適，但B-S模型無法有效解釋報酬率不對稱高狹峰、波動度微笑、波動度叢聚、長記憶性的性質。Merton (1976)認為不尋常的訊息來臨會影響股價不連續跳躍，因此發展B-S模型加入不連續跳躍風險項的跳躍擴散模型，該模型可同時描述報酬率不對稱高狹峰和波動度微笑兩性質。Charles, Fuh and Lin (2011)加以考慮市場狀態提出狀態轉換跳躍模型，除了保留跳躍擴散模型可描述報酬率不對稱高狹峰和波動度微笑，更可以敘述報酬率的波動度叢聚和長記憶性。本文進一步拓展狀態轉換跳躍模型，考慮不連續跳躍風險項的帄均數與市場狀態相關，提出狀態轉換跳躍相關模型。並以道瓊工業指數與S&;P 500指數1999年至2010年股價指數資料，採用EM和SEM分別估計參數與估計參數共變異數矩陣。使用概似比檢定結果顯示狀態轉換跳躍相關模型比狀態轉換跳躍獨立模型更適合描述股價指數報酬率。並驗證狀態轉換跳躍相關模型也可同時描述報酬率不對稱高狹峰、波動度微笑、波動度叢聚、長記憶性。最後利用Esscher轉換法計算股價指數選擇權定價公式，以敏感度分析模型參數對於定價結果的影響，並且市場驗證顯示狀態轉換跳躍相關模型會有最小的定價誤差。

Black and Scholes (1973) proposed B-S model to fit asset return, but B-S model can’t effectively explain some asset return properties, such as leptokurtic, volatility smile, volatility clustering and long memory. Merton (1976) develop jump diffusion model (JDM) that consider abnormal information of market will affect the stock price, and this model can explain leptokurtic and volatility smile of asset return at the same time. Charles, Fuh and Lin (2011) extended the JDM and proposed regime-switching jump independent model (RSJIM) that consider jump rate is related to market states. RSJIM not only retains JDM properties but describes volatility clustering and long memory. In this paper, we extend RSJIM to regime-switching jump dependent model (RSJDM) which consider jump size and jump rate are both related to market states. We use EM and SEM algorithm to estimate parameters and covariance matrix, and use LR test to compare RSJIM and RSJDM. By using 1999 to 2010 Dow-Jones industrial average index and S&P 500 index as empirical evidence, RSJDM can explain index return properties said before. Finally, we calculate index option price formulation by Esscher transformation and do sensitivity analysis and market validation which give the smallest error of option prices by RSJDM.

1. 介 紹 …………………………………………………………………….1
2. 文獻回顧…………………………………………………………………………5
2.1 股價指數選擇權…………………………………………………………..5
2.2 跳躍擴散模型…………………………………………………………….6
2.3 隨機波動度模型…………………………………………………………11
2.4 Esscher 轉換…………………………………………………………….13
3. 股價指數模型…………………………………………………………………15
3.1 跳躍擴散模型……………………………………………………………15
3.2 狀態轉換跳躍獨立模型…………………………………………………16
3.3 狀態轉換跳躍相關模型…………………………………………………18
3.4 馬可夫調控普瓦松過程…………………………………………………20
3.5 狀態轉換跳躍相關模型之估計與檢定…………………………………21
4. 股價指數選擇權定價…………………………………………………………25
4.1 Esscher 轉換……………………………………………………………25
4.1.1 跳躍擴散模型下Esscher 轉換…………………………………26
4.1.2 狀態轉換跳躍獨立模型下Esscher 轉換………………………27
4.1.3 狀態轉換跳躍相關模型下Esscher 轉換………………………30
4.2股價指數選擇權定價……………………………………………………33
4.2.1 跳躍擴散模型下股價指數選擇權定價…………………………33
4.2.2 狀態轉換跳躍獨立模型下股價指數選擇權定價………………34
4.2.3 狀態轉換跳躍相關模型下股價指數選擇權定價………………35
5. 實證分析、敏感度分析與市場驗證………………………………………36
5.1 實證分析…………………………………………………………………36
5.1.1 估計與檢定 ………………………………………………………36
5.1.2 模型經濟分析……………………………………………………39
5.1.3 峰態與偏態………………………………………………………40
5.1.4 波動叢聚…………………………………………………………41
5.1.5 波動度微笑………………………………………………………41
5.2 敏感度分析………………………………………………………………42
5.3 市場驗證…………………………………………………………………44
6. 結論……………………………………………………………………………45
參考文獻……………………………………………………………………………47
中文文獻………………………………………………………………………47
英文文獻………………………………………………………………………47
附錄…………………………………………………………………………………51
附錄A：跳躍擴散模型歐式買權定價公式………………………………51
附錄B：狀態轉換跳躍獨立模型歐式買權定價公式………………………56
附錄C：狀態轉換跳躍相關模型歐式買權定價公式……………………62
附錄D：狀態轉換跳躍相關模型自我相關函數公式………………………68
表目錄
表1.1：1999年至2010年道瓊工業指數報酬率敘述統計表.................................2
表5.1：道瓊工業指數與S&P 500模型參數估計與檢定…………………69
表5.2：跳躍擴散模型動差公式………………………………………………70
表5.3：狀態轉換跳躍獨立動差公式……………………………………71
表5.4：狀態轉換跳躍相關模型動差公式…………………………………72
表5.5：道瓊工業指數與S&P 500模型動差估計……………………………73
表5.6：轉移機率敏感度分析…..……………………………………………74
表5.7：布朗運動項標準差與跳躍幅度標準差敏感度分析……………74
表5.8：跳躍幅度平均數敏感度分析………………………………………74
表5.9：跳躍頻率敏感度分析………………………………………………74
表5.10：道瓊工業指數歐式買權樣本外定價誤差…………………….75
表5.11：S&P 500指數歐式買權樣本外定價誤差……………………………75
圖目錄
圖1.1：芝加哥選擇權交易所各年度指數選擇權交易量…………………1
圖1.2：道瓊工業指數報酬率動態圖………………………………3
圖5.1：道瓊工業指數股價、報酬率、狀態及跳躍機率動態圖...........76
圖5.2：S&P 500指數股價、報酬率、狀態及跳躍機率動態圖...........77
圖5.3：道瓊工業指數報酬率動態、資料與RSJDM自我相關函數.........78
圖5.4：S&P 500指數報酬率動態、資料與RSJDM自我相關函數.........78
圖5.5：道瓊工業指數隱含波動度微笑………....................79
圖5.6：S&P 500指數隱含波動度微笑………………..……………..……79

中文文獻
[1] 吳聲杰，(2009)。狀態轉換跳躍模型下Supplemented Expectation Maximization 演算法與Gibbs Sampling演算法之參數估計之變異數估計，國立高雄大學統計學研究所碩士論文。
[2] 林晉煜，(2010)。狀態轉換跳躍模型下權益指數年金之評價公式：股價指數之實證，國立高雄大學統計學研究所碩士論文。
[3] 康怡禎，(2008)。跳躍幅度與跳躍頻率相依下馬可夫跳躍擴散模型在財務金融之實證分析，國立東華大學應用數學系碩士論文。
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