The analysis of log returns using copula-based Markov models

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李建賞(Chien-Shang Lee) 查詢紙本館藏

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論文名稱

(The analysis of log returns using copula-based Markov models)

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摘要(中)

在現實生活中，我們學過股票的報酬隨著每天的股價變化而改變。本文中，我們透過copula之下的馬可夫鍊模型去探討股價的對數報酬的相關性。由於在股票市場上報酬有厚尾的性質，所以我們使用邊際分布為非標準學生T分配。但是自由度的最大概似估計量在此模型假設下無法求得。所以我們決定用貝氏理論方法去估計模型的參數。藉由馬可夫鍊蒙地卡羅法中的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。在實證分析中，我們將選用S&P 500指數作為我們的實證分析資料。

摘要(英)

In the real world, we learn that log returns change from the variety of the stock price everyday. In this paper, we propose a copula-based Markov model to perform the log return for the stock price. Owing to the fat tail feature in stock market, we select non-standardized Student′s t-distribution being the marginal distribution. However, the maximum likelihood estimator of the degree of freedom can not be found. We decide to use bayes inference to estimate the parameters of copula-based Markov model. Using Metropolis-Hastings algorithm within Markov chain Monte Carlo method can find the bayes estimator of our model and developing the algorithm for generating correlated data by extending the conditional approach to test bayes inference can work for the copula-based Markov model. In the empirical analysis, the S&P 500 Index is analyzed for illustration.

關鍵字(中)

★ 對數報酬
★ 非標準學生T分配
★ 貝氏理論
★ 馬可夫蒙地卡羅法

關鍵字(英)

★ Log returns
★ copula
★ non-standardized Student′s t-distribution
★ bayes inference
★ Markov chain Monte Carlo

論文目次

摘要i
Abstract ii
誌謝iii
1 Introduction 1
2 Copula-based Markov models 4
2.1 Non-standardized Student’s t-distribution . . . . . . . . . . . . . . . . . . 4
2.2 Background knowledge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
2.3 Model assumption . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
2.4 The likelihood function of Clayton copula . . . . . . . . . . . . . . . . . . 7
2.5 The difficulties on the MLE . . . . . . . . . . . . . . . . . . . . . . . . . . 8
3 Bayesian Inference 9
3.1 Selection of hyperparameters . . . . . . . . . . . . . . . . . . . . . . . . . . 9
3.2 Bayes estimate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
3.3 Metropolis-Hastings algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 12
4 Simulation Study 15
4.1 Simulation Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
4.2 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Empirical Study 24
5.1 Data description . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
5.2 Empirical result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
6 Conclusion 30
References 32

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指導教授

孫立憲

審核日期

2017-7-7

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