结构凸体的实用体积计算 在投资组合相关性和金融危机建模中的应用

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摘要翻译：
我们研究一般维多边形和更一般的凸体的体积计算，这些凸体定义为一个单纯形与一族平行超平面、另一族平行超平面或一族同心椭球的交集。这种凸体出现在对金融危机的建模和预测中。危机对经济（劳动、收入等）的影响使它的主要利益被发现。市场依赖的某些特征清楚地表明了动荡时期。我们用Copula描述了资产特征之间的关系；每个特征都是投资组合分量的线性或二次型，因此copula可以通过计算凸体的体积来构造。我们在精确和近似的情况下设计和实现了实用的算法，并对它们进行了实验并置，研究了准确性和精确度与速度之间的折衷。本文按通用性递增的顺序分析了以下几种方法：拒绝抽样法依赖于单纯形的均匀抽样，它是最快的方法，但对于小体积的方法是不准确的；基于概率分布函数积分计算的精确公式；一种优化的劳伦斯符号分解方法，因为手边的多边形是简单的；基于随机游动的Markov链Monte Carlo算法推广到非线性凸体，并依赖于计算包球的新方法；后者在实验上推广到非凸体，得到了非常令人鼓舞的结果。我们的C++软件，基于CGAL和Eigen，并在github上提供，被证明在多达100个维度上是非常有效的。我们的结果提供了一种新的、有效的计算投资组合依赖性的方法和一个金融危机的指标，它被证明能够正确地识别过去的危机。
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英文标题：
《Practical volume computation of structured convex bodies, and an
  application to modeling portfolio dependencies and financial crises》
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作者：
Ludovic Cales, Apostolos Chalkis, Ioannis Z.Emiris, Vissarion
  Fisikopoulos
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最新提交年份：
2018
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分类信息：

一级分类：Computer Science        计算机科学
二级分类：Computational Geometry        计算几何
分类描述：Roughly includes material in ACM Subject Classes I.3.5 and F.2.2.
大致包括ACM课程I.3.5和F.2.2中的材料。
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一级分类：Economics        经济学
二级分类：Econometrics        计量经济学
分类描述：Econometric Theory, Micro-Econometrics, Macro-Econometrics, Empirical Content of Economic Relations discovered via New Methods, Methodological Aspects of the Application of Statistical Inference to Economic Data.
计量经济学理论，微观计量经济学，宏观计量经济学，通过新方法发现的经济关系的实证内容，统计推论应用于经济数据的方法论方面。
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一级分类：Quantitative Finance        数量金融学
二级分类：General Finance        一般财务
分类描述：Development of general quantitative methodologies with applications in finance
通用定量方法的发展及其在金融中的应用
--

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英文摘要：
  We examine volume computation of general-dimensional polytopes and more general convex bodies, defined as the intersection of a simplex by a family of parallel hyperplanes, and another family of parallel hyperplanes or a family of concentric ellipsoids. Such convex bodies appear in modeling and predicting financial crises. The impact of crises on the economy (labor, income, etc.) makes its detection of prime interest. Certain features of dependencies in the markets clearly identify times of turmoil. We describe the relationship between asset characteristics by means of a copula; each characteristic is either a linear or quadratic form of the portfolio components, hence the copula can be constructed by computing volumes of convex bodies. We design and implement practical algorithms in the exact and approximate setting, we experimentally juxtapose them and study the tradeoff of exactness and accuracy for speed. We analyze the following methods in order of increasing generality: rejection sampling relying on uniformly sampling the simplex, which is the fastest approach, but inaccurate for small volumes; exact formulae based on the computation of integrals of probability distribution functions; an optimized Lawrence sign decomposition method, since the polytopes at hand are shown to be simple; Markov chain Monte Carlo algorithms using random walks based on the hit-and-run paradigm generalized to nonlinear convex bodies and relying on new methods for computing a ball enclosed; the latter is experimentally extended to non-convex bodies with very encouraging results. Our C++ software, based on CGAL and Eigen and available on github, is shown to be very effective in up to 100 dimensions. Our results offer novel, effective means of computing portfolio dependencies and an indicator of financial crises, which is shown to correctly identify past crises.
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PDF链接：
https://arxiv.org/pdf/1803.05861

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关键词：金融危机 投资组合 相关性 econometrics Experimental volume bodies 实用 dependencies 单纯形