Stochastic Portfolio Optimization Research Articles

Stochastic portfolio optimization: A regret-based approach on volatility risk measures: An empirical evidence from The New York stock market.

Portfolio optimization involves finding the ideal combination of securities and shares to reduce risk and increase profit in an investment. To assess the impact of risk in portfolio optimization, we utilize a significant volatility risk measure series. Behavioral finance biases play a critical role in portfolio optimization and the efficient allocation of stocks. Regret, within the realm of behavioral finance, is the feeling of remorse that causes hesitation in making significant decisions and avoiding actions that could lead to poor investment choices. This behavior often leads investors to hold onto losing investments for extended periods, refusing to acknowledge mistakes and accept losses. Ironically, by evading regret, investors may miss out on potential opportunities. in this paper, our purpose is to compare investment scenarios in the decision-making process and calculate the amount of regret obtained in each scenario. To accomplish this, we consider volatility risk metrics and utilize stochastic optimization to identify the most suitable scenario that not only maximizes yield in the investment portfolio and minimizes risk, but also minimizes resulting regret. To convert each multi-objective model into a single objective, we employ the augmented epsilon constraint (AEC) method to establish the Pareto efficiency frontier. As a means of validating the solution of this method, we analyze data spanning 20, 50, and 100 weeks from 150 selected stocks in the New York market based on fundamental analysis. The results show that the selection of the mad risk measure in the time horizon of 100 weeks with a regret rate of 0.104 is the most appropriate research scenario. this article recommended that investors diversify their portfolios by investing in a variety of assets. This can help reduce risk and increase overall returns and improve financial literacy among investors.

Scenario generation and risk-averse stochastic portfolio optimization applied to offshore renewable energy technologies

Scenario generation and risk-averse stochastic portfolio optimization applied to offshore renewable energy technologies

Strong Formulations for Distributionally Robust Chance-Constrained Programs with Left-Hand Side Uncertainty Under Wasserstein Ambiguity

Distributionally robust chance-constrained programs (DR-CCPs) over Wasserstein ambiguity sets exhibit attractive out-of-sample performance and admit big-M–based mixed-integer programming reformulations with conic constraints. However, the resulting formulations often suffer from scalability issues as problem size increases. To address this shortcoming, we derive stronger formulations that scale well with respect to the problem size. Our focus is on ambiguity sets under the so-called left-hand side uncertainty, where the uncertain parameters affect the coefficients of the decision variables in the linear inequalities defining the safety sets. The interaction between the uncertain parameters and the variable coefficients in the safety set definition causes challenges in strengthening the original big-M formulations. By exploiting the connection between nominal chance-constrained programs and DR-CCP, we obtain strong formulations with significant enhancements. In particular, through this connection, we derive a linear number of valid inequalities, which can be immediately added to the formulations to obtain improved formulations in the original space of variables. In addition, we suggest a quantile-based strengthening procedure that allows us to reduce the big-M coefficients drastically. Furthermore, based on this procedure, we propose an exponential class of inequalities that can be separated efficiently within a branch-and-cut framework. The quantile-based strengthening procedure can be expensive. Therefore, for the special case of covering and packing type problems, we identify an efficient scheme to carry out this procedure. We demonstrate the computational efficacy of our proposed formulations on two classes of problems, namely stochastic portfolio optimization and resource planning. Funding: This work was supported by the DARPA Lagrange Program [Grant N660011824020], the National Science Foundation [Grant 1740707], the Office of Naval Research [Grant N00014-19-1-2321], and the Institute for Basic Science [Grants IBS-R029-C1 and IBS-R029-Y2].

A new method to solve the Hamilton-Jacobi-Bellman equation for a stochastic portfolio optimization model with boundary memory

<p style='text-indent:20px;'>Portfolio optimization problem with memory effect, namely, taking the past performance of portfolio into account, which is motivated by the fact that the decisions of the investors are influenced by the historic performance of the portfolio, has recently been of interest to researchers. Due to the memory effect, this type of problem is challenging. The main purpose of this paper is to propose a new approach to solve the Hamilton-Jacobi-Bellman (HJB) equation for the stochastic portfolio optimization model, which is more simple and generalized. We study the portfolio management problem under single investor and multi-investor framework. We establish certain conditions under which the problem can be reduced to the classical stochastic control problem and give some examples to show the advantages of our approach.</p>

Two-stage bond portfolio optimization and its application to Saudi Sukuk Market

We consider a two-stage stochastic bond portfolio optimization problem, where an investor aims to optimize the cost of bond portfolio under different scenarios while ensuring predefined liabilities during a given planning horizon. The investor needs to optimally decide whether to buy, hold, or sell bonds based upon present market conditions under different scenarios and varying assumptions, where the scenarios are determined based on interest rates and buying prices of the bonds. Three stochastic integer programming models are proposed and tested using real data from Saudi Sukuk (Bond) Market. The obtained results demonstrate the varying optimal decisions made to manage bond portfolio over the two stages. In addition, the three stochastic programming models for bond portfolio optimization are tested on a large set of randomly generated instances similar to the Saudi Sukuk (Bond) Market. The results of computational experiments attest the efficiency of the proposed models.

A hybrid combinatorial approach to a two-stage stochastic portfolio optimization model with uncertain asset prices

Portfolio optimization is one of the most important problems in the finance field. The traditional Markowitz mean-variance model is often unrealistic since it relies on the perfect market information. In this work, we propose a two-stage stochastic portfolio optimization model with a comprehensive set of real-world trading constraints to address this issue. Our model incorporates the market uncertainty in terms of future asset price scenarios based on asset return distributions stemming from the real market data. Compared with existing models, our model is more reliable since it encompasses real-world trading constraints and it adopts CVaR as the risk measure. Furthermore, our model is more practical because it could help investors to design their future investment strategies based on their future asset price expectations. In order to solve the proposed stochastic model, we develop a hybrid combinatorial approach, which integrates a hybrid algorithm and a linear programming (LP) solver for the problem with a large number of scenarios. The comparison of the computational results obtained with three different metaheuristic algorithms and with our hybrid approach shows the effectiveness of the latter. The superiority of our model is mainly embedded in solution quality. The results demonstrate that our model is capable of solving complex portfolio optimization problems with tremendous scenarios while maintaining high solution quality in a reasonable amount of time and it has outstanding practical investment implications, such as effective portfolio constructions.

Flexible Options for Greenhouse Gas-Emitting Energy Producer

The reduction of emissions from deforestation and forest degradation (REDD) constitutes part of the international climate agreements and contributes to the Sustainable Development Goals. This research is motivated by the risks associated with the future C O 2 price uncertainty in the context of the offsetting of carbon emissions by regulated entities. The research asked whether it is possible to reduce these financial risks. In this study, we consider the bilateral interaction of a REDD supplier and a greenhouse gas (GHG)-emitting energy producer in an incomplete emission offsets market. Within this setting, we explore an innovative financial instrument—flobsion—a flexible option with benefit-sharing. For the quantitative assessment, we used a research method based on a two-stage stochastic technological portfolio optimization model established in earlier studies. First, we obtain an important result that the availability of REDD offsets does not increase the optimal emissions of the electricity producer under any future C O 2 price realization. Moreover, addressing concerns about a possible “crowding–out” effect of REDD-based offsets, we demonstrate that the emissions and offsetting cost will decrease and increase, respectively. Second, we demonstrate the flexibility of the proposed instrument by analyzing flobsion contracts with respect to the benefit-sharing ratio and strike price within the risk-adjusted supply and demand framework. Finally, we perform a sensitivity analysis with respect to C O 2 price distributions and the opportunity costs of the forest owner supplying REDD offsets. Our results show that flobsion’s flexibility has advantages compared to a standard option, which can help GHG-emitting energy producers with managing their compliance risks, while at the same time facilitating the development of REDD programs. In this study we limited our analysis to the case of the same C O 2 price distributions foreseen by both parties; the flobsion pricing under asymmetric information could be considered in the future.

Portfolio Optimization for Assets with Stochastic Yields and Stochastic Volatility

In this paper, we consider a stochastic portfolio optimization model for investment on a risky asset with stochastic yields and stochastic volatility. The problem is formulated as a stochastic control problem, and the goal is to choose the optimal investment and consumption controls to maximize the investor’s expected total discounted utility. The Hamilton–Jacobi–Bellman equation is derived by virtue of the dynamic programming principle, which is a second-order nonlinear equation. Using the subsolution–supersolution method, we establish the existence result of the classical solution of the equation. Finally, we verify that the solution is equal to the value function and derive and verify the optimal investment and consumption controls.

Dynamic intertemporal utility optimization by means of Riccati transformation of Hamilton–Jacobi–Bellman equation

In this paper we investigate a dynamic stochastic portfolio optimization problem involving both the expected terminal utility and intertemporal utility maximization. We solve the problem by means of a solution to a fully nonlinear evolutionary Hamilton–Jacobi–Bellman (HJB) equation. We propose the so-called Riccati method for transformation of the fully nonlinear HJB equation into a quasi-linear parabolic equation with non-local terms involving the intertemporal utility function. As a numerical method we propose a semi-implicit scheme in time based on a finite volume approximation in the spatial variable. By analyzing an explicit traveling wave solution we show that the numerical method is of the second experimental order of convergence. As a practical application we compute optimal strategies for a portfolio investment problem motivated by market financial data of German DAX 30 Index and show the effect of considering intertemporal utility on optimal portfolio selection.

Expected utility maximization and conditional value-at-risk deviation-based Sharpe ratio in dynamic stochastic portfolio

In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the results on the shape of a chosen utility function in regard to the associated risk aversion level. We define the Conditional value-at-risk deviation ($CVaRD$) based Sharpe ratio for measuring risk-adjusted performance of a dynamic portfolio. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index and we evaluate and analyze the dependence of the $CVaRD$-based Sharpe ratio on the utility function and the associated risk aversion level.

Multiperiod stochastic programming portfolio optimization for diversified funds

We investigate a multiperiod, stochastic portfolio optimization model for diversified funds choices associated with traditional 401K or 403B plans. This optimization model minimizes the L1‐norm for negative return rate risk (the downside mean absolute deviation) while examining parameters that maintain model feasibility. Important components of the model are the incorporation of appropriate time‐series components and evaluation of scenarios based on investor outlook. A case study experimentation of the model on five potential investment funds using historical data from 2003 to 2013 was conducted, and parameter constraints for diversification and minimum acceptable return rates were manipulated to produce contour plots. The maximum geometric rate of return investment strategy provided by the optimization would have resulted in a 9.7% geometric return rate in 2014 as compared with a 5.0% for a uniform distribution of investment funds across choices.

A Review on Ambiguity in Stochastic Portfolio Optimization

In mean-risk portfolio optimization, it is typically assumed that the assets follow a known distribution P0, which is estimated from observed data. Aiming at an investment strategy which is robust against possible misspecification of P0, the portfolio selection problem is solved with respect to the worst-case distribution within a Wasserstein-neighborhood of P0. We review tractable formulations of the portfolio selection problem under model ambiguity, as it is called in the literature. For instance, it is known that high model ambiguity leads to equally-weighted portfolio diversification. However, it often happens that the marginal distributions of the assets can be estimated with high accuracy, whereas the dependence structure between the assets remains ambiguous. This leads to the problem of portfolio selection under dependence uncertainty. We show that in this case portfolio concentration becomes optimal as the uncertainty with respect to the estimated dependence structure increases. Hence, distributionally robust portfolio optimization can have two very distinct implications: Diversification on the one hand and concentration on the other hand.

A stochastic portfolio optimization model with complete memory

ABSTRACTIn this article, we consider a portfolio optimization problem of the Merton’s type with complete memory over a finite time horizon. The problem is formulated as a stochastic control problem on a finite time horizon and the state evolves according to a process governed by a stochastic process with memory. The goal is to choose investment and consumption controls such that the total expected discounted utility is maximized. Under certain conditions, we derive the explicit solutions for the associated Hamilton–Jacobi–Bellman (HJB) equations in a finite-dimensional space for exponential, logarithmic, and power utility functions. For those utility functions, verification results are established to ensure that the solutions are equal to the value functions, and the optimal controls are also derived.

Optimal diversification, stochastic dominance, and sampling error

PurposeThe purpose of this paper is to contribute to the literature in three ways: first, the authors investigate the impact of the sampling errors on optimal portfolio weights and on financial investment decision. Second, the authors advance a comparative analysis between various domestic and international diversification strategies to define a stochastic optimal choice. Third, the authors propose a new methodology combining the re-sampling method, stochastic optimization algorithm, and nonparametric stochastic dominance (SD) approach to analyze a stochastic optimal portfolio choice for risk-averse American investors who care about benefits of domestic diversification relative to international diversification. The authors propose a new portfolio optimization model involving SD constraints on the portfolio return rate. The authors define a portfolio with return dominating the benchmark portfolio return in the second-order stochastic dominance (SSD) and having maximum expected return. The authors combine re-sampling procedure and stochastic optimization to establish more flexibility in the investment decision rule.Design/methodology/approachThe authors apply the re-sampling procedure to consider the sampling error in the optimization process. The authors try to resolve the problem of the stochastic optimal investment strategy choice using the nonparametric SD test by Linton et al. (2005) based on sub-sampling simulated p values. The authors apply the stochastic portfolio optimization algorithm with SSD constraints to define optimal diversified portfolios beating benchmark indices.FindingsFirst, the authors find that reducing sampling error increases the dominance relationships between different portfolios, which, in turn, alters portfolio investment decisions. Though international diversification is preferred in some cases, the study’s results show that for risk-averse US investors, in general, there is no difference between the diversification strategies; this implies that there is no increase in the expected utility of international diversification for the period before and after the 2007-2008 financial crisis. Nevertheless, the authors find that stochastic diversification in domestic, global, and Europe, Australasia, and Far East markets delivers better risk returns for the US risk averters during the crisis period.Originality/valueThe originality of the idea in this paper is to introduce a new methodology combining the concept of portfolio re-sampling, stochastic portfolio optimization with SSD constraints, and the nonparametric SD test by Linton et al. (2005) based on subsampling simulated p values to analyze the impact of sampling errors on optimal portfolio returns and to investigate the problem of stochastic optimal choice between international and domestic diversification strategies. The authors try to prove more coherence in the portfolio choice with the stochastically and the uncertainty characters of the paper.

Investigation of Multistage Stochastic Portfolio Optimization Problems

Multistage stochastic portfolio optimization problems are investigated with the use of Kusuoki's and spectral coherent risk measures. To evaluate the performance of models, a backtesting process is developed using initial (historical) data. A proportionate share of portfolio assets at different time stages is found for models with Kusuoki's and spectral coherent risk measures. Multistage stochastic portfolio optimization models are measured by testing in terms of the value of a portfolio. Minimum portfolio losses are calculated for different time stages.

Random fuzzy mean-absolute deviation models for portfolio optimization problem with hybrid uncertainty

Absolute deviation is a commonly used risk measure, which has attracted more attentions in portfolio optimization. The existing mean-absolute deviation models are devoted to either stochastic portfolio optimization or fuzzy one. However, practical investment decision problems often involve the mixture of randomness and fuzziness such as stochastic returns with fuzzy information. Thus it is necessary to model portfolio selection problem in such a hybrid uncertain environment. In this paper, we employ random fuzzy variable to describe the stochastic return on individual security with ambiguous information. We first define the absolute deviation of random fuzzy variable and then employ it as risk measure to formulate mean-absolute deviation portfolio optimization models. To find the optimal portfolio, we design random fuzzy simulation and simulation-based genetic algorithm to solve the proposed models. Finally, a numerical example for synthetic data is presented to illustrate the validity of the method.

Risk Management of Interest Rate Derivative Portfolios: A Stochastic Control Approach

In this paper we formulate the Risk Management Control problem in the interest rate area as a constrained stochastic portfolio optimization problem. The utility that we use can be any continuous function and based on the viscosity theory, the unique solution of the problem is guaranteed. The numerical approximation scheme is presented and applied using a single factor interest rate model. It is shown how the whole methodology works in practice, with the implementation of the algorithm for a specific interest rate portfolio. The recent financial crisis showed that risk management of derivatives portfolios especially in the interest rate market is crucial for the stability of the financial system. Modern Value at Risk (VAR) and Conditional Value at Risk (CVAR) techniques, although very useful and easy to understand, fail to grasp the need for on-line controlling and monitoring of derivatives portfolio. The portfolios should be designed in a way that risk and return be quantified and controlled in every possible state of the world. We hope that this methodology contributes towards this direction.

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.

A multi-period stochastic portfolio optimization model applied for an airline company in the EU ETS

This paper aims to set up and solve a multi-period stochastic portfolio optimization model from an airline company’s point of view, considering all the specific European Union Emissions Trading Scheme (EU ETS) regulatory, managerial and trading constraints (i.e. physical constraints). Our contribution to existing academic literature is multiple. As the first ever case, we apply this technique to the aviation sector, a newly included sector within the EU ETS. More than mainly incorporating physical and technical (‘engineering’) features and focusing on short-term planning issues, we particularly address financial features and focus on mid-term planning issues. Therefore, instead of using spot prices, we run Monte Carlo simulations of correlated geometric Brownian motions (GBM) for traded futures prices of various emission allowance types for different CO2 delivery time periods. We thereby specifically refer to the existing exchange-traded emission allowance types EU Emission Allowance (EUA) and Certified Emission Reduction (CER). By implementing actually valid and real-world-oriented regulatory constraints for EU ETS, namely managerial and trading constraints, our model implies a real-life application. We also highlight the possibility of banking and borrowing of emission allowances between CO2 compliance periods, which is a crucial regulatory feature of EU ETS.

Warm-Start Heuristic for Stochastic Portfolio Optimization with Fixed and Proportional Transaction Costs

We consider a probabilistic portfolio optimization model including fixed and proportional transaction costs. We derive a deterministic equivalent of the probabilistic model for fat-tailed portfolio returns. We develop a method which finds provably near-optimal solutions in minimal amount of time for industry-sized (up to 2000 assets) problems. To solve the mixed-integer nonlinear programming (MINLP) deterministic formulation equivalent to the stochastic problem, we design a mathematical programming-based warm-start heuristic. The tests show the computational efficiency of the heuristic which is more than an order of magnitude faster than Cplex in finding high-quality solutions.