Exploring financial data analysis in high dimensions

Abstract

Financial data analysis is crucial for making informative decisions based on accurate and reliable forecasting and structural analysis. With the advancement of modern technology, researchers can collect a vast amount of financial data and pursue more complex structures within the data, leading to emerging interests in high-dimensional modeling.

The first part of this thesis models realized volatilities for high-frequency data. A multilinear low-rank heterogeneous autoregressive (HAR) model is proposed by using tensor techniques, where a data-driven method is adopted to automatically select the heterogeneous components. In addition, HAR-It\^o models are introduced to interpret the corresponding high-frequency dynamics, as well as those of other HAR-type models. Moreover, non-asymptotic properties of the high-dimensional HAR modeling are established, and a projected gradient descent algorithm with theoretical justifications is suggested to search for estimates. Theoretical and computational properties of the proposed method are verified by simulation experiments and real data analysis.

Secondly, this thesis proposes a sample-average approximation-based portfolio strategy to tackle the existing difficulties in portfolio management with cardinality constraints. Our strategy bypasses the estimation of mean and covariance, the Great Wall in high-dimensional scenarios. Empirical results on S\&P 500 and Russell 2000 show that an appropriate number of carefully chosen assets leads to better out-of-sample mean-variance efficiency.

The third part of this thesis extends the results of the second part to risk management. Focusing on Conditional Value-at-Risk (CVaR) portfolio optimization problems in the expanding global markets, this thesis analyzes sparsity-induced portfolio strategies. The equivalence of the regularized CVaR minimization problem and its $l_0$-constrained counterpart in the norm ball is analyzed and the non-asymptotic error rate is established. The numerical experiments demonstrate the risk aversion preference and robustness of the proposed portfolio strategy on both synthetic data and the S\&P 500 dataset.