財務極端事件分析方法:二元柏拉圖連結函數與二元Zipf時間序列模型

An Analytic Method for Financial Extremal Events:Bivariate Pareto Copula and Zipf Time Series Model

胡以賢(Yi-Hsien Hu)

指導教授 ： 葉小蓁

國立臺灣大學/管理學院/財務金融學研究所/碩士(2007年)

https://doi.org/10.6342/NTU.2007.03013

摘要

此篇論文根據「柏瑞圖分配」(Pareto distribution)、「連結函數」(copula function)以及「點過程理論」(point processes theory)提出一個對於財務金融極端事件分析方法，共分為兩大部分：靜態分析和動態應用。在靜態分析部分，探討的二元柏瑞圖連結函數特性，包含了其脆弱性(frailty)、混合以及阿基米德(Archimedean)結構、尾部相關、收斂性質、蒙地卡羅(Monte Carlo)模擬方法等；並使用電腦模擬來驗證上述數學性質。在動態應用部分，本文將二元柏瑞圖連結函數嵌入一離散型時間序列模型。接著探討其相關性質、馬可夫結構(Markov structure)、相鄰n期聯合分配等；接著利用動態模擬來探討重要的參數影響作用。

關鍵字

財務極端事件 ； 二元柏瑞圖分配 ； 連結函數 ； 阿基米德結構 ； 尾部相關 ； 脆弱性結構

並列摘要

An analytic method for financial extremal events is presented in this paper. This method is based on Pareto distribution, copula theory, and a point process. This paper is divided into two parts: static analysis and dynamic application. In the static part, we explore some properties of bivariate Pareto copula, including its frailty, mixture, Archimedean structure, tail dependence, convergence property, Monte Carlo simulation method etc., perform static simulations, and explain the applications in finance. In the dynamic part, bivariate Pareto copula is embedded into a discrete time series model. Then the correlation structure, Markov structure, and the joint distribution of n-periods etc. are discussed. The effects of parameters and correlation characteristics are examined in the dynamic simulations.

並列關鍵字

financail extremal events ； bivariate Pareto distribution ； copula function ； Archimedean structure ； tail dependence ； frailty structure ； Zipf time series model

參考文獻

2. Axtell, R. L. (2001). Zipf distribution of U.S. Firm Sizes. Science 293, 1818-1820.

3. Berchtold A. and Raftery A. E. (2002). The mixture transition distribution model for high-order Markov chains and non-Gaussian time series. Statistical Science 17, 328-356.

7. Embrechts, P., Klüppelberg, C., Mikosch, T., 1997. Modelling Extremal Events for Insurance and Finance. Springer, Berlin.

8. Fujiwara Y. (2004). Zipf law in frms bankruptcy. Physica A 337, 219-230.

9.Genest, C. and Mackay, J. (1986) The joy of copulas: Bivariate distributions with uniform margins. The American Statistician 40, 280-283.