Yiyong Feng

Normalization of Linear Support Vector Machines

In this paper, we start with the standard support vector machine (SVM) formulation and extend it by considering a general SVM formulation with normalized margin. This results in a unified convex framework that allows many different variations in the formulation with very diverse numerical performance. The proposed unified framework can capture the existing methods, i.e., standard soft-margin SVM, l1-SVM, and SVMs with standardization, feature selection, scaling, and many more SVMs, as special cases. Furthermore, our proposed framework can not only provide us with more insights on different SVMs from the “energy” and “penalty” point of views, which help us understand the connections and differences between them in a unified way, but also enable us to propose more SVMs that outperform the existing ones under some scenarios.

Optimal order execution for algorithmic trading: A CVaR approach

Order execution for algorithmic trading has been studied in the literature as a means of determine the optimal strategy by minimizing a trade-off between expected execution cost and risk. Usually, the variance of the execution cost is taken as a proxy of risk. The problem of this approach is that variance being a symmetric measure of risk disregard the fact that investors only considered as risky cost realizations that are higher than an expected target value. This fact becomes even more sensitive when the return distribution is nonnormal, negatively skewed, or leptokurtic. In this paper, we propose the use of the conditional value-at-risk of the execution cost as risk measure, which allows for taking only the unfavorable part of the return distribution, or equivalently unwanted high cost, into consideration.

SCRIP: Successive Convex Optimization Methods for Risk Parity Portfolio Design

The traditional Markowitz portfolio optimization proposed in the 1950s has not been embraced by practitioners despite its theoretical elegance. Recently, an alternative risk parity portfolio design has been receiving significant attention from both the theoretical and practical sides due to its advantage in diversification of (ex-ante) risk contributions among assets. Such risk contributions can be deemed good predictors for the (ex-post) loss contributions, especially when there exist huge losses. Most of the existing specific problem formulations on risk parity portfolios are highly nonconvex and are solved via standard off-the-shelf numerical optimization methods, e.g., sequential quadratic programming and interior point methods. However, for nonconvex risk parity formulations, such standard numerical approaches may be highly inefficient and may not provide satisfactory solutions. In this paper, we first propose a general risk parity portfolio problem formulation that can fit most of the existing specific risk parity formulations, and then propose a family of simple and efficient successive convex optimization methods for the general formulation. The numerical results show that our proposed methods significantly outperform the existing ones.

Robust Optimization of Order Execution

Order execution for algorithmic trading has been studied in the literature to determine the optimal strategy by minimizing a trade-off between expected execution cost and risk. Usually, the variance of the execution cost is taken as a proxy of risk due to mathematical tractability. However, the variance has been recognized not to be practical since it is a symmetric measure of risk and, hence, penalizes the low-cost events. In this paper, we propose the use of the conditional value-at-risk (CVaR) of the execution cost as risk measure, which allows to take into consideration only the unfavorable part of the return distribution, or, equivalently, unwanted high cost. In addition, due to the parameter estimation errors in the price model, the naive strategies given by the nominal problem may perform badly in the real market, and hence it is extremely important to take such parameters estimation errors into consideration. To deal with this, we extend both the traditional mean-variance approach and our proposed CVaR approach to their robust design counterparts.

Robust order execution under box uncertainty sets

Order execution for algorithmic trading has been studied in the literature as a means of determining the optimal strategy by minimizing a trade-off between expected execution cost and risk. However, the variance has been recognized not to be practical since it is a symmetric measure of risk and, hence, penalizes the low-cost events. In this paper, we propose the use of the conditional value-at-risk (CVaR) of the execution cost as risk measure for the multiple assets case order execution problem. In addition, for the particular box-type parameter estimation errors, we extend both the existing mean-variance approach and our proposed CVaR approach to their robust designs.

A Signal Processing Perspective on Financial Engineering

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. A Signal Processing Perspective of Financial Engineering 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.

Sparse Portfolios for High-Dimensional Financial Index Tracking

Index tracking is a popular passive portfolio management strategy that aims at constructing a portfolio that replicates or tracks the performance of a financial index. The tracking error can be minimized by purchasing all the assets of the index in appropriate amounts. However, to avoid small and illiquid positions and large transaction costs, it is desired that the tracking portfolio consists of a small number of assets, i.e., a sparse portfolio. The optimal asset selection and capital allocation can be formulated as a combinatorial problem. A commonly used approach is to use mixed-integer programming (MIP) to solve small sized problems. Nevertheless, MIP solvers can fail for high-dimensional problems while the running time can be prohibiting for practical use. In this paper, we propose efficient and fast index tracking algorithms that automatically perform asset selection and capital allocation under a set of general convex constraints. A special consideration is given to the case of the nonconvex holding constraints and to the downside risk tracking measure. Furthermore, we derive specialized algorithms with closed-form updates for particular sets of constraints. Numerical simulations show that the proposed algorithms match or outperform existing methods in terms of performance, while their running time is lower by many orders of magnitude.

Optimization Methods for Financial Index Tracking: From Theory to Practice

Index tracking is a very popular passive investment strategy.Since an index cannot be traded directly, index trackingrefers to the process of creating a portfolio that approximatesits performance. A straightforward way to do that isto purchase all the assets that compose an index in appropriatequantities. However, to simplify the execution, avoidsmall and illiquid positions, and large transaction costs, it isdesired that the tracking portfolio consists of a small numberof assets, i.e., we wish to create a sparse portfolio.Although index tracking is driven from the financial industry,it is in fact a pure signal processing problem: a regression ofthe financial historical data subject to some portfolio constraintswith some caveats and particularities. Furthermore, the sparse index tracking problem is similar to many sparsityformulations in the signal processing area in the sense thatit is a regression problem with some sparsity requirements.In its original form, sparse index tracking can be formulatedas a combinatorial optimization problem. A commonly usedapproach is to use mixed-integer programming MIP tosolve small sized problems. Nevertheless, MIP solvers are notapplicable for high-dimensional problems since the runningtime can be prohibiting for practical use.The goal of this monograph is to provide an in-depth overviewof the index tracking problem and analyze all the caveats andpractical issues an investor might have, such as the frequentrebalancing of weights, the changes in the index composition,the transaction costs, etc. Furthermore, a unified frameworkfor a large variety of sparse index tracking formulations isprovided. The derived algorithms are very attractive forpractical use since they provide efficient tracking portfoliosorders of magnitude faster than MIP solvers.

Portfolio optimization with asset selection and risk parity control

After the 2008 financial crisis, risk management has become more important than performance management and an alternative portfolio design, referred to as risk parity portfolio, has been receiving significant attention from both theoretical and practical fields due to its advantage in diversification of (ex-ante) risk contributions among assets. Usually, this approach results in a portfolio with nonzero weights in all the assets. Investors, however, could not lay out the capital among all the assets listed on the markets, which results in unrealistically high transaction costs, and therefore, reduction of the return of the designed portfolio. To overcome this drawback, in this paper, we propose a method to jointly select only some of the assets and distribute the capital among the selected assets such that the risk is diversified enough.

Linear support vector machines with normalizations

In this paper, we start with the standard support vector machine (SVM) formulation and extend it by proposing a general SVM that allows many different variations captured by normalizations in the formulation with very diverse numerical performance. The proposed formulation can not only capture the existing work, i.e., standard soft-margin SVM, ℓ1-SVM, as special cases, but also enable us to propose more SVMs that outperform the existing ones under some scenarios.