VIX Index Analysis using Copula-Based Markov Chain Models

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吳家融(Jia-Rong Wu) 查詢紙本館藏

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統計研究所

論文名稱

(VIX Index Analysis using Copula-Based Markov Chain Models)

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摘要(中)

VIX 波動率指數，全名為「芝加哥選擇權交易所波動率指數」（CBOE Volatility Index），是芝加哥選擇權交易所根據標普 500 指數選擇權的價格，推算隱含波動率，再經過加權平均後所得的指標。我們透過 copula 之下的馬可夫鍊模型去探討波動率指數的相關性。由於波動率指數恆正，我們使用邊際分布為伽瑪分配。因為模型設定帶有累積機率函數的部分，伽瑪分配的累積機率函數並不為封閉式，以及在對模型進行偏微分所產生的問題，我們決定用貝氏理論方法去估計模型的參數。藉由 Metropolis-Hastigs 演算法可以估計出我們模型中的參數，再來利用條件機率的方式去產生有相關性的模擬資料驗證貝氏理論方法在 copula 之下的馬可夫鍊模型下是可以去估計的。最後，我們選用 VIX指數作為我們的實證分析資料。

摘要(英)

The VIX volatility index, which is known as the "CBOE Volatility Index," is the index that Chicago Option Exchange based on S&P 500 Index to calculate the implied volatility, then obtain through weighted average. We use the copula based Markov chain model to explore the relevance of the volatility index. Since the volatility index is positive, we use gamma distribution as the marginal distribution. Because the model sets the part with the cumulative probability function, the cumulative probability function of the gamma allocation is not closed, and the problems arising from partial differentiation of the model, we decided to use Bayesian theory to estimate the parameters of the model. The Metropolis-Hastings algorithm can be used to estimate the parameters in our model, and then use the conditional probability method to generate correlated simulation data to verify the Bayesian theory method under the copula-based Markov chain model can be estimated. In empirical analysis, we use the VIX index as our empirical analysis data.

關鍵字(中)

★ Copula
★ 伽瑪分配
★ VIX指數
★ 馬可夫鏈-蒙地卡羅演算法

關鍵字(英)

★ Copula
★ Gamma distribution
★ VIX index
★ Markov Chain Monte Carlo algorithm

論文目次

摘要 i
abstract ii
致謝 iii
. 1 Introduction 1
. 2 Copula-Based Markov models 5
. 2.1 Gamma distribution ........................ 5
. 2.2 Background knowledge....................... 5
. 2.3 Model assumption .......................... 7
. 2.4 The likelihood function of Clayton copula............ 8
. 3 Bayesian Inference 10
. 3.1 Selection of hyperparameters 1............. 10
. 3.2 Selection of hyperparameters 2 ............ 12
. 3.3 Bayesian estimate.......................... 13
3.4 Metropolis-Hastingsalgorithm.................... 14
. 4 Simulation Study 17
. 4.1 Simulation Methods ........................ 17
. 4.2 Simulation result ......................... 19
. 5 Empirical Study 28
. 5.1 VIX index.................................. 28
. 5.2 Data description .......................... 29
. 5.3 Empirical result........................... 29
. 6 Conclusion 31
Reference 33

參考文獻

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指導教授

孫立憲(Li-Hsien Sun)

審核日期

2018-7-24

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