Portfolio Optimization Problem Research Articles

A hybrid framework for mean-CVaR portfolio selection under jump-diffusion processes: Combining cross-entropy method with beluga whale optimization

<abstract><p>In this paper, a new hybrid meta-heuristic algorithm called CEBWO (cross-entropy method and beluga whale optimization) is presented to solve the mean-CVaR portfolio optimization problem based on jump-diffusion processes. The proposed CEBWO algorithm combines the advantages of the cross-entropy method and beluga whale optimization algorithm with the help of co-evolution technology to enhance the performance of portfolio selection. The method is evaluated on 29 unconstrained benchmark functions from CEC 2017, where its performance is compared against several state-of-the-art algorithms. The results demonstrate the superiority of the hybrid method in terms of solution quality and convergence speed. Finally, Monte Carlo simulation is employed to generate scenario paths based on the jump-diffusion model. Empirical results further confirm the effectiveness of the hybrid meta-heuristic algorithm for mean-CVaR portfolio selection, highlighting its potential for real-world applications.</p></abstract>

Portfolio optimization based on bi-objective linear programming

In this study, we deal with a portfolio optimization problem including both risky and risk-free assets. We use the infinity norm criterion to measure portfolio risk and formulate the problem as a bi-objective linear optimization problem. Then, a single objective linear program is considered related to the bi-objective optimization problem. Using the well-known Karush-Kuhn-Tucker optimality conditions, we obtain analytic formula for an optimal solution. Moreover, we determine the whole efficient frontier by multi-criteria optimization techniques. Based on the theoretical results, two algorithms are proposed for finding the portfolio weights and the efficient frontier. Numerical examples are given for illustrating the new models and algorithms. Additionally, a simulation study has been conducted to assess the performance of the proposed method.

带遗产计划和非流动性资产的DC型养老金的最优投资组合问题

带遗产计划和非流动性资产的DC型养老金的最优投资组合问题

Application of Asymptotic Analysis of a High-Dimensional HJB Equation to Portfolio Optimization

In this paper, we consider a portfolio optimization problem where the wealth consists of investing into a risky asset with a slow mean-reverting volatility and receiving an uncontrollable stochastic cash flow under the exponential utility. The Hamilton–Jacobi–Bellman equation formulated from the optimal investment problem is a high-dimensional nonlinear partial differential equation and difficult to find its analytical or numerical solutions. The paper provides a tractable asymptotic approach which treats the initial problem as a perturbation around the constant volatility problem. In this paper, we present a formal derivation of asymptotic approximation and prove the accuracy of the value function. Moreover, an illustrative example is provided to assess our approximate strategy and value function.

Credibilistic portfolio optimization with higher-order moments using coherent triangular fuzzy numbers

Financial portfolio formation is usually a multi-objective decision-making problem concerning return and risk on the investment. In this study, we make use of an extension of regular triangular fuzzy numbers, known as coherent triangular fuzzy numbers, to describe portfolio returns in a credibility-based framework. The paper’s novelty lies in proposing and deriving a crisp equivalent for computing the coherent triangular fuzzy number-based credibilistic semivariance, credibilistic skewness, and credibilistic semikurtosis. Utilizing these analytical expressions, along with additional formulations found in existing literature, we present three multi-objective portfolio optimization models involving the practical constraints related to investment decisions. All the proposed analytical expressions, when used with the returns of the portfolio as a whole, help to overcome the computationally expensive process of simulating results using the returns of individual assets. The proposed models differ with respect to different risk measures, viz. semivariance, Mean-Absolute-Semi-Deviation (MASD), and Conditional Value-at-Risk (CVaR). These models are solved using an adaptation of an efficient Multi-Objective Genetic Algorithm (MOGA) specifically designed to solve portfolio optimization problems with practical constraints. Data from the National Stock Exchange (NSE) in Mumbai, India and the New York Stock Exchange (NYSE) in New York, USA, are used to demonstrate the effectiveness of the proposed portfolio optimization models and solution methodology. All the proposed models are compared with respect to each other and the benchmarks considered in this study to bring out the performance stability.

A monotone numerical integration method for mean–variance portfolio optimization under jump-diffusion models

We develop a highly efficient, easy-to-implement, and strictly monotone numerical integration method for Mean-Variance (MV) portfolio optimization. This method proves very efficient in realistic contexts, which involve jump-diffusion dynamics of the underlying controlled processes, discrete rebalancing, and the application of investment constraints, namely no-bankruptcy and leverage.A crucial element of the MV portfolio optimization formulation over each rebalancing interval is a convolution integral, which involves a conditional density of the logarithm of the amount invested in the risky asset. Using a known closed-form expression for the Fourier transform of this conditional density, we derive an infinite series representation for the conditional density where each term is strictly positive and explicitly computable. As a result, the convolution integral can be readily approximated through a monotone integration scheme, such as a composite quadrature rule typically available in most programming languages. The computational complexity of our proposed method is an order of magnitude lower than that of existing monotone finite difference methods. To further enhance efficiency, we propose an implementation of this monotone integration scheme via Fast Fourier Transforms, exploiting the Toeplitz matrix structure.The proposed monotone numerical integration scheme is proven to be both ℓ∞-stable and pointwise consistent, and we rigorously establish its pointwise convergence to the unique solution of the MV portfolio optimization problem. We also intuitively demonstrate that, as the rebalancing time interval approaches zero, the proposed scheme converges to a continuously observed impulse control formulation for MV optimization expressed as a Hamilton–Jacobi–Bellman Quasi-Variational Inequality. Numerical results show remarkable agreement with benchmark solutions obtained through finite differences and Monte Carlo simulation, underscoring the effectiveness of our approach.

Alignment between initial state and mixer improves QAOA performance for constrained optimization

Quantum alternating operator ansatz (QAOA) has a strong connection to the adiabatic algorithm, which it can approximate with sufficient depth. However, it is unclear to what extent the lessons from the adiabatic regime apply to QAOA as executed in practice with small to moderate depth. In this paper, we demonstrate that the intuition from the adiabatic algorithm applies to the task of choosing the QAOA initial state. Specifically, we observe that the best performance is obtained when the initial state of QAOA is set to be the ground state of the mixing Hamiltonian, as required by the adiabatic algorithm. We provide numerical evidence using the examples of constrained portfolio optimization problems with both low (p ≤ 3) and high (p = 100) QAOA depth. Additionally, we successfully apply QAOA with XY mixer to portfolio optimization on a trapped-ion quantum processor using 32 qubits and discuss our findings in near-term experiments.

Equilibrium multi-agent model with heterogeneous views on fundamental risks

This paper investigates an equilibrium-based multi-agent optimal consumption and portfolio problem incorporating uncertainties on fundamental risks, where multiple agents have heterogeneous (conservative, neutral, aggressive) views on the risks represented by Brownian motions. Each agent maximizes its expected utility on consumption under its subjective probability measure. Specifically, we formulate the individual optimization problem as a sup–sup–inf problem, which is an optimal consumption and portfolio problem with a choice of a probability measure. Moreover, we provide an expression of the state-price density process in a market equilibrium, which derives the representations of the interest rate and the market price of risk. To the best of our knowledge, this is the first attempt to investigate the multi-agent model with heterogeneous views on the risks by considering a market equilibrium and solving sup–sup–inf problems on the choice of a probability measure. We emphasize that the setting, where each agent has heterogeneous views on different risks, incorporates a special case where each agent has only conservative or neutral views on risks with different degrees of conservativeness. Also, the setting includes the case where the agents have aggressive views on risks, commonly observed as bullish sentiments in the financial markets in the monetary easing after the global financial crisis and particularly in the COVID-19 pandemic. Finally, we present numerical examples of the interest rate model, which show how heterogeneous views of the multiple agents on the risks affect the shape of the yield curve.

Best practices for portfolio optimization by quantum computing, experimented on real quantum devices

In finance, portfolio optimization aims at finding optimal investments maximizing a trade-off between return and risks, given some constraints. Classical formulations of this quadratic optimization problem have exact or heuristic solutions, but the complexity scales up as the market dimension increases. Recently, researchers are evaluating the possibility of facing the complexity scaling issue by employing quantum computing. In this paper, the problem is solved using the Variational Quantum Eigensolver (VQE), which in principle is very efficient. The main outcome of this work consists of the definition of the best hyperparameters to set, in order to perform Portfolio Optimization by VQE on real quantum computers. In particular, a quite general formulation of the constrained quadratic problem is considered, which is translated into Quadratic Unconstrained Binary Optimization by the binary encoding of variables and by including constraints in the objective function. This is converted into a set of quantum operators (Ising Hamiltonian), whose minimum eigenvalue is found by VQE and corresponds to the optimal solution. In this work, different hyperparameters of the procedure are analyzed, including different ansatzes and optimization methods by means of experiments on both simulators and real quantum computers. Experiments show that there is a strong dependence of solutions quality on the sufficiently sized quantum computer and correct hyperparameters, and with the best choices, the quantum algorithm run on real quantum devices reaches solutions very close to the exact one, with a strong convergence rate towards the classical solution, even without error-mitigation techniques. Moreover, results obtained on different real quantum devices, for a small-sized example, show the relation between the quality of the solution and the dimension of the quantum processor. Evidences allow concluding which are the best ways to solve real Portfolio Optimization problems by VQE on quantum devices, and confirm the possibility to solve them with higher efficiency, with respect to existing methods, as soon as the size of quantum hardware will be sufficiently high.

A constrained swarm optimization algorithm for large-scale long-run investments using Sharpe ratio-based performance measures

We study large-scale portfolio optimization problems in which the aim is to maximize a multi-moment performance measure extending the Sharpe ratio. More specifically, we consider the adjusted for skewness Sharpe ratio, which incorporates the third moment of the returns distribution, and the adjusted for skewness and kurtosis Sharpe ratio, which exploits in addition the fourth moment. Further, we account for two types of real-world trading constraints. On the one hand, we impose stock market restrictions through cardinality, buy-in thresholds, and budget constraints. On the other hand, a turnover threshold restricts the total allowed amount of trades in the rebalancing phases. To deal with these asset allocation models, we embed a novel hybrid constraint-handling procedure into an improved dynamic level-based learning swarm optimizer. A repair operator maps candidate solutions onto the set characterized by the first type of constraints. Then, an adaptive ℓ1\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\ell _1$$\\end{document}-exact penalty function manages turnover violations. The focus of the paper is to highlight the importance of including higher-order moments in the performance measures for long-run investments, in particular when the market is turbulent. We carry out empirical tests on two worldwide sets of assets to illustrate the scalability and effectiveness of the proposed strategies, and to evaluate the performance of our investments compared to the strategy maximizing the Sharpe ratio.

Bi-directional learning particle swarm optimization for large-scale optimization

Large-scale optimization problems (LSOPs) are an essential research topic in the evolutionary computation (EC) community with two challenges: slow convergence in the huge search space and the trap in massive locally optimal solutions. To tackle these two challenges, this paper proposes a bi-directional learning particle swarm optimization (BLPSO) with two learning strategies, called diversity learning strategy (DLS) and convergence learning strategy (CLS). In DLS, we first propose a diversity evaluation mechanism based on locally sensitive hashing (LSH) to measure the diversity of individuals. Then the density individuals will learn from other dispersed individuals with good diversity to enhance the diversity of the population. In CLS, the inferior individuals will learn from other superior individuals with excellent evolution information to help the population accelerate convergence speed. These two learning strategies act in different roles and complement each other. With these two learning strategies, BLPSO achieves a balance between sufficient diversity and fast convergence in solving LSOPs. Two large-scale test suites, IEEE CEC2010 and IEEE CEC2013, are used to test the performance between BLPSO and other state-of-the-art algorithms. The experimental results show that BLPSO outperforms other algorithms on both test suites, including the winner algorithms of the IEEE CEC2010 and IEEE CEC2012 competitions. Finally, BLPSO is applied to a large-scale portfolio optimization problem to illustrate its application capability.

Multi-period portfolio optimization using a deep reinforcement learning hyper-heuristic approach

Portfolio optimization concerns with periodically allocating the limited funds to invest in a variety of potential assets in order to satisfy investors’ appetites for risk and return goals. Recently, Deep Reinforcement Learning (DRL) has shown its promising capabilities in sequential decision making problems. However, traditional DRL algorithms directly operate in the space of low-level actions, which exhibits poor scalability and becomes intractable in real-world problem instances when the dimensionality of the environment increases. To deal with this, in this work, a novel DRL hyper-heuristic framework is proposed for multi-period portfolio optimization problem. Instead of exploiting the entire action domain, our proposed approach is more effective by searching for low-level well-developed trading strategies. In addition, our proposed approach is data-driven and respects the nature of the problem by taking advantage of expert domain knowledge and posing it multidimensional states to further leverage additional diverse information from alternative views of the environment. The proposed approach is evaluated on five real-world capital market problem instances and numerous experimental results demonstrate our proposed method can achieve notable performance gains compared to state-of-art trading strategies as well as traditional DRL baseline method. The data we used are from five stock indices, covering the period from the 2012 to 2022. Our study can have salient policy implications for investment strategy formulation and effective regulatory frameworks establishment.

POLYNOMIAL UTILITY

We approximate the utility function by polynomial series and solve the related dynamic portfolio optimization problems. We study the quality of the Taylor and Bernstein series approximation in response to the points and degrees of the expansions and generalize from earlier expansions applied to portfolio optimization. The issue of time inconsistency, arising from a dynamically adapted center of the expansion, is approached by equilibrium theory. We present new ways of constructing polynomial utility functions and study their pitfalls and potentials. In the numerical study, we focus on two specific utility functions: For power utility, access to the optimal portfolio allows for a complete illustration of the approximations; for the S-shaped utility function of prospect theory, the use of equilibrium theory allows for approximating the solution to the (obviously interesting but yet unsolved) case of current wealth as a dynamic reference point.

A Low-Cost Alternating Projection Approach for a Continuous Formulation of Convex and Cardinality Constrained Optimization

We consider convex constrained optimization problems that also include a cardinality constraint. In general, optimization problems with cardinality constraints are difficult mathematical programs which are usually solved by global techniques from discrete optimization. We assume that the region defined by the convex constraints can be written as the intersection of a finite collection of convex sets, such that it is easy and inexpensive to project onto each one of them (e.g., boxes, hyper-planes, or half-spaces). Taking advantage of a recently developed continuous reformulation that relaxes the cardinality constraint, we propose a specialized penalty gradient projection scheme combined with alternating projection ideas to compute a solution candidate for these problems, i.e., a local (possibly non-global) solution. To illustrate the proposed algorithm, we focus on the standard mean-variance portfolio optimization problem for which we can only invest in a preestablished limited number of assets. For these portfolio problems with cardinality constraints, we present a numerical study on a variety of data sets involving real-world capital market indices from major stock markets. In many cases, we observe that the proposed scheme converges to the global solution. On those data sets, we illustrate the practical performance of the proposed scheme to produce the effective frontiers for different values of the limited number of allowed assets.

A multi-population evolutionary algorithm for multi-objective constrained portfolio optimization problem

A multi-population evolutionary algorithm for multi-objective constrained portfolio optimization problem

Optimal reinsurance design under solvency constraints

We consider the optimal risk transfer from an insurance company to a reinsurer. The problem formulation considered in this paper is closely connected to the optimal portfolio problem in finance, with some crucial distinctions. In particular, the insurance company's surplus is here (as is routinely the case) approximated by a Brownian motion, as opposed to the geometric Brownian motion used to model assets in finance. Furthermore, risk exposure is dialled ‘down’ via reinsurance, rather than ‘up’ via risky investments. This leads to interesting qualitative differences in the optimal designs. In this paper, using the martingale method, we derive the optimal design as a function of proportional, non-cheap reinsurance design that maximises the quadratic utility of the terminal value of the insurance surplus. We also consider several realistic constraints on the terminal value: a strict lower boundary, the probability (Value at Risk) constraint, and the expected shortfall (conditional Value at Risk) constraints under the P and Q measures, respectively. In all cases, the optimal reinsurance designs boil down to a combination of proportional protection and option-like protection (stop-loss) of the residual proportion with various deductibles. Proportions and deductibles are set such that the initial capital is fully allocated. Comparison of the optimal designs with the optimal portfolios in finance is particularly interesting. Results are illustrated.

Futuristic portfolio optimization problem: wavelet based long short-term memory

PurposeThis paper aims to propose an improved version of portfolio optimization model through the prediction of the future behavior of stock returns using a combined wavelet-based long short-term memory (LSTM).Design/methodology/approachFirst, data are gathered and divided into two parts, namely, “past data” and “real data.” In the second stage, the wavelet transform is proposed to decompose the stock closing price time series into a set of coefficients. The derived coefficients are taken as an input to the LSTM model to predict the stock closing price time series and the “future data” is created. In the third stage, the mean-variance portfolio optimization problem (MVPOP) has iteratively been run using the “past,” “future” and “real” data sets. The epsilon-constraint method is adapted to generate the Pareto front for all three runes of MVPOP.FindingsThe real daily stock closing price time series of six stocks from the FTSE 100 between January 1, 2000, and December 30, 2020, is used to check the applicability and efficacy of the proposed approach. The comparisons of “future,” “past” and “real” Pareto fronts showed that the “future” Pareto front is closer to the “real” Pareto front. This demonstrates the efficacy and applicability of proposed approach.Originality/valueMost of the classic Markowitz-based portfolio optimization models used past information to estimate the associated parameters of the stocks. This study revealed that the prediction of the future behavior of stock returns using a combined wavelet-based LSTM improved the performance of the portfolio.

Diversification quotients based on VaR and ES

The diversification quotient (DQ) is recently introduced for quantifying the degree of diversification of a stochastic portfolio model. It has an axiomatic foundation and can be defined through a parametric class of risk measures. Since the Value-at-Risk (VaR) and the Expected Shortfall (ES) are the most prominent risk measures widely used in both banking and insurance, we investigate DQ constructed from VaR and ES in this paper. In particular, for the popular models of elliptical and multivariate regular varying (MRV) distributions, explicit formulas are available. The portfolio optimization problems for the elliptical and MRV models are also studied. Our results further reveal favorable features of DQ, both theoretically and practically, compared to traditional diversification indices based on a single risk measure.

Z-Number-Based Maximum Expected Linear Programming Model with Applications

In research of a better description for information uncertainty, Z-numbers, which are related to both the objective information and the subjective criticism, were first conceptualized by Zadeh. Because of its neologism, there have been multitudinous attempts toward continuation and expansion of the prototype. In this paper, we mainly study varieties of theoretical preparations for classical Z-numbers and derive the maximum expected linear programming model of Z-numbers, which are constructed on the basis of reliability conversion factors and proliferation on applications due to their simplicity. Firstly, by means of transforming Z-numbers into LR fuzzy intervals through their reliability variable, the credibility distribution and inverse distribution of converted Z-numbers are stated precisely. Then, the operational law of independent variables and its expected value can be derived via credibility distribution. The maximum expected Z-number linear programming model is determined on the basis of previous theoretical preparations, and it transforms from a classical Z-number chance-constrained model into a crisp one. Finally, with the aim of improving the programming method, its application in pragmatic practice with the realistic examples of a supplier section and optimal portfolio problems are enumerated to interpret the effectiveness of our model.

HPOSS: A hierarchical portfolio optimization stacking strategy to reduce the generalization error of ensembles of models.

Surrogate models are frequently used to replace costly engineering simulations. A single surrogate is frequently chosen based on previous experience or by fitting multiple surrogates and selecting one based on mean cross-validation errors. A novel stacking strategy will be presented in this paper. This new strategy results from reinterpreting the model selection process based on the generalization error. For the first time, this problem is proposed to be translated into a well-studied financial problem: portfolio management and optimization. In short, it is demonstrated that the individual residues calculated by leave-one-out procedures are samples from a given random variable ϵi, whose second non-central moment is the i-th model's generalization error. Thus, a stacking methodology based solely on evaluating the behavior of the linear combination of the random variables ϵi is proposed. At first, several surrogate models are calibrated. The Directed Bubble Hierarchical Tree (DBHT) clustering algorithm is then used to determine which models are worth stacking. The stacking weights can be calculated using any financial approach to the portfolio optimization problem. This alternative understanding of the problem enables practitioners to use established financial methodologies to calculate the models' weights, significantly improving the ensemble of models' out-of-sample performance. A study case is carried out to demonstrate the applicability of the new methodology. Overall, a total of 124 models were trained using a specific dataset: 40 Machine Learning models and 84 Polynomial Chaos Expansion models (which considered 3 types of base random variables, 7 least square algorithms for fitting the up to fourth order expansion's coefficients). Among those, 99 models could be fitted without convergence and other numerical issues. The DBHT algorithm with Pearson correlation distance and generalization error similarity was able to select a subgroup of 23 models from the 99 fitted ones, implying a reduction of about 77% in the total number of models, representing a good filtering scheme which still preserves diversity. Finally, it has been demonstrated that the weights obtained by building a Hierarchical Risk Parity (HPR) portfolio perform better for various input random variables, indicating better out-of-sample performance. In this way, an economic stacking strategy has demonstrated its worth in improving the out-of-sample capabilities of stacked models, which illustrates how the new understanding of model stacking methodologies may be useful.