Dynamics of top eigenvalues of empirical covariance matrices of financial data
- Covariance matrices of financial data and their inverses play an important role in algorithmic portfolio optimization and risk management. For example, PCA (principal component analysis) based strategies in algorithmic portfolio trading, and Markowitz portfolio theory in risk control and portfolio construction. However, since the true covariance matrix is neither perfectly known nor constant in time, the dynamics of the empirical covariance covariances is of great importance, as the volatilities and correlations evolve with time. It is obviously too heavy to carry the whole empirical covariance matrices all the time, especially when they contain a lot of noise. As PCA strategies suggest, most of the meaningful economic information is contained in the large eigenvalues and eigenvectors of covariance matrices, especially, the largest eigenvalue and eigenvector correspond to a collective market mode. As also suggested in risk control, the largest eigenvalue and eigenvector represent the most risky direction in a financial context. The largest eigenvalue of the empirical covariance matrix and the corresponding eigenvector are of considerable importance and we want to analyze their stability over time and characterize their fluctuations. We thus study the dynamics of the top eigenvalues and eigenvectors, instead of the whole matrices. In the thesis, we use a one factor continuous time model and study the dynamics of the top eigenvalues of the empirical covariance matrices. We first model the dynamics of stock returns and construct the empirical covariance matrices through an exponential moving average of the returns. We then derive a stochastic differential equation for the deviation part of the empirical covariance matrix from the true covariance matrix. We also establish a weak convergence result for the top eigenvalue of the deviation matrix at the equilibrium level, and show that the dynamics of the top eigenvalues of the deviation matrices satisfies in probability a reflecting stochastic differential equation for large N (size of covariance matrix) and a large true top eigenvalue. Numerical results are presented at the end to validate our theoretical results.
|Type of resource
|electronic; electronic resource; remote
|1 online resource.
|Stanford University, Institute for Computational and Mathematical Engineering.
|Glynn, Peter W
|Glynn, Peter W
|Statement of responsibility
|Submitted to the Institute for Computational and Mathematical Engineering.
|Thesis (Ph.D.)--Stanford University, 2013.
- © 2013 by Lijia Wang
- This work is licensed under a Creative Commons Attribution Non Commercial 3.0 Unported license (CC BY-NC).
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