基于group SCAD惩罚的非对称乘法copula模型选择及其应用

捕捉变量间相依结构中的非对称性有助于把握其间的地位关系.非对称乘法copula模型常用于刻画非对称相依结构,但在应用中面临模型选择问题.文章首次将正则化思想与非对称乘法copula模型相结合,并依据权重参数的群组结构对模型中各copula成分的权重施加group SCAD惩罚,构建惩罚似然函数,采用单步LLA算法得到权重的稀疏估计,自动剔除对模型整体贡献较小的copula成分,实现模型选择.同时,文章还给出了惩罚似然估计量的收敛率及其证明.在数值模拟中,文章所构建的模型选择方法具有较高的准确性与精度,在模型误设定的情形下则会选择出与真实模型最为接近的copula成分组合.在实证分析中,文章应用非对称乘法copula模型分析医药产业板块的横向与纵向关联,从结果来看本文的模型选择方法能较好地应对实际数据,选择出合适的copula成分组合,刻画板块相依结构中潜在的非对称性.     

中国股市相依结构测定初探

提出了中国股市测定copula相依结构的一般方法,并结合中国股市的实际数据作了分析.在假定边际分布为正态分布时,得到了描述工业指数与商业指数相依结构的较好copula结构为正态copula族.     

时长不等数据的vine-copula建模及多资产组合VaR分析

传统的copula模型在对二维以上相依结构建模时存在参数过少的缺陷,vine copula理论基本弥补了这一缺陷.介绍了vine copula理论以及其相对于传统多元模型的优势,尤其提出了vine copula对于时长不一致的数据进行建模具有数据利用率较高的特性,给出了这类数据vine copula的建模步骤以及基于极大似然估计的统计推断.最后对国内A股市场的五种金融股票的联合分布进行建模,并利用蒙特卡罗方法对资产组合的VaR进行了模拟.     

极值copula的统计推断及实证研究

为了分析由极端事件所引起的巨额损失变量之间的相依关系,本文引入了比一般copula函数更有效的极值copula和上尾copula。我们介绍copula的对角截面以确定上尾相依系数。基于极大似然法,讨论了关于这些copula函数类的半参数估计方法。通过构建Cramer-Von Mises统计量对copula的拟合优度进行假设检验。在实证分析部分,我们通过具体的实例来说明,在应用研究中该如何选取最优的copula以描述变量之间的相关性。     

基于R-Vine Copula模型的情景模拟研究

相关性分析是多变量分析中的一个中心问题,而压力情景模拟则是确定某变量在多变量系统中重要性的常用方法.文章针对灵活刻画多变量相依结构的R-vine copula模型,提出了R-vine结构下的情景模拟算法,并以德国五公司收益率序列为样本系统,发现了系统内个体相依结构,模拟了个体在上、下尾极值情景下,其他个体及整个系统的响应情况,发现了不同行情下的系统重要性企业,实证结果与现实基本一致.所提出的算法可用于金融管理领域的风险传染、系统重要性机构的识别以及宏观审慎监管等方面的研究.     

C~(A,B)copula在CreditMetrics中的应用

利用copula方法初步探讨了CreditMetrics中相关性假定对计算Value-atrisk结果的影响.将相关文献提出的C~(A,B) copula应用到CreditMetrics中来替代传统的正态copula函数.在理论上探讨了C~(A,B) copula的系数的确定方法,并针对二元情形进行了数值分析.     

基于Copula的金融市场相关分析

本文采用连接函数(copula)方法研究上证和深证市场的相关性。对上证指数和深证成指收益率的边缘分布分别用正则逆Gamma分布(Normal inverse gamma mixture)、偏T分布(Skew Student-t,SST)进行拟合,然后在此基础上采用copula函数方法建立两者的联合分布。其中的copula函数分别用Gumbel-Hougaard copula、Frank copula和Clayton copula,相依参数应用推断函数法(method of inference functions,IFM方法)估计。结果表明沪深两证券市场具有相关性。     

一个基于pair-copula法构建高维相依结构的研究

用pair-copula构建高维相依结构,将n维联合密度函数转化为若干个pair-copula密度函数相乘。在pair-copula的选择上,本文构造了能描述非对称尾部相关性的混合copula函数——M-copula。并在实证分析部分用该方法探索了上证市场上四个板块的相依关系,得到了比较理想的结果。     

基于广义FGM Copula的相依和扰动风险模型下的Gerber-Shiu函数分析（英文）

该文考虑了带扰动的相依风险模型,并以一类广义的Farlie-Gumbel-Morgenstern copula定义了索赔额和索赔时间间隔之间的相依结构.首先,该模型下期望折扣罚金函数所满足的积分方程、拉普拉斯变换和瑕疵更新方程被给出.最后当索赔额分布为指数分布时,给出了期望折扣罚金函数所满足的解析解和破产概率的数值实例.     

基于pair-copula情景生成和广义熵约束的CVaR投资组合模型研究

通过GARCH模型对收益率序列的边缘分布建模,结合copula构建收益率的联合分布函数,并由蒙特卡洛模拟生成收益率的情景,得到的结果代入广义熵约束的CVaR模型中,由此得到最优的投资权重.实证表明,在考虑不同资产之间的相依结构基础上得到的最优化结果相比传统的M-V模型具有明显的优势,在分散化和收益性上的到很好的效果.     

Estimation of nonstrict Archimedean copulas and its application to quantum networks

Bivariate nonstrict Archimedean copulas form a subclass of Archimedean copulas and are able to model the dependence structure of random variables that do not take on low quantiles simultaneously; i.e. their domain includes a set, the so‐called zero set, with positive Lebesgue measure but zero probability mass. Standard methods to fit a parametric Archimedean copula, e.g. classical maximum likelihood estimation, are either getting computationally more involved or even fail when dealing with this subclass. We propose an alternative method for estimating the parameter of a nonstrict Archimedean copula that is based on the zero set and the functional form of its boundary curve. This estimator is fast to compute and can be applied to absolutely continuous copulas but also allows singular components. In a simulation study, we compare its performance to that of the standard estimators. Finally, the estimator is applied when modeling the dependence structure of quantities describing the quality of transmission in a quantum network, and it is shown how this model can be used effectively to detect potential intruders in this network. Copyright © 2014 John Wiley & Sons, Ltd.     

半参数阿基米德Copula的实证研究

半参数阿基米德Copula族的生成元可由现有阿基米德Copula生成元得到,由于有独特的构造方式,该Copula族具有灵活的相关结构,能"自适应"地描述数据中包含的相关结构.外汇市场的实证分析证实了该Copula族在描述相关结构时的灵活性,对选择何种Copula描述金融资产间的相关结构有一定的参考意义.     

Efficient algorithms for basket default swap pricing with multivariate Archimedean copulas

We introduce a new importance sampling method for pricing basket default swaps employing exchangeable Archimedean copulas and nested Gumbel copulas. We establish more realistic dependence structures than existing copula models for credit risks in the underlying portfolio, and propose an appropriate density for importance sampling by analyzing multivariate Archimedean copulas. To justify efficiency and accuracy of the proposed algorithms, we present numerical examples and compare them with the crude Monte Carlo simulation, and finally show that our proposed estimators produce considerably smaller variances.     

Modeling defaults with nested Archimedean copulas

In 2001, Schönbucher and Schubert extended Li’s well-known Gaussian copula model for modeling dependent defaults to allow for tail dependence. Instead of the Gaussian copula, Schönbucher and Schubert suggested to use Archimedean copulas. These copulas are able to capture tail dependence and therefore allow a standard intensity-based default model to have a positive probability of joint defaults within a short time period. As can be observed in the current financial crisis, this is an indispensable feature of any realistic default model. Another feature, motivated by empirical observations but rarely taken into account in default models, is that modeled portfolio components affected by defaults show significantly different levels of dependence depending on whether they belong to the same industry sector or not. The present work presents an extension of the model suggested by Schönbucher and Schubert to account for this fact. For this, nested Archimedean copulas are applied. As an application, the pricing of collateralized debt obligations is treated. Since the resulting loss distribution is not analytical tractable, fast sampling algorithms for nested Archimedean copulas are developed. Such algorithms boil down to sampling certain distributions given by their Laplace-Stieltjes transforms. For a large range of nested Archimedean copulas, efficient sampling techniques can be derived. Moreover, a general transformation of an Archimedean generator allows to construct and sample the corresponding nested Archimedean copulas.     

Hierarchical Archimedean copulas through multivariate compound distributions

In this paper, we propose a new hierarchical Archimedean copula construction based on multivariate compound distributions. This new imbrication technique is derived via the construction of a multivariate exponential mixture distribution through compounding. The absence of nesting and marginal conditions, contrarily to the nested Archimedean copulas approach, leads to major advantages, such as a flexible range of possible combinations in the choice of distributions, the existence of explicit formulas for the distribution of the sum, and computational ease in high dimensions. A balance between flexibility and parsimony is targeted. After presenting the construction technique, properties of the proposed copulas are investigated and illustrative examples are given. A detailed comparison with other construction methodologies of hierarchical Archimedean copulas is provided. Risk aggregation under this newly proposed dependence structure is also examined.     

Tail order and intermediate tail dependence of multivariate copulas

In order to study copula families that have tail patterns and tail asymmetry different from multivariate Gaussian and t copulas, we introduce the concepts of tail order and tail order functions. These provide an integrated way to study both tail dependence and intermediate tail dependence. Some fundamental properties of tail order and tail order functions are obtained. For the multivariate Archimedean copula, we relate the tail heaviness of a positive random variable to the tail behavior of the Archimedean copula constructed from the Laplace transform of the random variable, and extend the results of Charpentier and Segers [7] [A. Charpentier, J. Segers, Tails of multivariate Archimedean copulas, Journal of Multivariate Analysis 100 (7) (2009) 1521–1537] for upper tails of Archimedean copulas. In addition, a new one-parameter Archimedean copula family based on the Laplace transform of the inverse Gamma distribution is proposed; it possesses patterns of upper and lower tails not seen in commonly used copula families. Finally, tail orders are studied for copulas constructed from mixtures of max-infinitely divisible copulas.     

From Archimedean to Liouville copulas

We use a recent characterization of the d-dimensional Archimedean copulas as the survival copulas of d-dimensional simplex distributions (McNeil and Nešlehová (2009) [1]) to construct new Archimedean copula families, and to examine the relationship between their dependence properties and the radial parts of the corresponding simplex distributions. In particular, a new formula for Kendall’s tau is derived and a new dependence ordering for non-negative random variables is introduced which generalises the Laplace transform order. We then generalise the Archimedean copulas to obtain Liouville copulas, which are the survival copulas of Liouville distributions and which are non-exchangeable in general. We derive a formula for Kendall’s tau of Liouville copulas in terms of the radial parts of the corresponding Liouville distributions.     

Construction of asymmetric multivariate copulas

In this paper we introduce two methods for the construction of asymmetric multivariate copulas. The first is connected with products of copulas. The second approach generalises the Archimedean copulas. The resulting copulas are asymmetric and may have more than two parameters in contrast to most of the parametric families of copulas described in the literature. We study the properties of the proposed families of copulas such as the dependence of two components (Kendall’s tau, tail dependence), marginal distributions and the generation of random variates.     

Constructing hierarchical Archimedean copulas with Lévy subordinators

A probabilistic interpretation for hierarchical Archimedean copulas based on Lévy subordinators is given. Independent exponential random variables are divided by group-specific Lévy subordinators which are evaluated at a common random time. The resulting random vector has a hierarchical Archimedean survival copula. This approach suggests an efficient sampling algorithm and allows one to easily construct several new parametric families of hierarchical Archimedean copulas.     

Lower tail dependence for Archimedean copulas: Characterizations and pitfalls

Tail dependence copulas provide a natural perspective from which one can study the dependence in the tail of a multivariate distribution. For Archimedean copulas with continuously differentiable generators, regular variation of the generator near the origin is known to be closely connected to convergence of the lower tail dependence copulas to the Clayton copula. In this paper, these characterizations are refined and extended to the case of generators which are not necessarily continuously differentiable. Moreover, a counterexample is constructed showing that even if the generator of a strict Archimedean copula is continuously differentiable and slowly varying at the origin, then the lower tail dependence copulas still do not need to converge to the independent copula.