Factor models, mean-reversion time, and statistical arbitrage [electronic resource]
Keywords: Computational and Mathematical Engineering.

Financial markets are well-defined complex systems, which are continuously monitored and recorded, and generate tremendous amounts of data. This thesis is an attempt to analyze financial markets as high-dimensional and dynamic systems. In dealing with high-dimensional data sets, factor models are often useful for dimension reduction. In the first part of this thesis, we present a new approach to estimate high-dimensional factor models, using the empirical spectral density of residuals. The spectrum of covariance matrices from financial data typically exhibits two characteristic aspects: a few spikes and bulk. The former represent factors that mainly drive the features and the latter arises from idiosyncratic noise. Motivated by these two aspects, we consider a minimum distance between two spectrums; one from a covariance structure model and the other from real residuals of financial data that are obtained by subtracting principal components. Our method simultaneously provides estimators of the number of factors and information about correlation structures in residuals. Using free random variable techniques, the proposed algorithm can be implemented and controlled effectively. Monte Carlo simulations confirm that our method is robust to noise or the presence of weak factors. Furthermore, the application to financial time-series shows that our estimators capture essential aspects of market dynamics. In the second part of this thesis, we turn to pay attention to more practical applications using residual dynamics. In this part, we study the risk of mean-reversions in statistical arbitrage. The basic concept of statistical arbitrage is to exploit short-term deviations in returns from a long-term equilibrium across several assets. This kind of strategy heavily relies on the assumption of mean-reversion of idiosyncratic returns - reverting to a long-term mean after a certain amount of time, but literature on the assessment of risk on this belief is rare. In this chapter, we propose a scheme that controls the risk on mean-reversions, via portfolio selections and screenings. Realizing that each residual has a different mean-reversion time, the residuals that are fast mean-reverting are selected to form portfolios. In addition, further control is imposed by allowing the trading activity only when the goodness-of-fit of the estimation for trading signals is sufficiently high. We design the dynamic asset allocation with the market- and dollar- neutrality conditions as a constrained optimization problem, which is solved numerically. The improved reliability and robustness of the strategy of our method is demonstrated through back-testing with real data. It is found that the performance of this investment framework is robust to market changes. We further provide answers to puzzles regarding relations among the number of factors, length of estimation window, and transaction costs, which are crucial parameters that have direct impacts on the investment strategy. In the third part, we consider the Wishart processes, which is a good candidate for modeling a stochastic covariance matrix. Specifically, we use perturbation theory to derive the eigenvalues and eigenvectors of the Wishart processes. These processes have some interesting features where each eigenvalue is part of a multi-dimensional stochastic systems of non-colliding particles, which has been widely studied in mathematical physics. In addition, we implement a numerical scheme to effectively simulate eigenvalues of the square Ornstein-Uhlenbeck processes. In this scheme, the Yosida approximation was exploited to resolve the difficulty that arises from the singular drift terms. The results in this chapter contribute to the dynamic modeling of stochastic covariance matrices and their financial applications.