Sparse Optimization Methods and Statistical Modeling with Applications to Finance

It is well known that the out-of-sample performance of Markowitz's mean-variance portfolio criterion can be negatively affected by estimation errors in the mean and covariance. In this dissertation we examine methods to address this problem through application of methods and techniques from sparse optimization and modeling. Two new techniques are developed with the aim of improving the performance of mean-variance portfolio optimization.In the first technique a pairwise weighted elastic net penalized mean-variance criterion for portfolio design in proposed. Here we motivate the use of this penalty through a robust optimization interpretation. This interpretation is then employed to develop a bootstrap calibration technique for the pairwise elastic net. The benefit of the pairwise weighted elastic net and calibration is shown in portfolio performance results using recent U.S. stock market data.In the second application robust Kalman filtering techniques are applied to return covariance estimation from high frequency financial price data. The methods developed address three factors which make covariance estimation from high frequency data difficult: 1) microstructure noise, 2) asynchronous trading, and 3) jumps. The performance of these robust Kalman filtering techniques are tested against simulated high frequency data and are compared with other existing covariance estimators. The results indicate that the robust Kalman filtering techniques substantially improve covariance estimation performance versus other approaches