A memetic algorithm for cardinality-constrained portfolio optimization with transaction costs

A memetic approach that combines a genetic algorithm (GA) and quadratic programming is used to address the problem of optimal portfolio selection with cardinality constraints and piecewise linear transaction costs. The framework used is an extension of the standard Markowitz mean–variance model that incorporates realistic constraints, such as upper and lower bounds for investment in individual assets and/or groups of assets, and minimum trading restrictions. The inclusion of constraints that limit the number of assets in the final portfolio and piecewise linear transaction costs transforms the selection of optimal portfolios into a mixed-integer quadratic problem, which cannot be solved by standard optimization techniques. We propose to use a genetic algorithm in which the candidate portfolios are encoded using a set representation to handle the combinatorial aspect of the optimization problem. Besides specifying which assets are included in the portfolio, this representation includes attributes that encode the trading operation (sell/hold/buy) performed when the portfolio is rebalanced. The results of this hybrid method are benchmarked against a range of investment strategies (passive management, the equally weighted portfolio, the minimum variance portfolio, optimal portfolios without cardinality constraints, ignoring transaction costs or obtained with L₁ regularization) using publicly available data. The transaction costs and the cardinality constraints provide regularization mechanisms that generally improve the out-of-sample performance of the selected portfolios.     

Continuous-time mean variance portfolio with transaction costs: a proximal approach involving time penalization

This paper proposes a new continuous-time optimization solution that enables the computation of the portfolio problem (based on the utility option pricing and the shortfall risk minimization). We first propose a dynamical stock price process, and then, we transform the solution to a continuous-time discrete-state Markov decision processes. The market behavior is characterized by considering arbitrage-free and assessing transaction costs. To solve the problem, we present a proximal optimization approach, which considers time penalization in the transaction costs and the utility. In order to include the restrictions of the market, as well as those that imposed by the continuous-time space, we employ the Lagrange multipliers approach. As a result, we obtain two different equations: one for computing the portfolio strategies and the other for computing the Lagrange multipliers. Each equation in the portfolio is an optimization problem, for which the necessary condition of a maximum/minimum is solved employing the gradient method approach. At each step of the iterative proximal method, the functional increases and finally converges to a final portfolio. We show the convergence of the method. A numerical example showing the effectiveness of the proposed approach is also developed and presented.     

Large scale portfolio optimization with piecewise linear transaction costs

We consider the fundamental problem of computing an optimal portfolio based on a quadratic mean-variance model for the objective function and a given polyhedral representation of the constraints. The main departure from the classical quadratic programming formulation is the inclusion in the objective function of piecewise linear, separable functions representing the transaction costs. We handle the non-smoothness in the objective function by using spline approximations. The problem is first solved approximately using a primal-dual interior-point method applied to the smoothed problem. Then, we crossover to an active set method applied to the original non-smooth problem to attain a high accuracy solution. Our numerical tests show that we can solve large scale problems efficiently and accurately.     

Growth optimal investment in discrete-time markets with proportional transaction costs

We investigate how and when to diversify capital over assets, i.e., the portfolio selection problem, from a signal processing perspective. To this end, we first construct portfolios that achieve the optimal expected growth in i.i.d. discrete-time two-asset markets under proportional transaction costs. We then extend our analysis to cover markets having more than two stocks. The market is modeled by a sequence of price relative vectors with arbitrary discrete distributions, which can also be used to approximate a wide class of continuous distributions. To achieve the optimal growth, we use threshold portfolios, where we introduce a recursive update to calculate the expected wealth. We then demonstrate that under the threshold rebalancing framework, the achievable set of portfolios elegantly form an irreducible Markov chain under mild technical conditions. We evaluate the corresponding stationary distribution of this Markov chain, which provides a natural and efficient method to calculate the cumulative expected wealth. Subsequently, the corresponding parameters are optimized yielding the growth optimal portfolio under proportional transaction costs in i.i.d. discrete-time two-asset markets. As a widely known financial problem, we also solve the optimal portfolio selection problem in discrete-time markets constructed by sampling continuous-time Brownian markets. For the case that the underlying discrete distributions of the price relative vectors are unknown, we provide a maximum likelihood estimator that is also incorporated in the optimization framework in our simulations.     

A linear programming algorithm for optimal portfolio selection with transaction costs

We study the optimal portfolio selection problem with transaction costs. In general, the efficient frontier can be determined by solving a parametric non-quadratic programming problem. In a general setting, the transaction cost is a V-shaped function of difference between the existing and the new portfolio. We show how to transform this problem into a quadratic programming model. Hence a linear programming algorithm is applicable by establishing a linear approximation on the utility function of return and variance.     

基于改进粒子群算法的投资组合选择模型

研究了在实际投资决策中存在交易成本(税收和交易费用)和投资数量约束下的投资组合选择问题,并进一步设计了一种求解该问题的改进粒子群算法.最后,给出了一个数值例子,说明该模型和方法的有效性.     

Mean-variance-skewness model for portfolio selection with transaction costs

A mean-variance-skewness model is proposed for portfolio selection with transaction costs. It is assumed that the transaction cost is a V-shaped function of the difference between the existing portfolio and a new one. The mean-variance-skewness model is a non-smooth programming problem. To overcome the difficulty arising from non-smoothness, the problem was transformed into a linear programming problem. Therefore, this technique can be used to solve large-scale portfolio selection problems. A numerical example is used to illustrate that the method can be efficiently used in practice.     

An optimization model of the portfolio adjusting problem with fuzzy return and a SMO algorithm

Based on possibilistic mean and variance theory, this paper deals with the portfolio adjusting problem for an existing portfolio under the assumption that the returns of risky assets are fuzzy numbers and there exist transaction costs in portfolio adjusting precess. We propose a portfolio optimization model with V-shaped transaction cost which is associated with a shift from the current portfolio to an adjusted one. A sequential minimal optimization (SMO) algorithm is developed for calculating the optimal portfolio adjusting strategy. The algorithm is based on deriving the shortened optimality conditions for the formulation and solving 2-asset sub-problems. Numerical experiments are given to illustrate the application of the proposed model and the efficiency of algorithm. The results also show clearly the influence of the transaction costs in portfolio selection.     

基于灵敏度分析的含比例型手续费的投资组合优化

研究含比例型手续费的离散时间投资组合优化问题. 基于马尔可夫决策过程模型和性能灵敏度分析方法, 推导两个不同投资策略之间的资产长期平均增值率的差分公式, 利用差分公式的结构特点, 证明了最优性方程, 并设计出可在线应用的策略迭代算法. 仿真实例验证了所提出算法的有效性.

     

Sparse portfolio rebalancing model based on inverse optimization

This paper considers a sparse portfolio rebalancing problem in which rebalancing portfolios with minimum number of assets are sought. This problem is motivated by the need to understand whether the initial portfolio is worthwhile to adjust or not, inducing sparsity on the selected rebalancing portfolio to reduce transaction costs (TCs), out-of-sample performance and small changes in portfolio. We propose a sparse portfolio rebalancing model by adding an l₁ penalty item into the objective function of a general portfolio rebalancing model. In this way, the model is sparse with low TCs and can decide whether and which assets to adjust based on inverse optimization. Numerical tests on four typical data sets show that the optimal adjustment given by the proposed sparse portfolio rebalancing model has the advantage of sparsity and better out-of-sample performance than the general portfolio rebalancing model.     

Dynamic option hedging with transaction costs: A stochastic model predictive control approach

This paper proposes stochastic model predictive control as a tool for hedging derivative contracts (such as plain vanilla and exotic options) in the presence of transaction costs. The methodology combines stochastic scenario generation for the prediction of asset prices at the next rebalancing interval with the minimization of a stochastic measure of the predicted hedging error. We consider 3 different measures to minimize in order to optimally rebalance the replicating portfolio: a trade‐off between variance and expected value of hedging error, conditional value at risk, and the largest predicted hedging error. The resulting optimization problems require solving at each trading instant a quadratic program, a linear program, and a (smaller‐scale) linear program, respectively. These can be combined with 3 different scenario generation schemes: the lognormal stock model with parameters recursively identified from data, an identification method based on support vector regression, and a simpler scheme based on perturbation noise. The hedging performance obtained by the proposed stochastic model predictive control strategies is illustrated on real‐world data drawn from the NASDAQ‐100 composite, evaluated for a European call and a barrier option, and compared with delta hedging.     

Robust portfolio optimization via solution to the Hamilton–Jacobi–Bellman equation

We consider a problem of dynamic stochastic portfolio optimization modelled by a fully non-linear Hamilton–Jacobi–Bellman (HJB) equation. Using the Riccati transformation, the HJB equation is transformed to a simpler quasi-linear partial differential equation. An auxiliary quadratic programming problem is obtained, which involves a vector of expected asset returns and a covariance matrix of the returns as input parameters. Since this problem can be sensitive to the input data, we modify the problem from fixed input parameters to worst-case optimization over convex or discrete uncertainty sets both for asset mean returns and their covariance matrix. Qualitative as well as quantitative properties of the value function are analysed along with providing illustrative numerical examples. We show application to robust portfolio optimization for the German DAX30 Index.     

有交易费的折算资产优化性质和可达性

针对有交易费的多股票资产模型, 引入了资产折算函数, 并利用辅助鞅和凸分析方法, 讨论了该模型下折算资产优化性质和优化的可达性.     

Formulation of Closed-Loop Min–Max MPC as a Quadratically Constrained Quadratic Program

In this note, we show that min-max model predictive control (MPC) for linearly constrained polytopic systems with quadratic cost can be cast as a quadratically constrained quadratic program (QCQP). We use the rigorous closed loop formulation of min-max MPC, and show that any such min-max MPC problem with convex costs and constraints can be cast as a finite dimensional convex optimization problem, with the QCQP arising from quadratic costs as a special case. At the base of the proof is a lemma showing the convexity of the dynamic programming cost-to-go, which implies that the worst case on an infinite polytopic set is assumed on one of its finitely many vertices. As the approach is based on a scenario tree formulation, the number of variables in this problem grows exponentially with the horizon length. Fortunately, the QCQP is tree structured, and can thus be efficiently solved by specially tailored interior-point methods whose computational costs are linear in the number of variables. The new formulation as a tree sparse QCQP promises to facilitate online solution of the rigorous min-max MPC problem with quadratic costs     

Quadratic programming with transaction costs

We consider the problem of maximizing the mean-variance utility function of n$n$ assets. Associated with a change in an asset's holdings from its current or target value is a transaction cost. These must be accounted for in practical problems. A straightforward way of doing so results in a 3n$3n$-dimensional optimization problem with 3n$3n$ additional constraints. This higher dimensional problem is computationally expensive to solve. We present an algorithm for solving the 3n$3n$-dimensional problem by modifying an active set quadratic programming (QP) algorithm to solve the 3n$3n$-dimensional problem as an n$n$-dimensional problem accounting for the transaction costs implicitly rather than explicitly. The method is based on deriving the optimality conditions for the higher dimensional problem solely in terms of lower dimensional quantities and requires substantially less computational effort than any active set QP algorithm applied directly on a 3n$3n$-dimensional problem.     

Models and Simulations for Portfolio Rebalancing

In 1950 Markowitz first formalized the portfolio optimization problem in terms of mean return and variance. Since then, the mean-variance model has played a crucial role in single-period portfolio optimization theory and practice. In this paper we study the optimal portfolio selection problem in a multi-period framework, by considering fixed and proportional transaction costs and evaluating how much they affect a re-investment strategy. Specifically, we modify the single-period portfolio optimization model, based on the Conditional Value at Risk (CVaR) as measure of risk, to introduce portfolio rebalancing. The aim is to provide investors and financial institutions with an effective tool to better exploit new information made available by the market. We then suggest a procedure to use the proposed optimization model in a multi-period framework. Extensive computational results based on different historical data sets from German Stock Exchange Market (XETRA) are presented.     

Robust multiperiod portfolio management in the presence of transaction costs

We study the viability of different robust optimization approaches to multiperiod portfolio selection. Robust optimization models treat future asset returns as uncertain coefficients in an optimization problem, and map the level of risk aversion of the investor to the level of tolerance of the total error in asset return forecasts. We suggest robust optimization formulations of the multiperiod portfolio optimization problem that are linear and computationally efficient. The linearity of the optimization problems is an advantage when complex additional requirements need to be imposed on the portfolio structure, e.g., limitations on positions in certain assets or tax constraints. We compare the performance of our robust formulations to the performance of the traditional single period mean-variance formulation frequently employed in the financial industry.     

基于PSO的考虑完整费用的证券组合优化研究

通过分析中国证券市场证券交易不可拆分、不能卖空的特点以及现存的各种交易费用,建立一个考虑完整交易费用的证券投资组合优化模型,同时给出一个应用粒子群算法(PSO)求解的实例。结果证明该证券投资组合优化模型的完整性和有效性,也表明PSO算法可以快速准确地求解证券投资组合优化问题。     

Optimal Portfolio Hedging with Nonlinear Derivatives and Transaction Costs

We consider the problem of dynamically hedging a fixed portfolio of assets in the presence of non-linear instruments and transaction costs, as well as constraints on feasible hedging positions. We assume an investor maximizing the expected utility of his terminal wealth over a finite holding period, and analyse the dynamic portfolio optimization problem when the trading interval is fixed. An approximate solution is obtained from a two-stage numerical procedure. The problem is first transformed into a nonlinear programming problem which utilizes simulated coefficient matrices. The nonlinear programming problem is then solved numerically using standard constrained optimization techniques.     

Sharpe-ratio pricing and hedging of contingent claims in incomplete markets by convex programming

We analyze the problem of pricing and hedging contingent claims in a financial market described by a multi-period, discrete-time, finite-state scenario tree using an arbitrage-adjusted Sharpe-ratio criterion. We show that the writer’s and buyer’s pricing problems are formulated as conic convex optimization problems which allow to pass to dual problems over martingale measures and yield tighter pricing intervals compared to the interval induced by the usual no-arbitrage price bounds. An extension allowing proportional transaction costs is also given. Numerical experiments using S&P 500 options are given to demonstrate the practical applicability of the pricing scheme.