Index tracking is a very popular passive investment strategy. Since an index cannot be traded directly, index tracking refers to the process of creating a portfolio that approximates its performance. A straightforward way to do that is to purchase all the assets that compose an index in appropriate quantities. However, to simplify the execution, avoid small and illiquid positions and large transaction costs, it is desired that the tracking portfolio consists of a small number of assets. Although index tracking is driven from the financial industry, it is in fact a pure signal processing problem: a regression of the financial historical data subject to some portfolio constraints with some caveats and particularities. Furthermore, the sparse index tracking problem is similar to many sparsity formulations in the signal processing area in the sense that it is a regression problem with some sparsity requirements. In its original form, sparse index tracking can be formulated as a combinatorial optimization problem. A commonly used approach is to use mixed-integer programming (MIP) to solve small sized problems. Nevertheless, MIP solvers are not applicable for high-dimensional problems since the running time can be prohibiting for practical use. This monograph provides an in-depth overview of the index tracking problem and analyzes all the caveats and practical issues an investor might have. Furthermore, it provides a unified framework for a large variety of sparse index tracking formulations. The derived algorithms are very attractive for practical use since they provide efficient tracking portfolios orders of magnitude faster than MIP solvers.

Despite the different nature of financial engineering and electrical engineering, both areas are intimately connected on a mathematical level. The foundations of financial engineering lie on the statistical analysis of numerical time series and the modeling of the behavior of the financial markets in order to perform predictions and systematically optimize investment strategies. Similarly, the foundations of electrical engineering, for instance, wireless communication systems, lie on statistical signal processing and the modeling of communication channels in order to perform predictions and systematically optimize transmission strategies. Both foundations are the same in disguise. It is often the case in science that the same or very similar methodologies are developed and applied independently in different areas. A Signal Processing Perspective of Financial Engineering is about investment in financial assets treated as a signal processing and optimization problem. It explores such connections and capitalizes on the existing mathematical tools developed in wireless communications and signal processing to solve real-life problems arising in the financial markets in an unprecedented way. It provides straightforward and systematic access to financial engineering for researchers in signal processing and communications so that they can understand problems in financial engineering more easily and may even apply signal processing techniques to handle some financial problems.