寻找最接近的协方差矩阵:外汇市场案例

A. Minabutdinov, I. Manaev, Maxim Bouev
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引用次数: 1

摘要

我们考虑在给定一个初始非正半定(non-PSD)估计的外汇市场上找到一个有效协方差矩阵的问题。一般的无套利假设对这样的矩阵施加了额外的线性约束,不可避免地使它们成为奇异的。因此,即使是最先进的数值技术也会在看似标准的优化任务中退缩。究其原因,问题是不恰当的,而其PSD解决方案也不是严格可行的。为了解决这个问题,我们描述了包含可行集的PSD锥的低维面。在将初始问题投射到这张脸上之后,我们得到了一个简化的问题,它既具有良好的定位,又具有较小的规模。我们证明,在解出约简问题后,初始问题的解可以在一步内唯一地恢复。我们进行了大量的数值实验来比较不同算法在解决约简问题方面的性能,并展示了处理约简问题相对于原始问题的优势。简化问题的较小规模意味着它的解可以通过应用几乎任何数值方法有效地找到。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Finding the Nearest Covariance Matrix: The Foreign Exchange Market Case
We consider the problem of finding a valid covariance matrix in the foreign exchange market given an initial nonpositively semidefinite (non-PSD) estimate of such a matrix. The common no-arbitrage assumption imposes additional linear constraints on such matrixes, inevitably making them singular. As a result, even the most advanced numerical techniques will predictably balk at a seemingly standard optimization task. The reason is that the problem is ill posed, while its PSD solution is not strictly feasible. In order to deal with this issue we describe a low-dimensional face of the PSD cone that contains the feasible set. After projecting the initial problem onto this face, we come out with a reduced problem, which is both well posed and of a smaller scale. We show that, after solving the reduced problem, the solution to the initial problem can be recovered uniquely in one step. We run numerous numerical experiments to compare the performance of different algorithms in solving the reduced problem and to demonstrate the advantages of dealing with the reduced problem as opposed to the original one. The smaller scale of the reduced problem implies that its solution can effectively be found by the application of virtually any numerical method.
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