Financial networks model based on random matrix

In the diversified and complex environment of financial markets, effective investment strategy prediction has become particularly crucial. Especially in the bond market, due to its close connection with macroeconomic factors, the accuracy of predictions is of great significance for investment returns and risk management. This article delves into the application and strategic analysis of financial neural network models in bond investment prediction. We first reviewed the basic architecture of financial neural network models and emphasized their advantages in capturing nonlinear market dynamics. Then, we describe in detail how to use this model to predict the bond market, including the impact of interest rate changes, credit risk and macroeconomic factors. Furthermore, we propose a bond investment strategy framework based on neural network prediction. This framework considers various factors such as market liquidity, bond duration, and credit rating to optimize the returns and risks of investment portfolios. With the further development of deep learning and neural network technology in the financial field, its application potential in bond market prediction and strategy formulation will be more widely recognized and utilized.

When studying financial markets, we often look at estimating a correlation matrix from asset returns. These tend to be noisy, with many more dimensions than samples, so often the resulting correlation matrix is filtered. Popular methods to do this include the minimum spanning tree, planar maximally filtered graph and the triangulated maximally filtered graph, which involve using the correlation network as the adjacency matrix of a graph and then using tools from graph theory. These assume the data fits some form of shape. We do not necessarily have a reason to believe that the data does fit into this shape, and there have been few empirical investigations comparing how the methods perform. In this paper we look at how the filtered networks are changed from the original networks using stock returns from the US, UK, German, Indian and Chinese markets, and at how these methods affect our ability to distinguish between datasets created from different correlation matrices using a graph embedding algorithm. We find that the relationship between the full and filtered networks depends on the data and the state of the market, and decreases as we increase the size of networks, and that the filtered networks do not provide an improvement in classification accuracy compared to the full networks.

Correlation matrix plays a key role in many multivariate methods (e.g., graphical model estimation and factor analysis). The current state-of-the-art in estimating large correlation matrices focuses on the use of Pearson's sample correlation matrix. Although Pearson's sample correlation matrix enjoys various good properties under Gaussian models, its not an effective estimator when facing heavy-tail distributions with possible outliers. As a robust alternative, Han and Liu (2013b) advocated the use of a transformed version of the Kendall's tau sample correlation matrix in estimating high dimensional latent generalized correlation matrix under the transelliptical distribution family (or elliptical copula). The transelliptical family assumes that after unspecified marginal monotone transformations, the data follow an elliptical distribution. In this paper, we study the theoretical properties of the Kendall's tau sample correlation matrix and its transformed version proposed in Han and Liu (2013b) for estimating the population Kendall's tau correlation matrix and the latent Pearson's correlation matrix under both spectral and restricted spectral norms. With regard to the spectral norm, we highlight the role of "effective rank" in quantifying the rate of convergence. With regard to the restricted spectral norm, we for the first time present a "sign subgaussian condition" which is sufficient to guarantee that the rank-based correlation matrix estimator attains the optimal rate of convergence. In both cases, we do not need any moment condition.

Default Count-Based Network Models for Credit Contagion

We investigate network formation for a set of financial institutions represented as nodes. Linkages are source of income, and at the same time they bear the risk of contagion. The optimal connectivity of the nodes results from a game, in which the risk of contagion depends on the choices of all nodes in the system. Our financial network model can be interpreted as a set of banks connected through funding relations, in which a node’s threshold to contagion is represented by its external funding capacity. A second interpretation is that of a set of insurers connected through reinsurance contracts, in which the threshold to contagion is represented by their capital. Our results show that when the threshold distribution across the nodes has higher variance, then, in equilibrium, the average connectivity is in general increasing, but the link failure probability decreases. This suggests that financial stability is best described in terms of the mechanism of network formation than in terms of simple statistics of the network topology like the average connectivity.

Community structure is ubiquitous in real-world complex networks. The task of community detection over these networks is of paramount importance in a variety of applications. Recently, nonnegative matrix factorization (NMF) has been widely adopted for community detection due to its great interpretability and its natural fitness for capturing the community membership of nodes. However, the existing NMF-based community detection approaches are shallow methods. They learn the community assignment by mapping the original network to the community membership space directly. Considering the complicated and diversified topology structures of real-world networks, it is highly possible that the mapping between the original network and the community membership space contains rather complex hierarchical information, which cannot be interpreted by classic shallow NMF-based approaches. Inspired by the unique feature representation learning capability of deep autoencoder, we propose a novel model, named Deep Autoencoder-like NMF (DANMF), for community detection. Similar to deep autoencoder, DANMF consists of an encoder component and a decoder component. This architecture empowers DANMF to learn the hierarchical mappings between the original network and the final community assignment with implicit low-to-high level hidden attributes of the original network learnt in the intermediate layers. Thus, DANMF should be better suited to the community detection task. Extensive experiments on benchmark datasets demonstrate that DANMF can achieve better performance than the state-of-the-art NMF-based community detection approaches.

Asymptotic normal distributions of eigenvalue-normed eigenvectors (Pearson-Hotelling principal-component vectors) of sample variance and correlation matrices are derived. Each distribution follows from the asymptotic distribution of the whole matrix of eigenvectors, also obtained in the paper. The results are presented for the general case, where the existence of fourth-order finite population moments is assumed, and also specified for normal and elliptical populations. Population variance and correlation matrices are supposed to be nonsingular and without multiple eigenvalues. A comparison with unit-length eigenvectors (Anderson principal-component vectors) is being made. Special attention is being paid to asymptotic variances of eigenvectors of both sample variance and correlation matrices in the two approaches. It is shown that the asymptotic variance matrices of principal-component vectors for sample variance and correlation matrices are singular in the Anderson approach. In the Pearson-Hotelling approach, however, the asymptotic variance of eigenvectors of the sample variance matrix is nonsingular, whereas in the sample correlation matrix case the asymptotic variance matrix of eigenvectors is nonsingular under certain conditions.

Testing the equality of two or multiple correlation or covariance matrices is an important problem in biology, finance and many other areas. High dimensionality, where the number of features is much larger than the sample size, causes conventional procedures to perform poorly, as they are often based on limiting distributions of test statistics in the classical large sample size setting. Moreover, their performance is often contingent on whether the matrix of differences is sparse or dense, while such information is rarely available. In this article, we develop a new family of permutation testing procedures to tackle these challenges. The introduced tests are demonstrated to outperform many other competing procedures in terms of size control and power under various settings. In particular, using our variance-stabilizing transformation, the proposed methods provide the best performance for testing correlation or covariance matrix differences in both sparse and dense settings. We establish non-asymptotic guarantees on the power of our test, which ensure its reliability for sparse and dense differential correlation matrices. Through the analysis of gene-expression and brain imaging data, we showcase the high power and accurate size control of our test in high-dimensional statistical applications.

Executive Summary. The fact that portfolio variances must be positive implies that the correlation matrix for asset returns should be positive definite. As such, a test for positive definiteness in the correlation matrix should be routinely implemented prior to any portfolio optimization exercise. A program is developed to test for positive definiteness in correlation matrices and to detect correlation coefficients that violate the positive definiteness condition. The correlation matrices from five randomly selected papers are tested and violations revealed in larger correlation matrices. Where violations are detected, the portfolio manager or analyst should proceed with the optimization exercise only if the alternative bounds on the correlation coefficients computed by our program are acceptable.

The Nearest Correlation Matrix Problem: Solution by Differential Evolution Method of Global Optimization

In financial credit networks, prepayments enable a firm to settle its debt obligations ahead of an agreed-upon due date. Prepayments have a transformative impact on the structure of networks, influencing the financial well-being (utility) of individual firms. This study investigates prepayments from both theoretical and empirical perspectives. We first establish the computational complexity of finding prepayments that maximize welfare, assuming global coordination among firms in the financial network. Subsequently, our focus shifts to understanding the strategic behavior of individual firms in the presence of prepayments. We introduce a prepayment game where firms strategically make prepayments, delineating the existence of pure strategy Nash equilibria and analyzing the price of anarchy (stability) within this game. Recognizing the computational challenges associated with determining Nash equilibria in prepayment games, we use a simulation-based approach, known as empirical game-theoretic analysis (EGTA). Through EGTA, we are able to find Nash equilibria among a carefully-chosen set of heuristic strategies. By examining the equilibrium behavior of firms, we outline the characteristics of high-performing strategies for strategic prepayments and establish connections between our empirical and theoretical findings.

Let $\widetilde{X}_{M\times N}$ be a rectangular data matrix with independent real-valued entries $[\widetilde{x}_{ij}]$ satisfying $\mathbb {E}\widetilde{x}_{ij}=0$ and $\mathbb {E}\widetilde{x}^2_{ij}=\frac{1}{M}$, $N,M\to\infty$. These entries have a subexponential decay at the tails. We will be working in the regime $N/M=d_N,\lim_{N\to\infty}d_N\neq0,1,\infty$. In this paper we prove the edge universality of correlation matrices ${X}^{\dagger}X$, where the rectangular matrix $X$ (called the standardized matrix) is obtained by normalizing each column of the data matrix $\widetilde{X}$ by its Euclidean norm. Our main result states that asymptotically the $k$-point ($k\geq1$) correlation functions of the extreme eigenvalues (at both edges of the spectrum) of the correlation matrix ${X}^{\dagger}X$ converge to those of the Gaussian correlation matrix, that is, Tracy-Widom law, and, thus, in particular, the largest and the smallest eigenvalues of ${X}^{\dagger}X$ after appropriate centering and rescaling converge to the Tracy-Widom distribution. The asymptotic distribution of extreme eigenvalues of the Gaussian correlation matrix has been worked out only recently. As a corollary of the main result in this paper, we also obtain that the extreme eigenvalues of Gaussian correlation matrices are asymptotically distributed according to the Tracy-Widom law. The proof is based on the comparison of Green functions, but the key obstacle to be surmounted is the strong dependence of the entries of the correlation matrix. We achieve this via a novel argument which involves comparing the moments of product of the entries of the standardized data matrix to those of the raw data matrix. Our proof strategy may be extended for proving the edge universality of other random matrix ensembles with dependent entries and hence is of independent interest.

Accurately capturing the spatial correlations of traffic network significantly benefits short-term traffic forecasting. Some existing works represent spatial correlations in a simple one-dimensional space, but they cannot represent the real spatial correlations among sensors comprehensively. The other existing works represent the spatial correlations through grid-based method, but the local correlation of constructed spatial map is too superficial to extract deep spatial features effectively. Therefore, a novel deep learning model is proposed, which aims to represent the spatial correlations more effectively through a new correlation matrix structure. In the proposed model, the correlations among sensors are calculated from multiple perspectives to construct the speed, volume, and occupancy correlation matrices respectively. Then, considering that highly correlated sensors are close in the spatial dimension, an enhanced heuristic optimization algorithm is proposed to evolve these three correlation matrices into optimal ones by reorganizing the highly correlated sensors into each others neighborhood. Finally, the three optimal correlation matrices are combined to form a three-dimensional multivariate correlation matrix characterized by locally high correlation, which is beneficial to exploit the deep spatial features of traffic network. The experiments show that the proposed model has better accuracy and stability than other commonly used baseline models.

Most simulation studies in factor analysis follow a process of constructing population correlation matrices from the common-factor model and generating sample correlation matrices from the population matrices. In the common-factor model, the population correlation matrix is perfectly fit by the model's containing common and unique factors. However, since no mathematical model accounts exactly for the real-world phenomena that it is intended to represent, the Tucker-Koopman-Linn model (1969) is more realistic for generating correlation matrices than the conventional common-factor model because the former incorporates model error. In this paper, a procedure for generating population and sample correlation matrices with model error by combining the Tucker-Koopman-Linn model and Wijsman's algorithm (1959) is presented. The SAS/IML program for generating correlation matrices is described, and an example is also provided.

The simplest way to describe the dependence for a set of financial assets is their correlation matrix. This correlation matrix can be improper when it is specified element-wise. We describe a new method for obtaining a positive definite correlation matrix starting from an improper one. The expert's opinion and trust in each pairwise correlation is described by a beta distribution. Then, by combining these individual distributions, a joint distribution over the space of positive definite correlation matrices is obtained using Cholesky factorization, and its mode constitutes the new proper correlation matrix. The optimization is complemented by a visual representation of the entries that were most affected by the legalization procedure. We also sketch a Bayesian approach to the same problem.