Kernel Search: a new heuristic framework for?portfolio?selection

In this paper we propose a new heuristic framework, called Kernel Search, to solve the complex problem of portfolio selection with real features. The method is based on the identification of a restricted set of promising securities (kernel) and on the exact solution of the MILP problem on this set. The continuous relaxation of the problem solved on the complete set of available securities is used to identify the initial kernel and a sequence of integer problems are then solved to identify further securities to insert into the kernel. We analyze the behavior of several heuristic algorithms as implementations of the Kernel Search framework for the solution of the analyzed problem. The proposed heuristics are very effective and quite efficient. The Kernel Search has the advantage of being general and thus easily applicable to a variety of combinatorial problems.     

Dynamic Index Tracking and Risk Exposure Control Using Derivatives

We develop a methodology for index tracking and risk exposure control using financial derivatives. Under a continuous-time diffusion framework for price evolution, we present a pathwise approach to construct dynamic portfolios of derivatives in order to gain exposure to an index and/or market factors that may be not directly tradable. Among our results, we establish a general tracking condition that relates the portfolio drift to the desired exposure coefficients under any given model. We also derive a slippage process that reveals how the portfolio return deviates from the targeted return. In our multi-factor setting, the portfolio’s realized slippage depends not only on the realized variance of the index but also the realized covariance among the index and factors. We implement our trading strategies under a number of models, and compare the tracking strategies and performances when using different derivatives, such as futures and options.     

Mixed-integer programming approaches for index tracking and enhanced indexation

We consider the problem of reproducing the performance of a stock market index, but without purchasing all of the stocks that make up the index, index tracking. We also consider the problem of out-performing the index, enhanced indexation. We present mixed-integer linear programming formulations of these problems. Our formulations include transaction costs, a constraint limiting the number of stocks that can be purchased and a limit on the total transaction cost that can be incurred. As our formulations of these problems are mixed-integer linear programs we can use a standard solver (Cplex). Numeric results are presented for eight data sets drawn from major markets. The largest of these data sets involves over 2000 stocks.     

Kernel search for the capacitated facility location problem

The Capacitated Facility Location Problem (CFLP) is among the most studied problems in the OR literature. Each customer demand has to be supplied by one or more facilities. Each facility cannot supply more than a given amount of product. The goal is to minimize the total cost to open the facilities and to serve all the customers. The problem is $\mathcal{NP}$ -hard. The Kernel Search is a heuristic framework based on the idea of identifying subsets of variables and in solving a sequence of MILP problems, each problem restricted to one of the identified subsets of variables. In this paper we enhance the Kernel Search and apply it to the solution of the CFLP. The heuristic is tested on a very large set of benchmark instances and the computational results confirm the effectiveness of the Kernel Search framework. The optimal solution has been found for all the instances whose optimal solution is known. Most of the best known solutions have been improved for those instances whose optimal solution is still unknown.     

Optimal construction and rebalancing of index-tracking portfolios

Index funds aim to track the performance of a financial index, such as, e.g., the Standard?&?Poor’s?500 index. Index funds have become popular because they offer attractive risk-return profiles at low costs. The index-tracking problem considered in this paper consists of rebalancing the composition of the index fund’s tracking portfolio in response to new market information and cash deposits and withdrawals from investors such that the index fund’s tracking accuracy is maximized. In a frictionless market, maximum tracking accuracy is achieved by investing the index fund’s entire capital in a tracking portfolio that has the same normalized value development as the index. In the presence of transaction costs, which reduce the fund’s capital, one has to manage the trade-off between transaction costs and similarity in terms of normalized value developments. Existing mathematical programing formulations for the index-tracking problem do not optimize this trade-off explicitly, which may result in substantial transaction costs or tracking portfolios that differ considerably from the index in terms of normalized value development. In this paper, we present a mixed-integer linear programing formulation with a novel optimization criterion that directly considers the trade-off between transaction costs and similarity in terms of normalized value development. In an experiment based on a set of real-world problem instances, the proposed formulation achieves a considerably higher tracking accuracy than state-of-the-art formulations.     

Simultaneous pursuit of out-of-sample performance and sparsity in index tracking portfolios

Index tracking is a passive investment strategy in which a fund (e.g., an ETF: exchange traded fund) manager purchases a set of assets to mimic a market index. The tracking error, i.e., the difference between the performances of the index and the portfolio, may be minimized by buying all the assets contained in the index. However, this strategy results in a considerable transaction cost and, accordingly, decreases the return of the constructed portfolio. On the other hand, a portfolio with a small cardinality may result in poor out-of-sample performance. Of interest is, thus, constructing a portfolio with good out-of-sample performance, while keeping the number of assets invested in small (i.e., sparse). In this paper, we develop a tracking portfolio model that addresses the above conflicting requirements by using a combination of L0- and L2-norms. The L2-norm regularizes the overdetermined system to impose smoothness (and hence has better out-of-sample performance), and it shrinks the solution to an equally-weighted dense portfolio. On the other hand, the L0-norm imposes a cardinality constraint that achieves sparsity (and hence a lower transaction cost). We propose a heuristic method for estimating portfolio weights, which combines a greedy search with an analytical formula embedded in it. We demonstrate that the resulting sparse portfolio has good tracking and generalization performance on historic data of weekly and monthly returns on the Nikkei 225 index and its constituent companies.     

Exact and heuristic approaches for the index tracking problem with UCITS constraints

Index tracking aims at determining an optimal portfolio that replicates the performance of an index or benchmark by investing in a smaller number of constituents or assets. The tracking portfolio should be cheap to maintain and update, i.e., invest in a smaller number of constituents than the index, have low turnover and low transaction costs, and should avoid large positions in few assets, as required by the European Union Directive UCITS (Undertaking for Collective Investments in Transferable Securities) rules. The UCITS rules make the problem hard to be satisfactorily modeled and solved to optimality: no exact methods but only heuristics have been proposed so far. The aim of this paper is twofold. First, we present the first Mixed Integer Quadratic Programming (MIQP) formulation for the constrained index tracking problem with the UCITS rules compliance. This allows us to obtain exact solutions for small- and medium-size problems based on real-world datasets. Second, we compare these solutions with the ones provided by the state-of-art heuristic Differential Evolution and Combinatorial Search for Index Tracking (DECS-IT), obtaining information about the heuristic performance and its reliability for the solution of large-size problems that cannot be solved with the exact approach. Empirical results show that DECS-IT is indeed appropriate to tackle the index tracking problem in such cases. Furthermore, we propose a method that combines the good characteristics of the exact and of the heuristic approaches.     

Quantile regression for index tracking and enhanced indexation

Quantile regression differs from traditional least-squares regression in that one constructs regression lines for the quantiles of the dependent variable in terms of the independent variable. In this paper we apply quantile regression to two problems in financial portfolio construction, index tracking and enhanced indexation. Index tracking is the problem of reproducing the performance of a stock market index, but without purchasing all of the stocks that make up the index. Enhanced indexation deals with the problem of out-performing the index. We present a mixed-integer linear programming formulation of these problems based on quantile regression. Our formulation includes transaction costs, a constraint limiting the number of stocks that can be in the portfolio and a limit on the total transaction cost that can be incurred. Numeric results are presented for eight test problems drawn from major world markets, where the largest of these test problems involves over 2000 stocks.     

基于LARS-Lasso的指数跟踪及其在股指期货套利策略中的应用

本文将多元线性回归选择变量的Lasso方法引入到指数跟踪和股指期货套利策略研究,提出运用LARS算法实现非负限制下的Lasso选择现货组合问题,为业界给出了一种选择构造现货组合的股票的新方法。实证表明:采用本文提出的方法得到的现货组合,在组合含有较少数量股票的情况下,得到较文献中已有方法更小的跟踪误差。同时,利用本文的方法对沪深300仿真交易的期现套利进行研究,得到有重要市场价值的结果。     

基于下半概率风险度量并兼顾收益分布厚尾性的新型金融指数跟踪模型

考虑到投资者通常采取安全第一的准则,采用跟踪偏差的下半概率作为跟踪风险的度量;而为在恰当描述证券收益分布的厚尾特性的同时克服机会约束对模型求解所造成的困难,假设风险资产的收益服从多元t分布,由此建立了新型金融指数跟踪模型.在分析所建立模型结构特性的基础上,文中还导出了该模型的解析最优解.实证结果表明了新模型的有效性和实用价值.     

A heuristic for BILP problems: The Single Source Capacitated Facility Location Problem

In the Single Source Capacitated Facility Location Problem (SSCFLP) each customer has to be assigned to one facility that supplies its whole demand. The total demand of customers assigned to each facility cannot exceed its capacity. An opening cost is associated with each facility, and is paid if at least one customer is assigned to it. The objective is to minimize the total cost of opening the facilities and supply all the customers. In this paper we extend the Kernel Search heuristic framework to general Binary Integer Linear Programming (BILP) problems, and apply it to the SSCFLP. The heuristic is based on the solution to optimality of a sequence of subproblems, where each subproblem is restricted to a subset of the decision variables. The subsets of decision variables are constructed starting from the optimal values of the linear relaxation. Variants based on variable fixing are proposed to improve the efficiency of the Kernel Search framework. The algorithms are tested on benchmark instances and new very large-scale test problems. Computational results demonstrate the effectiveness of the approach. The Kernel Search algorithm outperforms the best heuristics for the SSCFLP available in the literature. It found the optimal solution for 165 out of the 170 instances with a proven optimum. The error achieved in the remaining instances is negligible. Moreover, it achieved, on 100 new very large-scale instances, an average gap equal to 0.64% computed with respect to a lower bound or the optimum, when available. The variants based on variable fixing improved the efficiency of the algorithm with minor deteriorations of the solution quality.     

A DIRECT METHOD IN OPTIMAL PORTFOLIO AND CONSUMPTION CHOICE

In this paper, we use a direct method to solve the optimal portfolio and consumption choice problem in the security market for a specific case, in which the utility function is of a given homogenous form, i.e. the so-called CRRA case. The idea comes from the completion technique ever used in LQ optimal control.     

On the use of T-coercivity to study the interior transmission eigenvalue problem

In this Note, we investigate the so-called interior transmission problem using the T-coercivity approach. In particular, we prove that this problem, which appears when one is interested in the reconstruction of the support of an inclusion embedded in a homogeneous medium, is of Fredholm type and that so-called transmission eigenvalues form at most a discrete set. Our approach treats cases where the difference between the inclusion index and the background index can change sign, which are not covered by other techniques that can be found in the literature. We also provide Faber–Krahn type inequalities associated with this general case.     

A mean-field approach for the asymptotic tracking problem of moving continuum target clouds

We propose a new coupled kinetic system arising from the asymptotic tracking of a continuum target cloud, and study its asymptotic tracking property. For the proposed kinetic system, we present an energy functional which is monotonic and distance between particle trajectories corresponding to kinetic equations for target, and tracking ensembles tend to zero asymptotically under a suitable sufficient framework. The framework is formulated in terms of system parameters and initial data.     

指数跟踪问题的广义双线性规划模型

本文对指数跟踪问题建立了一种广义的双线性规划模型,其中考虑了交易费用、持仓限制与重平衡问题.根据该模型的特殊结构,本文给出了近似规划算法,通过逐次逼近的线性规划求解最优指数跟踪问题.     

Hierarchical and conditional combination of belief functions induced by visual tracking

In visual tracking, sources of information are often disrupted and deliver imprecise or unreliable data leading to major data fusion issues. In the Dempster-Shafer framework, such issues can be addressed by attempting to design robust combination rules. Instead of introducing another rule, we propose to use existing ones as part of a hierarchical and conditional combination scheme. The sources are represented by mass functions which are analysed and labelled regarding unreliability and imprecision. This conditional step divides the problem into specific sub-problems. In each of these sub-problems, the number of constraints is reduced and an appropriate rule is selected and applied. Two functions are thus obtained and analysed, allowing another rule to be chosen for a second (and final) fusion level. This approach provides a fast and robust way to combine disrupted sources using contextual information brought by a particle filter. Our experiments demonstrate its efficiency on several visual tracking situations.     

A linear time algorithm for edge coloring of binomial trees

We consider the problem of efficient coloring of the edges of a so-called binomial tree T, i.e. acyclic graph containing two kinds of edges: those which must have a single color and those which are to be colored with L consecutive colors, where L is an arbitrary integer greater than 1. We give an O(n) time algorithm for optimal coloring of such a tree, where n is the number of vertices of T. Also, we give simple bounds on the chromatic index of T and a division of all binomial trees into two classes depending on their chromaticity.     

具有不确定交易价格的指数跟踪资产组合再平衡问题

本文考虑具有某种不确定性的交易费用及预算约束的指数跟踪资产组合的再平衡问题。在已有的模型中,预算约束中使用的交易价格通常是一个确定值。而在再平衡过程中,股票的实际交易价格是不确定的。本文使用有限状态的离散时间马尔柯夫链模型处理交易价格的不确定性,并基于情景分析方法建立了具有不确定预算关系式的再平衡模型,然后使用股票市场的实际样本数据进行了数值实验,模拟结果说明本文的模型是可行的。     

A causal framework for surrogate endpoints with semi-competing risks data

In this note, we address the problem of surrogacy using a causal modelling framework that differs substantially from the potential outcomes model that pervades the biostatistical literature. The framework comes from econometrics and conceptualizes direct effects of the surrogate endpoint on the true endpoint. While this framework can incorporate the so-called semi-competing risks data structure, we also derive a fundamental non-identifiability result. Relationships to existing causal modelling frameworks are also discussed.     

Inverse spectral problem for differential operator pencils on noncompact spatial networks

We study boundary value problems on noncompact cycle-free graphs (i.e., trees) for second-order ordinary differential equations with a nonlinear dependence on the spectral parameter. We establish properties of the spectrum and analyze the inverse problem of reconstructing the coefficients of a differential equation on the basis of the so-called Weyl functions. For this inverse problem, we prove a uniqueness theorem and obtain a procedure for constructing the solution by the method of spectral mapping.