Portfolio Allocation Problem Research Articles

Multivariate Overnight GARCH-Ito Model with Applications in Large Volatility Matrix Estimation and Prediction

This paper introduces a unified multivariate overnight GARCH-It\^o model for volatility matrix estimation and prediction both in the low- and high-dimensional set-up. To account for whole-day market dynamics in the financial market, the proposed model has two different instantaneous volatility processes for the open-to-close and close-to-open periods, while each embeds the discrete-time multivariate GARCH model structure. We call it the multivariate overnight GARCH-It\^o (MOGI) model. Based on the connection between the discrete-time model structure and the continuous-time diffusion process, we propose a weighted least squares estimation procedure for estimating model parameters with open-to-close high-frequency and close-to-open low-frequency financial data, and establish its asymptotic theorems. We further discuss the prediction of future vast volatility matrices and study its asymptotic properties. A simulation study is conducted to check the finite sample performance of the proposed estimation and prediction methods. The empirical analysis is carried out to compare the performance of the proposed MOGI model with other benchmark methods in portfolio allocation problems.

Distributionally robust stochastic programs with side information based on trimmings

We consider stochastic programs conditional on some covariate information, where the only knowledge of the possible relationship between the uncertain parameters and the covariates is reduced to a finite data sample of their joint distribution. By exploiting the close link between the notion of trimmings of a probability measure and the partial mass transportation problem, we construct a data-driven Distributionally Robust Optimization (DRO) framework to hedge the decision against the intrinsic error in the process of inferring conditional information from limited joint data. We show that our approach is computationally as tractable as the standard (without side information) Wasserstein-metric-based DRO and enjoys performance guarantees. Furthermore, our DRO framework can be conveniently used to address data-driven decision-making problems under contaminated samples. Finally, the theoretical results are illustrated using a single-item newsvendor problem and a portfolio allocation problem with side information.

Portfolio Optimization Under Regime Switching and Transaction Costs: Combining Neural Networks and Dynamic Programs

The contributions of this paper are threefold. First, by combining dynamic programs and neural networks, we provide an efficient numerical method to solve a large multiperiod portfolio allocation problem under regime-switching market and transaction costs. Second, the performance of our combined method is shown to be close to optimal in a stylized case. To our knowledge, this is the first paper to carry out such a comparison. Last, the superiority of the combined method opens up the possibility for more research on financial applications of generic methods, such as neural networks, provided that solutions to simplified subproblems are available via traditional methods. The research on combining fast starts with neural networks began about four years ago. We observed that Professor Weinan E’s approach for solving systems of differential equations by neural networks had much improved performance when starting close to an optimal solution and could stall if the current iterate was far from an optimal solution. As we all know, this behavior is common with Newton- based algorithms. As a consequence, we discovered that combining a system of differential equations with a feedforward neural network could much improve overall computational performance. In this paper, we follow a similar direction for dynamic portfolio optimization within a regime-switching market with transaction costs. It investigates how to improve efficiency by combining dynamic programming with a recurrent neural network. Traditional methods face the curse of dimensionality. In contrast, the running time of our combined approach grows approximately linearly with the number of risky assets. It is inspiring to explore the possibilities of combined methods in financial management, believing a careful linkage of existing dynamic optimization algorithms and machine learning will be an active domain going forward. Relationship of the authors: Professor John M. Mulvey is Xiaoyue Li’s doctoral advisor.

Linear-Quadratic Stochastic Delayed Control and Deep Learning Resolution

We consider a simple class of stochastic control problems with a delayed control, in both the drift and the diffusion part of the state stochastic differential equation. We provide a new characterization of the solution in terms of a set of Riccati partial differential equations. Existence and uniqueness of a solution are obtained under a sufficient condition expressed directly as a relation between the time horizon, the drift, the volatility and the delay. Furthermore, a deep learning scheme (The code is available in a IPython notebook.) is designed and used to illustrate the effect of the delay feature on the Markowitz portfolio allocation problem with execution delay.

Portfolio optimization with a prescribed terminal wealth distribution

This paper studies a portfolio allocation problem, where the goal is to reach a prescribed wealth distribution at a final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which is solved with a gradient descent algorithm. This involves solving an associated Hamilton–Jacobi–Bellman and Fokker–Planck equations with a finite difference method. Numerical examples for various prescribed terminal distributions are given, showing that we can successfully reach attainable targets. We then consider adding consumption during the investment process, to take into account distributions that are either not attainable, or sub-optimal.

Multi-criteria group decision-making for portfolio allocation with consensus reaching process under interval type-2 fuzzy environment

The portfolio allocation problem is a hot topic in modern economics and management science. It involves many immeasurable criteria and uncertainties. Given the advantage of interval type-2 fuzzy sets (IT2FSs) in modeling complexities and uncertainties, this study provides an integrated methodology to deal with portfolio allocation problem based on the multiple criteria group decision-making (MCGDM) method considering the consensus reaching process in an interval type-2 fuzzy environment. First, some new information measures for IT2FSs include the weighted mean, and the weighted semi-absolute deviation measures are defined. They are used to describe the return and risk of the portfolio allocation, respectively. Second, a two-stage MCGDM framework is constructed for portfolio allocation: one is to achieve the group consensus, and the other is to obtain the optimal portfolio ratios. Finally, an example on calculating the optimal portfolio ratios for four newly listed stocks in China is offered to elaborate on the performance of our approach. Sensitivity and comparative analyses are also given to show the effectiveness and advantages of the proposed method.

First Passage Time Covariance Matrix Estimators

We devise a new high-frequency covariance matrix estimator based on price durations which is guaranteed to be positive-definite. Both non-parametric and parametric versions are proposed. A comprehensive Monte Carlo simulation shows that this class of estimators are less biased, more efficient, and generate lower RMSE as well as QLIKE errors. Empirically, we apply both estimators to a global minimum variance portfolio allocation problem and find they can generate comparably low portfolio variance, higher Sharpe ratios, but with considerably lower portfolio turnovers. This matrix estimator is also shown empirically to be more well-conditioned.

Economic Evaluation of Asset Pricing Models Under Predictability

This paper performs an out-of-sample comparison of linear factor asset pricing models from an economic perspective under predictability. I assess the economic value added of several factor models when a Bayesian investor is faced with a portfolio allocation problem whereby each model imposes cross-sectional restrictions on the parameters of a predictive stock return regression. The empirical framework explicitly accounts for investor skepticism about the model, i.e., mispricing uncertainty. Using several US portfolios as test assets, I find that the q5 model of Hou et al. (2020), as well as the behavioral factor models of Stambaugh and Yuan (2017) and Daniel et al. (2020) outperform competing models across investment horizons. At the longest evaluated horizon (one year), a benchmark portfolio built using historical data produces larger portfolio gains than all the factor models, but in the short run (at the one-month horizon), their performance is comparable.

Ambiguity with Machine Learning: An Application to Portfolio Choice

To characterize ambiguity we use machine learning to impose guidance and discipline on the formulation of expectations in a data-rich environment. In addition, we use the bootstrap to generate plausible synthetic samples of data not seen in historical real data to create statistics of interest pertaining to uncertainty. While our approach is generic we focus on robust portfolio allocation problems as an application and study the impact of risk versus uncertainty in a dynamic mean-variance setting. We show that a mean-variance optimizing investor achieves economically meaningful wealth gains (33%) across our sample from 1996-2019 by internalizing our uncertainty measure during portfolio formation.

Robust portfolio optimization: a categorized bibliographic review

Robust portfolio optimization refers to finding an asset allocation strategy whose behavior under the worst possible realizations of the uncertain inputs, e.g., returns and covariances, is optimized. The robust approach is in contrast to the classical approach, where one estimates the inputs to a portfolio allocation problem and then treats them as certain and accurate. In this paper we provide a categorized bibliography on the application of robust mathematical programming to the portfolio selection problem. With no similar surveys available, one of the aims of this review is to provide quick access for those interested, but maybe not yet in the area, so they know what the area is about, what has been accomplished and where everything can be found. Toward this end, a total of 148 references have been compiled and classified in various ways. Additionally, the number of Scopus© citations by contribution and journal is recorded. Finally, a brief discussion of the review’s major findings is provided and some solid leads on future directions are given.

Portfolio Selection Under Systemic Risk

This paper proposes a novel methodology to construct optimal portfolios that explicitly incorporates the occurrence of systemic events. Investors maximize a modified Sharpe ratio that is conditional on a systemic event, with the latter interpreted as a low market return environment. We solve the portfolio allocation problem analytically under the absence of short-selling restrictions and numerically when short-selling restrictions are imposed. This approach for obtaining an optimal portfolio allocation is made operational by embedding it in a multivariate dynamic setting using dynamic conditional correlation and copula models. We evaluate the out-of-sample performance of our portfolio empirically on the US stock market over the period 2007 to 2020 using ex-post wealth paths and systemic risk metrics against mean-variance, equally-weighted, and global minimum variance portfolios. Our portfolio maximizing a modified Sharpe ratio outperforms all competitors under market distress and remains competitive in non-crisis periods.

Portfolio Optimization With a Prescribed Terminal Wealth Distribution

This paper studies a portfolio allocation problem, where the goal is to prescribe the wealth distribution at final time. We study this problem with the tools of optimal mass transport. We provide a dual formulation which we solve by a gradient descent algorithm. This involves solving an associated HJB and Fokker–Planck equation by a finite difference method. Numerical examples for various prescribed terminal distributions are given, showing that we can successfully reach attainable targets. We next consider adding consumption during the investment process, to take into account distribution that either not attainable, or sub-optimal.

Solving quantum stochastic LQR optimal control problem in Fock space and its application in finance

This paper is an attempt for solving operator-valued quantum stochastic optimal control problems, in Fock space. For this purpose, the dynamics of the classical system state is described by Hudson–Parthasarathy type Quantum Stochastic Differential Equation (QSDE) in Fock space and then by associating a quadratic performance criterion with the QSDE, a Quantum Stochastic Linear Quadratic Regulator (QS-LQR) optimal control problem is formulated. Also, an algorithm for solving the QS-LQR optimal control problem is designed. For solving the resulting optimal control problem, a new HJB equation is obtained. Thereby, the operator valued control process is obtained. Two theorems are proved to facilitate the algorithm. In this paper, for the first time, the optimal strategy for trading stock is designed via the presented method. For this purpose, Merton portfolio allocation problem is solved. The simulation results show that portfolio optimal performances, minimum risk and maximum return are achieved via presented method.

Large Scale Continuous-Time Mean-Variance Portfolio Allocation via Reinforcement Learning

We propose to solve large scale Markowitz mean-variance (MV) portfolio allocation problem using reinforcement learning (RL). By adopting the recently developed continuous-time exploratory control framework, we formulate the exploratory MV problem in high dimensions. We further show the optimality of a multivariate Gaussian feedback policy, with time-decaying variance, in trading off exploration and exploitation. Based on a provable policy improvement theorem, we devise a scalable and data-efficient RL algorithm and conduct large scale empirical tests using data from the S&P 500 stocks. We found that our method consistently achieves over 10% annualized returns and it outperforms econometric methods and the deep RL method by large margins, for both long and medium terms of investment with monthly and daily trading.

Neural Stochastic Control Application : Optimal Portfolio Allocation

Neural Control (NC) is a growing field in stochastic optimal control applied to various dynamical systems such as quadcopter. NC is a non-parametric, learning based computational scheme, capable of handling high dimensional control problems. In this paper, “NC” principles have been applied to the problem of optimal investment portfolio allocation, which is commonly solved by the usage of covariance matrix in a Markowitz framework. Therefore, the portfolio quadratic optimization problem has been first re-cast into a generic stochastic control framework with non-linear dependencies and then solved using various Neural Networks (NNs). In NC context, literature on stacked Feed-Forward for time independent control has been recalled to formulate the properties of new NN architectures: Long Short-Term Memory (LSTM) and Attention LSTM for time-dependent control. Final section presents the numerical results of control weights convergence and dynamics using Monte Carlo simulation of portfolio asset prices (Geometric Brownian Motion) through various scenarios: first using stationary regime of volatility and correlation, second using non-stationary regime. Thus, behaviour of three NN architectures has been studied in those scenarios. Results suggest the further development of NC in domain of portfolio allocation as alternative of covariance matrix based solutions and related finance, economic supply and demand problems.

Portfolio allocation problems between risky and ambiguous assets

This paper considers a portfolio allocation problem between a risky asset and an ambiguous asset, and investigates how greater ambiguity aversion influences the optimal proportion invested in the two assets. We derive several sufficient conditions under which greater ambiguity aversion decreases the optimal proportion invested in the ambiguous asset. Furthermore, we consider an international diversification problem as an application and show that ambiguity aversion partially resolves the home bias puzzle.

Systemic Portfolio Diversification

We study the portfolio allocation problem of banks in the presence of fire-sale spillovers. We show that the joint portfolio selection problem may be framed as a potential game, and prove the existence of an equilibrium outcome. Our analysis suggests that while sacrificing individual diversification benefits to reduce portfolio commonality increases the probability that a sale occurs, it also lowers the probability of a costly widespread sell-off. We prove that if heterogeneity in leverage is higher, banks have stronger incentives to achieve systemic diversification. This in turn reduces the expected aggregate vulnerability of the system.

Properties of the Margrabe Best-of-two strategy to tactical asset allocation

The Margrabe Best-of-two (MBo2) strategy is a rule-based dynamic investment solution for the two-asset allocation problem. Its typical implementation involves yearly rebalancing the portfolio weights to 50–50 between a low-risk and high-risk asset. It uses intra-year weight adjustments to chase the momentum of the best performing asset by replicating the Margrabe formula for the value of a European option to exchange an asset for another asset at year-end. In practice, this means that the Margrabe portfolio allocation benefits from the upside potential of the high-risk asset and the downside protection from the low-risk asset. The MBo2 allocation depends on the assets' prices, their return volatilities, and correlation, as well as the remaining time until year-end. In this paper, we derive analytical formulae and use simulations to provide insights into the sensitivity of the strategy's weights and performance to these input parameters. We also report the results of an extensive out-of-sample evaluation for the MBo2 strategy applied to the bond–equity, real estate–equity, and world equity–emerging market equity portfolio allocation problems.

Optimal Investment with Correlated Stochastic Volatility Factors

The problem of portfolio allocation in the context of stocks evolving in random environments, that is with volatility and returns depending on random factors, has attracted a lot of attention. The problem of maximizing a power utility at a terminal time with only one random factor can be linearized thanks to a classical distortion transformation. In the present paper, we address the problem with several factors using a perturbation technique around the case where these factors are perfectly correlated reducing the problem to the case with a single factor. We illustrate our result with a particular model for which we have explicit formulas. A rigorous accuracy result is also derived using sub- and super-solutions of the HJB equation involved. In order to keep the notations as explicit as possible, we treat the case with one stock and two factors and we describe an extension to the case with two stocks and two factors.

Risk Sensitive Portfolio Optimization with Default Contagion and Regime-Switching

We study an open problem of risk-sensitive portfolio allocation in a regime-switching credit market with default contagion. The state space of the Markovian regime-switching process is assumed to b...