Robust portfolio selection involving options under a “ marginal+joint ” ellipsoidal uncertainty set

In typical robust portfolio selection problems, one mainly finds portfolios with the worst-case return under a given uncertainty set, in which asset returns can be realized. A too large uncertainty set will lead to a too conservative robust portfolio. However, if the given uncertainty set is not large enough, the realized returns of resulting portfolios will be outside of the uncertainty set when an extreme event such as market crash or a large shock of asset returns occurs. The goal of this paper is to propose robust portfolio selection models under so-called “ marginal+joint” ellipsoidal uncertainty set and to test the performance of the proposed models. A robust portfolio selection model under a “marginal + joint” ellipsoidal uncertainty set is proposed at first. The model has the advantages of models under the separable uncertainty set and the joint ellipsoidal uncertainty set, and relaxes the requirements on the uncertainty set. Then, one more robust portfolio selection model with option protection is presented by combining options into the proposed robust portfolio selection model. Convex programming approximations with second-order cone and linear matrix inequalities constraints to both models are derived. The proposed robust portfolio selection model with options can hedge risks and generates robust portfolios with well wealth growth rate when an extreme event occurs. Tests on real data of the Chinese stock market and simulated options confirm the property of both the models. Test results show that (1) under the “ marginal+joint” uncertainty set, the wealth growth rate and diversification of robust portfolios generated from the first proposed robust portfolio model (without options) are better and greater than those generated from Goldfarb and Iyengar’s model, and (2) the robust portfolio selection model with options outperforms the robust portfolio selection model without options when some extreme event occurs.     

证券投资组合优化问题的强健性的锥优化方法(英)

本文研究了具有强健性的证券投资组合优化问题.模型以最差条件在值风险为风险度量方法,并且考虑了交易费用对收益的影响.当投资组合的收益率概率分布不能准确确定但是在有界的区间内,尤其是在箱型区间结构和椭球区域结构内时,我们可以把具有强健性的证券投资组合优化问题的模型分别转化成线性规划和二阶锥规划形式.最后,我们用一个真实市场数据的算例来验证此方法.     

证券投资组合优化问题的强健性的锥优化方法(英文)

本文研究了具有强健性的证券投资组合优化问题.模型以最差条件在值风险为风险度量方法,并且考虑了交易费用对收益的影响.当投资组合的收益率概率分布不能准确确定但是在有界的区间内,尤其是在箱型区间结构和椭球区域结构内时,我们可以把具有强健性的证券投资组合优化问题的模型分别转化成线性规划和二阶锥规划形式.最后,我们用一个真实市场数据的算例来验证此方法.     

Robust multiobjective optimization & applications in portfolio optimization

Motivated by Markowitz portfolio optimization problems under uncertainty in the problem data, we consider general convex parametric multiobjective optimization problems under data uncertainty. For the first time, this uncertainty is treated by a robust multiobjective formulation in the gist of Ben-Tal and Nemirovski. For this novel formulation, we investigate its relationship to the original multiobjective formulation as well as to its scalarizations. Further, we provide a characterization of the location of the robust Pareto frontier with respect to the corresponding original Pareto frontier and show that standard techniques from multiobjective optimization can be employed to characterize this robust efficient frontier. We illustrate our results based on a standard mean–variance problem.     

Robust ranking and portfolio optimization

The portfolio optimization problem has attracted researchers from many disciplines to resolve the issue of poor out-of-sample performance due to estimation errors in the expected returns. A practical method for portfolio construction is to use assets’ ordering information, expressed in the form of preferences over the stocks, instead of the exact expected returns. Due to the fact that the ranking itself is often described with uncertainty, we introduce a generic robust ranking model and apply it to portfolio optimization. In this problem, there are n objects whose ranking is in a discrete uncertainty set. We want to find a weight vector that maximizes some generic objective function for the worst realization of the ranking. This robust ranking problem is a mixed integer minimax problem and is very difficult to solve in general. To solve this robust ranking problem, we apply the constraint generation method, where constraints are efficiently generated by solving a network flow problem. For empirical tests, we use post-earnings-announcement drifts to obtain ranking uncertainty sets for the stocks in the DJIA index. We demonstrate that our robust portfolios produce smaller risk compared to their non-robust counterparts.     

Worst-Case Violation of Sampled Convex Programs for Optimization with Uncertainty

A deterministic approach called robust optimization has been recently proposed to deal with optimization problems including inexact data, i.e., uncertainty. The basic idea of robust optimization is to seek a solution that is guaranteed to perform well in terms of feasibility and near-optimality for all possible realizations of the uncertain input data. To solve robust optimization problems, Calafiore and Campi have proposed a randomized approach based on sampling of constraints, where the number of samples is determined so that only a small portion of the original constraints is violated by the randomized solution. Our main concern is not only the probability of violation, but also the degree of violation, i.e., the worst-case violation. We derive an upper bound of the worst-case violation for the sampled convex programs and consider the relation between the probability of violation and the worst-case violation. The probability of violation and the degree of violation are simultaneously bounded by a prescribed value when the number of random samples is large enough. In addition, a confidence interval of the optimal value is obtained when the objective function includes uncertainty. Our method is applicable to not only a bounded uncertainty set but also an unbounded one. Hence, the scope of our method includes random sampling following an unbounded distribution such as the normal distribution.     

基于鲁棒二阶随机占优的投资组合优化模型研究

本文主要考虑一类经典的含有二阶随机占优约束的投资组合优化问题,其目标为最大化期望收益,同时利用二阶随机占优约束度量风险,满足期望收益二阶随机占优预定的参考目标收益。与传统的二阶随机占优投资组合优化模型不同,本文考虑不确定的投资收益率,并未知其精确的概率分布,但属于某一不确定集合,建立鲁棒二阶随机占优投资组合优化模型,借助鲁棒优化理论,推导出对应的鲁棒等价问题。最后,采用S&P 500股票市场的实际数据,对模型进行不同训练样本规模和不确定集合下的最优投资组合的权重、样本内和样本外不确定参数对期望收益的影响的分析。结果表明,投资收益率在最新的历史数据规模下得出的投资策略,能够获得较高的样本外期望收益,对未来投资更具参考意义。在保证样本内解的最优性的同时,也能取得较高的样本外期望收益和随机占优约束被满足的可行性。     

Regularized robust optimization: the optimal portfolio execution case

An uncertainty set is a crucial component in robust optimization. Unfortunately, it is often unclear how to specify it precisely. Thus it is important to study sensitivity of the robust solution to variations in the uncertainty set, and to develop a method which improves stability of the robust solution. In this paper, to address these issues, we focus on uncertainty in the price impact parameters in an optimal portfolio execution problem. We first illustrate that a small variation in the uncertainty set may result in a large change in the robust solution. We then propose a regularized robust optimization formulation which yields a solution with a better stability property than the classical robust solution. In this approach, the uncertainty set is regularized through a regularization constraint, defined by a linear matrix inequality using the Hessian of the objective function and a regularization parameter. The regularized robust solution is then more stable with respect to variation in the uncertainty set specification, in addition to being more robust to estimation errors in the price impact parameters. The regularized robust optimal execution strategy can be computed by an efficient method based on convex optimization. Improvement in the stability of the robust solution is analyzed. We also study implications of the regularization on the optimal execution strategy and its corresponding execution cost. Through the regularization parameter, one can adjust the level of conservatism of the robust solution.     

Robust portfolio optimization with derivative insurance guarantees

Robust portfolio optimization aims to maximize the worst-case portfolio return given that the asset returns are allowed to vary within a prescribed uncertainty set. If the uncertainty set is not too large, the resulting portfolio performs well under normal market conditions. However, its performance may substantially degrade in the presence of market crashes, that is, if the asset returns materialize far outside of the uncertainty set. We propose a novel robust optimization model for designing portfolios that include European-style options. This model trades off weak and strong guarantees on the worst-case portfolio return. The weak guarantee applies as long as the asset returns are realized within the prescribed uncertainty set, while the strong guarantee applies for all possible asset returns. The resulting model constitutes a convex second-order cone program, which is amenable to efficient numerical solution procedures. We evaluate the model using simulated and empirical backtests and analyze the impact of the insurance guarantees on the portfolio performance.     

期望和残差收益估计不可靠的鲁棒模型

投资优化问题的最优策略会随着输入参数的扰动而出现敏感的变化,针对投资优化问题中出现的随机变量的参数估计不可靠的情况,本文引入不确定集合描述随机收益的有关矩信息,提出了投资优化问题的一个鲁棒性模型,并采用数学规划的理论和方法,给出了该模型的最优策略和有效前沿的解析表示。本方法能够为采用保守策略的、对不确定性厌恶的投资者提供一种最优的投资策略。     

On robust mean-variance portfolios

We derive closed-form portfolio rules for robust mean–variance portfolio optimization where the return vector is uncertain or the mean return vector is subject to estimation errors, both uncertainties being confined to an ellipsoidal uncertainty set. We consider different mean–variance formulations allowing short sales, and derive closed-form optimal portfolio rules in static and dynamic settings.     

Robustness of optimal portfolios under risk and stochastic dominance constraints

Solutions of portfolio optimization problems are often influenced by a model misspecification or by errors due to approximation, estimation and incomplete information. The obtained results, recommendations for the risk and portfolio manager, should be then carefully analyzed. We shall deal with output analysis and stress testing with respect to uncertainty or perturbations of input data for static risk constrained portfolio optimization problems by means of the contamination technique. Dependence of the set of feasible solutions on the probability distribution rules out the straightforward construction of convexity-based global contamination bounds. Results obtained in our paper [Dupa?ová, J., & Kopa, M. (2012). Robustness in stochastic programs with risk constraints. Annals of Operations Research, 200, 55–74.] were derived for the risk and second order stochastic dominance constraints under suitable smoothness and/or convexity assumptions that are fulfilled, e.g. for the Markowitz mean–variance model. In this paper we relax these assumptions having in mind the first order stochastic dominance and probabilistic risk constraints. Local bounds for problems of a special structure are obtained. Under suitable conditions on the structure of the problem and for discrete distributions we shall exploit the contamination technique to derive a new robust first order stochastic dominance portfolio efficiency test.     

A note on issues of over-conservatism in robust optimization with cost uncertainty

We identify and discuss issues of hidden over-conservatism in robust linear optimization, when the uncertainty set is polyhedral with a budget of uncertainty constraint. The decision-maker selects the budget of uncertainty to reflect his degree of risk aversion, i.e. the maximum number of uncertain parameters that can take their worst-case value. In the first setting, the cost coefficients of the linear programming problem are uncertain, as is the case in portfolio management with random stock returns. We provide an example where, for moderate values of the budget, the optimal solution becomes independent of the nominal values of the parameters, i.e. is completely disconnected from its nominal counterpart, and discuss why this happens. The second setting focusses on linear optimization with uncertain upper bounds on the decision variables, which has applications in revenue management with uncertain demand and can be rewritten as a piecewise linear problem with cost uncertainty. We show in an example that it is possible to have more demand parameters equal their worst-case value than what is allowed by the budget of uncertainty, although the robust formulation is correct. We explain this apparent paradox.     

Robust option pricing

In this paper, we combine robust optimization and the idea of ?$?$-arbitrage to propose a tractable approach to price a wide variety of options. Rather than assuming a probabilistic model for the stock price dynamics, we assume that the conclusions of probability theory, such as the central limit theorem, hold deterministically on the underlying returns. This gives rise to an uncertainty set that the underlying asset returns satisfy. We then formulate the option pricing problem as a robust optimization problem that identifies the portfolio which minimizes the worst case replication error for a given uncertainty set defined on the underlying asset returns. The most significant benefits of our approach are (a) computational tractability illustrated by our ability to price multi-asset, American and Asian options using linear optimization; and thus the computational complexity of our approach scales polynomially with the number of assets and with time to expiry and (b) modeling flexibility illustrated by our ability to model different kinds of options, various levels of risk aversion among investors, transaction costs, shorting constraints and replication via option portfolios.     

A Midpoint-Radius approach to regression with interval data

In this paper, a revisited interval approach for linear regression is proposed. In this context, according to the Midpoint-Radius (MR) representation, the uncertainty attached to the set-valued model can be decoupled from its trend. The estimated interval model is built from interval input-output data with the objective of covering all available data. The constrained optimization problem is addressed using a linear programming approach in which a new criterion is proposed for representing the global uncertainty of the interval model. The potential of the proposed method is illustrated by simulation examples.     

International portfolio management with affine policies

While dynamic decision making has traditionally been represented as scenario trees, these may become severely intractable and difficult to compute with an increasing number of time periods. We present an alternative tractable approach to multiperiod international portfolio optimization based on an affine dependence between the decision variables and the past returns. Because local asset and currency returns are modeled separately, the original model is non-linear and non-convex. With the aid of robust optimization techniques, however, we develop a tractable semidefinite programming formulation of our model, where the uncertain returns are contained in an ellipsoidal uncertainty set. We add to our formulation the minimization of the worst case value-at-risk and show the close relationship with robust optimization. Numerical results demonstrate the potential gains from considering a dynamic multiperiod setting relative to a single stage approach.     

Robust international portfolio management

We present an international portfolio optimization model where we take into account the two different sources of return of an international asset: the local returns denominated in the local currency, and the returns on the foreign exchange rates. The explicit consideration of the returns on exchange rates introduces non-linearities in the model, both in the objective function (return maximization) and in the triangulation requirement of the foreign exchange rates. The uncertainty associated with both types of returns is incorporated directly in the model by the use of robust optimization techniques. We show that, by using appropriate assumptions regarding the formulation of the uncertainty sets, the proposed model has a semidefinite programming formulation and can be solved efficiently. While robust optimization provides a guaranteed minimum return inside the uncertainty set considered, we also discuss an extension of our formulation with additional guarantees through trading in quanto options for the foreign assets and in equity options for the domestic assets.     

Robust optimization and portfolio selection: The cost of robustness

Robust optimization is a tractable alternative to stochastic programming particularly suited for problems in which parameter values are unknown, variable and their distributions are uncertain. We evaluate the cost of robustness for the robust counterpart to the maximum return portfolio optimization problem. The uncertainty of asset returns is modelled by polyhedral uncertainty sets as opposed to the earlier proposed ellipsoidal sets. We derive the robust model from a min-regret perspective and examine the properties of robust models with respect to portfolio composition. We investigate the effect of different definitions of the bounds on the uncertainty sets and show that robust models yield well diversified portfolios, in terms of the number of assets and asset weights.     

Robust portfolio optimization with a hybrid heuristic algorithm

Estimation errors in both the expected returns and the covariance matrix hamper the construction of reliable portfolios within the Markowitz framework. Robust techniques that incorporate the uncertainty about the unknown parameters are suggested in the literature. We propose a modification as well as an extension of such a technique and compare both with another robust approach. In order to eliminate oversimplifications of Markowitz’ portfolio theory, we generalize the optimization framework to better emulate a more realistic investment environment. Because the adjusted optimization problem is no longer solvable with standard algorithms, we employ a hybrid heuristic to tackle this problem. Our empirical analysis is conducted with a moving time window for returns of the German stock index DAX100. The results of all three robust approaches yield more stable portfolio compositions than those of the original Markowitz framework. Moreover, the out-of-sample risk of the robust approaches is lower and less volatile while their returns are not necessarily smaller.