Variance Portfolio Selection Problem Research Articles

Mean–variance portfolio selection under no-shorting rules: A BSDE approach

This paper revisits the mean–variance portfolio selection problem in continuous-time within the framework of short-selling of stocks is prohibited via backward stochastic differential equation approach. To relax the strong condition in Li et al. (Li et al. 2002), the above issue is formulated as a stochastic recursive optimal linear–quadratic control problem. Due to no-shorting rules (namely, the portfolio taking non-negative values), the well-known “completion of squares” no longer applies directly. To overcome this difficulty, we study the corresponding Hamilton–Jacobi–Bellman (HJB, for short) equation inherently and derive the two groups of Riccati equations. On one hand, the value function constructed via Riccati equations is shown to be a viscosity solution of the HJB equation mentioned before; On the other hand, by means of these Riccati equations and backward semigroup, we are able to get explicitly the efficient frontier and efficient investment strategies for the recursive utility mean–variance portfolio optimization problem.

Mean--variance portfolio selection problem: Asset reduction via nondominated sorting

Due to recent globalization of financial markets, investors have access to large numbers of assets. It may be beneficial for them to focus on a limited number of assets filtered out by some meaningful procedure. This, however, can result in selecting portfolios which, in terms of reward and risk, are not efficient in the original set of assets. A challenge is to have ways for determining subsets of assets in which the effect of lost efficiency would be minimal. To meet this challenge, we propose a method for asset reduction, based on the notion of layers of maxima and the concept of nondominated sorting. We conduct experiments on large problems derived from the USA stock market data. Our approach resulted in a much smaller loss of efficiency compared to two representative asset reduction methods known from the literature. We test the approach viability via computational experiments on the mean–variance problem of portfolio selection, with and without the cardinality constraints, and real-life data consisting of up to 1000 assets.

Maximum principle for discrete-time stochastic control problem of mean-field type

In this work, a discrete-time mean-field type stochastic optimal control problem is studied. The goal is to derive the stochastic maximum principle with convex control domains. L-derivative is applied to handle the mean-field term and a technique of adjoint operator is used to overcome the difficulties of obtaining adjoint equations and duality relation. Then, the stochastic maximum principle for discrete-time mean-field type stochastic optimal control problem is established. Finally, as an illustration of our stochastic maximum principle, a discrete-time mean–variance portfolio selection problem is solved with the decoupling technique which is different from the continuous-time case.

Optimal multi-period transaction-cost-aware long-only portfolios and time consistency in efficiency

This paper studies a multi-period mean–variance (MV) portfolio selection problem in a market of one risk-free asset and one risky asset traded with proportional transaction costs and no-shorting constraint. A particular interest of this study is to investigate the time consistency in efficiency (TCIE) of the optimal MV portfolio in the presence of transaction costs. To this end, we derive a semi-closed-form solution of the optimal pre-committed dynamic MV policy in the no-shorting non-frictionless market with a combination of embedding and dynamic programming techniques, as well as its several analytical properties. We show that the optimal MV policy is always TCIE when no-shorting constraint is imposed, which gives right to the long-only portfolios in dynamic settings advocated in some empirical evidences. Numerically, we conduct sensitivity analyses of the efficient frontiers and the width of the no-transaction region with respect to the rates of transaction costs and initial wealth allocations. Moreover, we show the significance of the TCIE and the intuitive rationale behind it.

Optimal dynamic mean–variance portfolio subject to proportional transaction costs and no-shorting constraint

This paper studies mean–variance portfolio selection problem subject to proportional transaction costs and no-shorting constraint. We do not impose any distributional assumptions on the asset returns. By adopting dynamic programming, duality theory, and a comparison approach, we manage to derive a semi-closed form solution of the optimal dynamic investment policy with the boundaries of buying, no-transaction, selling, and liquidation regions. Numerically, we illustrate the properties of the optimal policy by depicting the corresponding efficient frontiers under different rates of transaction costs and initial wealth allocations. We find that the efficient frontier is distorted due to the transaction cost incurred. We also examine how the width of the no-transaction region varies with different transaction cost rates. Empirically, we show that our transaction-cost-aware policy outperforms the transaction-cost-unaware policy in a realistic trading environment that incurs transaction costs

Time-varying mean–variance portfolio selection problem solving via LVI-PDNN

It is widely acclaimed that the Markowitz mean–variance portfolio selection is a very important investment strategy. One approach to solving the static mean–variance portfolio selection (MVPS) problem is based on the usage of quadratic programming (QP) methods. In this article, we define and study the time-varying mean–variance portfolio selection (TV-MVPS) problem both in the cases of a fixed target portfolio’s expected return and for all possible portfolio’s expected returns as a time-varying quadratic programming (TVQP) problem. The TV-MVPS also comprises the properties of a moving average. These properties make the TV-MVPS an even greater analysis tool suitable to evaluate investments and identify trading opportunities across a continuous-time period. Using an originally developed linear-variational-inequality primal–dual neural network (LVI-PDNN), we also provide an online solution to the static QP problem. To the best of our knowledge, this is an innovative approach that incorporates robust neural network techniques to provide an online, thus more realistic, solution to the TV-MVPS problem. In this way, we present an online solution to a time-varying financial problem while eliminating static method limitations. It has been shown that when applied simultaneously to TVQP problems subject to equality, inequality and boundary constraints, the LVI-PDNN approaches the theoretical solution. Our approach is also verified by numerical experiments and computer simulations as an excellent alternative to conventional MATLAB methods.

Time-consistent mean-variance investment with unit linked life insurance contracts in a jump-diffusion setting

We consider a time-consistent mean–variance portfolio selection problem of an insurer and allow for the incorporation of basis (mortality) risk. The optimal solution is identified with a Nash subgame perfect equilibrium. We characterize an optimal strategy as solution of a system of partial integro-differential equations (PIDEs), a so called extended Hamilton–Jacobi–Bellman (HJB) system. We prove that the equilibrium is necessarily a solution of the extended HJB system. Under certain conditions we obtain an explicit solution to the extended HJB system and provide the optimal trading strategies in closed-form. A simulation shows that the previously found strategies yield payoffs whose expectations and variances are robust regarding the distribution of jump sizes of the stock. The same phenomenon is observed when the variance is correctly estimated, but erroneously ascribed to the diffusion components solely. Further, we show that differences in the insurance horizon and the time to maturity of a longevity asset do not add to the variance of the terminal wealth.

Mean–variance portfolio selection under partial information with drift uncertainty

ABSTRACT In this paper, we study the mean–variance portfolio selection problem under partial information with drift uncertainty. First we show that the market model is complete even in this case while the information is not complete and the drift is uncertain. Then, the optimal strategy based on partial information is derived, which reduces to solving a related backward stochastic differential equation (BSDE). Finally, we propose an efficient numerical scheme to approximate the optimal portfolio that is the solution of the BSDE mentioned above. Malliavin calculus and the particle representation play important roles in this scheme.

Cardinality-constrained programs with nonnegative variables and an SCA method

We study a cardinality-constrained optimisation problem with nonnegative variables in this paper. This problem is often encountered in practice. Firstly, we study some properties on the optimal solutions of this optimisation problem under some conditions. An equivalent reformulation of the problem under consideration is proposed. Based on the reformulation, we present a successive convex approximation method for the cardinality constrained optimisation problem. We prove that the method converges to a KKT point of the reformulation problem. Under some conditions, the KKT points of the reformulation problem are local optimisers of the original problem. Our numerical results on a limited diversified mean–variance portfolio selection problem demonstrate some promising results.

Indefinite mean-field type linear–quadratic stochastic optimal control problems

This paper focuses on indefinite stochastic mean-field linear–quadratic (MF-LQ, for short) optimal control problems, which allow the weighting matrices for state and control in the cost functional to be indefinite. The solvability of stochastic Hamiltonian system and Riccati equations is presented under indefinite case. The optimal controls in open-loop form and closed-loop form are derived, respectively. In particular, dynamic mean–variance portfolio selection problem can be formulated as an indefinite MF-LQ problem to tackle directly. Another example also sheds light on the theoretical results established.

Portfolio selection with parameter uncertainty under [formula omitted] maxmin mean–variance criterion

We consider a mean–variance portfolio selection problem with uncertain model parameters. We formulate the mean–variance problem under the α maxmin criterion, in which the investor has mixed ambiguity aversion and ambiguity seeking attitudes and solves a convex combination of max–min and max–max optimization problems. By the Lagrangian method, we obtain the efficient portfolio and quasi-efficient frontier in closed form. We provide comparative statics of the quasi-efficient frontier to various parameters.

Robust time-consistent mean–variance portfolio selection problem with multivariate stochastic volatility

This paper solves for the robust time-consistent mean–variance portfolio selection problem on multiple risky assets under a principle component stochastic volatility model. The model uncertainty is introduced to the drifts of the risky assets prices and the stochastic eigenvalues of the covariance matrix of asset returns. Using an extended dynamic programming approach, we manage to derive a semi-closed form solution of the desired portfolio via the solution to a coupled matrix Riccati equation. We provide the conditions, under which we prove the existence and the boundedness of the solution to the coupled matrix Riccati equation and derive the value function of the control problem. Moreover, we conduct numerical and empirical studies to perform sensitivity analyses and examine the losses due to ignoring model uncertainty or volatility information.

Mean–variance portfolio selection under a non-Markovian regime-switching model

In this paper, we investigate the mean–variance portfolio selection problem in a continuous-time setting. We assume that the coefficients in the model are random and adapted to the filtration generated by a Markov chain. Instead of using the embedding approach which is widely adopted in the existing literature, we study the problem from the viewpoint of mean-field formulation and provide a distinctive and straightforward approach. By introducing and discussing a new system of mean-field backward stochastic differential equations driven by a Markov chain, we obtain both the optimal strategy and the efficient frontier in explicit forms. In particular, we revisit the Markovian regime-switching model in which the coefficients are deterministic functions of the Markov chain.

Constrained LQ Problem with a Random Jump and Application to Portfolio Selection

This paper deals with a constrained stochastic linear-quadratic (LQ for short) optimal control problem where the control is constrained in a closed cone. The state process is governed by a controlled SDE with random coefficients. Moreover, there is a random jump of the state process. In mathematical finance, the random jump often represents the default of a counter party. Thanks to the Ito-Tanaka formula, optimal control and optimal value can be obtained by solutions of a system of backward stochastic differential equations (BSDEs for short). The solvability of the BSDEs is obtained by solving a recursive system of BSDEs driven by the Brownian motions. The author also applies the result to the mean variance portfolio selection problem in which the stock price can be affected by the default of a counterparty.

Portfolio selection in a regime switching market with a bankruptcy state and an uncertain exit-time: multi-period mean–variance formulation

This paper studies three versions of the multi-period mean–variance portfolio selection problem, that are: minimum variance problem, maximum expected return problem and the trade-off problem, in a Markovian regime switching market where the exit-time is uncertain and exogenous. The underlying Markov chain contains an absorbing state, which denotes the bankruptcy state. When the Markov chain switches to this state, the investors only get a random fraction, known as the recovery rate, taking values in [0, 1] of their wealth. Asset returns, as well as recovery rate, depend on the market state. Dynamic programming and Lagrange duality method are used to derive analytical expressions for optimal investment strategies and the mean–variance efficient frontier. It is shown that portfolio selection models with no bankruptcy state and certain exit-time can be considered as special cases of our model. Some numerical examples are provided to demonstrate the effect of the recovery rate and exit-probabilities.

Equilibrium Investment Strategy for a DC Plan With Partial Information and Mean–Variance Criterion

This paper studies a mean–variance portfolio selection problem with partial information for a defined-contribution (DC) pension scheme. We assume that the DC pension plan member can only observe the price of the risky asset but not the appreciation rate of it in the financial market. Moreover, inflation risk and salary risk are taken into account in our model. First, by virtue of the filtering theory, we transform the partially observable mean–variance portfolio selection problem to a completely observable one. Then, to look for an equilibrium investment strategy, we formulate and tackle the mean–variance problem within a game theoretic framework. By solving an extended Hamilton–Jacobi–Bellman (HJB) system of equations, closed-form expressions of the equilibrium investment strategy and corresponding equilibrium value function with partial information are derived for the DC pension plan. In addition, some numerical illustrations are provided to show the effects of parameters on the derived equilibrium investment strategies and the efficient frontier.

Explicit solutions for continuous time mean–variance portfolio selection with nonlinear wealth equations

This paper concerns the continuous time mean–variance portfolio selection problem with a special nonlinear wealth equation. This nonlinear wealth equation has a nonsmooth coefficient and the dual method developed in Ji (2010) does not work. We invoke the HJB equation of this problem and give an explicit viscosity solution of the HJB equation. Furthermore, via this explicit viscosity solution, we obtain explicitly the efficient portfolio strategy and efficient frontier for this problem. Finally, we show that our nonlinear wealth equation can cover three important cases.

A general characterization of the stochastic optimal combined control of mean field stochastic systems with application

In this paper, a general characterization of the optimal stochastic combined control for mean-field jump-systems is derived by applying mixed convex-spike perturbation method. The diffusion coefficient depends on the continuous control variable and the control domain is not necessary convex. In our combined mean-field control problem, we discuss two classes of jumps for the state processes, the inaccessible jumps which caused by Poisson martingale measure and the predictable ones which caused by the singularity of the control variable. Markowitz’s mean–variance portfolio selection problem with intervention control is discussed.

The Admissible Multiperiod Mean Variance Portfolio Selection Problem with Cardinality Constraints

Uncertain factors in finical markets make the prediction of future returns and risk of asset much difficult. In this paper, a model,assuming the admissible errors on expected returns and risks of assets, assisted in the multiperiod mean variance portfolio selection problem is built. The model considers transaction costs, upper bound on borrowing risk-free asset constraints, cardinality constraints and threshold constraints. Cardinality constraints limit the number of assets to be held in an efficient portfolio. At the same time, threshold constraints limit the amount of capital to be invested in each stock and prevent very small investments in any stock. Because of these limitations, the proposed model is a mix integer dynamic optimization problem with path dependence. The forward dynamic programming method is designed to obtain the optimal portfolio strategy. Finally, to evaluate the model, our result of a meaning example is compared to the terminal wealth under different constraints.

Mean–variance portfolio selection with regime switching under shorting prohibition

This paper investigates a mean–variance portfolio selection problem with regime switching under the constraint of short-selling being prohibited. By applying the dynamic programming approach, a system of Hamilton–Jacobi–Bellman (HJB) equations is constructed. Recognizing the features of the optimal wealth process, the optimal feedback control and verification theorem are obtained. The efficient portfolio and efficient frontier are explicitly derived through the Lagrange multiplier approach.