Systemic risk and topology

系统性风险和拓扑

• 批准号: 1312071
• 负责人: Henry Schenck
• 金额: $ 13.5万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-10-01 至 2016-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1312071&HistoricalAwards=false
• 关键词: Systemic; risk; topology

Schenck1312071 This project is focused on a novel application of recently developed tools in computational topology to certain problems of risk in financial networks. Complex problems often benefit from being viewed in a new light; the investigators use persistent homology to study behavior of financial trading networks. Persistent homology focuses on how a family of simplicial complexes D(t) vary with a real parameter t. Typically one starts with a discrete data set (point cloud data) corresponding to t=0, and then as t varies, points are replaced by balls of radius t, centered at the original point. Thus varying t yields a filtered family of simplicial complexes, the Rips-Vietoris complex of the associated topological space. In the context of trading networks, the parameter t represents margin; a low value for t corresponds to allowing highly leveraged trading. The aim of the project is to understand how changes in the topology of the D(t) associated to a trading network correspond in some way to systemic collapse. Put simply, the investigators explore the circumstances that cause a local event to cause global contagion. The problem of understanding interconnectedness is one of today's premier challenges. Connectivity is ubiquitous; interconnections often allow a system to be tapped to its full potential and enhance its robustness. The flip side of connectivity is that it gives rise to pathways for unexpected emergent behavior ("black swan" events) and even systemic collapse. The world is awash in data: how can we extract meaning (and understand connectivity) from it? One answer is via the mathematical discipline of algebraic topology, which is a tool to distill the study of complex objects or spaces into a simple form. For example, how can we distinguish between an orange and a donut? The obvious answer is that the donut has a "hole"; algebraic topology makes the natural intuition (which is not so natural in higher dimensions!) rigorous. In particular, algebraic topology is built to probe the relationship between local and global structures; this makes it an ideal tool for studying financial trading networks. The proliferation of problems where there is a confluence of local/global transitions and a large experimental data set has given rise to the field of applied algebraic topology, yielding a systematic way to unfold connections. The investigators interpret various notions of collapse of financial networks via this unfolding perspective. The work focuses on clearing networks, which encode trades and liabilities, and bring the tools developed in applied topology to bear on unfolding the interconnections in such networks. In particular, the investigators study how local events propagate: under what circumstances does contagion and systemic collapse result from a local event?

Schenck1312071这个项目专注于计算拓扑学中最新开发的工具在金融网络中的某些风险问题上的新应用。从新的角度来看待复杂的问题往往是有益的；调查人员使用持久的同源性来研究金融交易网络的行为。持久同调主要研究一族单纯复形D(T)如何随实参数t而变化。通常情况下，从对应于t=0的离散数据集(点云数据)开始，然后随着t的变化，将点替换为半径为t的球，并以原点为中心。因此，改变t产生一个经过过滤的单纯复合族，即相关拓扑空间的Rips-Vietoris复合族。在交易网络的背景下，参数t代表保证金；t的低值对应于允许高杠杆交易。该项目的目的是了解与交易网络相关的D(T)拓扑结构的变化如何以某种方式对应于系统崩溃。简而言之，调查人员探索了导致局部事件引发全球蔓延的情况。理解互联互通的问题是当今的首要挑战之一。连通性无处不在；互连往往使系统能够充分挖掘其潜力并增强其健壮性。连接的另一面是，它会引发意外的紧急行为(“黑天鹅”事件)，甚至导致系统崩溃。世界上充斥着数据：我们如何从数据中提取意义(并理解连接)？一个答案是通过代数拓扑学的数学学科，它是一种将复杂对象或空间的研究提炼成简单形式的工具。例如，我们如何区分橙子和甜甜圈？显而易见的答案是，甜甜圈有一个“洞”；代数拓扑造就了自然的直觉(这在更高的维度上并不是那么自然！)严谨。特别是，代数拓扑学的建立是为了探索局部结构和全局结构之间的关系，这使得它成为研究金融交易网络的理想工具。在局部/全局跃迁交汇和大量实验数据集的情况下，问题的激增已经引起了应用代数拓扑学领域的兴起，产生了一种系统地展开连接的方法。研究人员通过这一展开的视角解读了金融网络崩溃的各种概念。这项工作的重点是对交易和负债进行编码的清算网络，并利用应用拓扑学中开发的工具来揭示此类网络中的互联。特别是，研究人员研究了局部事件是如何传播的：在什么情况下，局部事件会导致传染和系统性崩溃？