英文标题：
《Model-free bounds on Value-at-Risk using extreme value information and
statistical distances》
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作者：
Thibaut Lux, Antonis Papapantoleon
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最新提交年份：
2018
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英文摘要：
We derive bounds on the distribution function, therefore also on the Value-at-Risk, of $\\varphi(\\mathbf X)$ where $\\varphi$ is an aggregation function and $\\mathbf X = (X_1,\\dots,X_d)$ is a random vector with known marginal distributions and partially known dependence structure. More specifically, we analyze three types of available information on the dependence structure: First, we consider the case where extreme value information, such as the distributions of partial minima and maxima of $\\mathbf X$, is available. In order to include this information in the computation of Value-at-Risk bounds, we utilize a reduction principle that relates this problem to an optimization problem over a standard Fr\\\'echet class, which can then be solved by means of the rearrangement algorithm or using analytical results. Second, we assume that the copula of $\\mathbf X$ is known on a subset of its domain, and finally we consider the case where the copula of $\\mathbf X$ lies in the vicinity of a reference copula as measured by a statistical distance. In order to derive Value-at-Risk bounds in the latter situations, we first improve the Fr\\\'echet--Hoeffding bounds on copulas so as to include this additional information on the dependence structure. Then, we translate the improved Fr\\\'echet--Hoeffding bounds to bounds on the Value-at-Risk using the so-called improved standard bounds. In numerical examples we illustrate that the additional information typically leads to a significant improvement of the bounds compared to the marginals-only case.
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中文摘要：
我们推导了$\\varphi（\\mathbf X）$的分布函数的界，因此也推导了风险值的界，其中$\\varphi$是一个聚合函数，$\\mathbf X=（X_1，dots，X_d）$是一个具有已知边缘分布和部分已知依赖结构的随机向量。更具体地说，我们分析了依赖结构上的三种可用信息：首先，我们考虑了极值信息的情况，例如$\\mathbf X$的部分最小值和最大值的分布。为了在计算风险值边界时包含此信息，我们利用了一个简化原则，将此问题与标准Fr趶echet类上的优化问题联系起来，然后可以通过重排算法或使用分析结果来解决该问题。其次，我们假设$\\mathbf X$的copula在其域的子集上是已知的，最后我们考虑$\\mathbf X$的copula位于由统计距离测量的参考copula附近的情况。为了推导出后一种情况下的风险值界限，我们首先改进了copulas上的Fr？echet-hoefffding界限，以便包含关于依赖结构的额外信息。然后，我们使用所谓的改进标准边界，将改进的Fr\\echet-hoeffing边界转换为风险值的边界。在数值例子中，我们说明了与仅边缘情况相比，附加信息通常会导致边界的显著改进。
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分类信息：

一级分类：Quantitative Finance 数量金融学
二级分类：Risk Management 风险管理
分类描述：Measurement and management of financial risks in trading, banking, insurance, corporate and other applications
衡量和管理贸易、银行、保险、企业和其他应用中的金融风险
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一级分类：Mathematics 数学
二级分类：Probability 概率
分类描述：Theory and applications of probability and stochastic processes: e.g. central limit theorems, large deviations, stochastic differential equations, models from statistical mechanics, queuing theory
概率论与随机过程的理论与应用：例如中心极限定理，大偏差，随机微分方程，统计力学模型，排队论
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