Spectral Theory of Sample Covariance Matrices from Discretized Lévy Processes

• Main Author: Zitelli, Gregory
• Format: Dissertation
• Language: English
• Published: ProQuest Dissertations & Theses 01-01-2019
• Subjects: Finance; Mathematics
• ISBN: 9798607315672

Asymptotic spectral techniques have become a powerful tool in estimating statistical properties of systems that can be well approximated by rectangular matrices with i.i.d. or highly structured entries. Starting in the mid 2000s, results from Random Matrix Theory were used along these lines to investigate problems related to financial data, particularly the out-of sample risk underestimation of portfolios constructed through mean-variance optimization. If the returns of assets to be held in a portfolio are assumed independent and stationary, then these results are universal in that they do not depend on the precise distribution of returns. This universality has been somewhat misrepresented in the literature, however, as asymptotic results require that an arbitrarily long time horizon be available before such predictions necessarily become accurate. This makes these methods ill-suited when moving to high frequency data, for example, where the number of data-points increases but the overall time horizon remains the same or even decreases. In order to reconcile these models with the highly non-Gaussian returns observed in financial data, a new ensemble of random rectangular matrices are introduced, modeled on the observations of independent Levy processes over a fixed time horizon. The key mathematical results describe the eigenvalues of these models' sample covariance matrices, which exhibit remarkably similar scaling behavior to what is seen when working with daily and intraday data on the S&P 500 and Nikkei 225.