Two stochastic dominance criteria based on tail comparisons

Actuarial risks and financial asset returns are typically heavy tailed. In this paper, we introduce 2 stochastic dominance criteria, called the right‐tail order and the left‐tail order, to compare these variables stochastically. The criteria are based on comparisons of expected utilities, for 2 classes of utility functions that give more weight to the right or the left tail (depending on the context) of the distributions. We study their properties, applications, and connections with other classical criteria, including the increasing convex and the second‐order stochastic dominance. Finally, we rank some parametric families of distributions and provide empirical evidence of the new stochastic dominance criteria with an example using real data.     

Option pricing bounds with standard risk aversion preferences

For a theoretical valuation of a financial option, various models have been proposed that require specific hypotheses regarding both the stochastic process driving the price behaviour of the underlying security and market efficiency. When some of these assumptions are removed, we obtain an uncertainty interval for the option price. Up to now, the most restrictive intervals for option prices have been obtained using the decreasing absolute risk aversion (DARA) rule in a state-preference approach. Precautionary saving entails the concept of prudence; in particular, decreasing absolute prudence is a necessary and sufficient condition that guarantees that the saving of wealthier people is less sensitive to the risk associated to future incomes. If this condition is coupled with the DARA assumption we obtain standard risk aversion (SRA), which guarantees on the one hand that introducing a zero-mean background risk to wealth makes people less willing to accept another independent risk and on the other hand that an increase in the risk of the returns distribution of an asset reduces the demand for this asset. The main idea of this contribution is to apply decreasing absolute prudence and SRA rules in a state-preference context in order to obtain efficient bounds for the value of European-style options portfolio strategies. Lower and upper bounds for the options portfolio value are obtained by solving non-linear optimization problems. The numerical experiments carried out show the efficiency of the technique proposed.     

Characterizations on Almost Stochastic Dominance Revisited

Almost stochastic dominance has been receiving more attention in the financial and economic literature. In this short note, we characterize the almost first- and second-degree stochastic dominance by requiring one distribution to be ``close to' a new distribution that dominates or is dominated by another distribution in the traditional sense of the first- and second-order stochastic dominance, respectively. We also investigate the concept of almost stochastic dominance for unbounded random variables.     

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??Almost stochastic dominance has been receiving more attention in the financial and economic literature. In this short note, we characterize the almost first- and second-degree stochastic dominance by requiring one distribution to be ``close to' a new distribution that dominates or is dominated by another distribution in the traditional sense of the first- and second-order stochastic dominance, respectively. We also investigate the concept of almost stochastic dominance for unbounded random variables.     

两类随机控制风险秩序的等价性分析

本文首先对比分析了两类风险秩序:随机控制秩序与对偶随机控制秩序.得到并证明了下述命题:(1)效用自由秩序等价于随机控制秩序;(2)畸变自由秩序等价于对偶随机控制秩序;(3)第一、第二阶随机控制秩序等价于第一、第二阶的对偶随机控制秩序,但对高于三阶的情况由实例说明不一定成立.     

Stochastic dominance of portfolio insurance strategies

The purpose of this article is to analyze and compare two standard portfolio insurance methods: Option-based Portfolio Insurance (OBPI) and Constant Proportion Portfolio Insurance (CPPI). Various stochastic dominance criteria up to third order are considered. We derive parameter conditions implying the second- and third-order stochastic dominance of the CPPI strategy. In particular, restrictions on the CPPI multiplier resulting from the spread between the implied volatility and the empirical volatility are analyzed.     

基于随机参照点的均值-风险分析

本文依据参照依赖偏好模型提出了基于随机参照点的风险度量方法,进而构建了均值-风险模型,并讨论了该决策方法与随机占优之间的一致性。研究发现,该决策方法不仅与一级随机占优是一致的而且与二级随机占优也是一致的。由于二级随机占优与期望效用理论的一致性,因而所构建的均值-风险模型与期望效用理论也是一致的。     

Asymptotic Multivariate Dominance: A Financial Application

We propose a multivariate stochastic dominance relation aimed at ranking different financial markets/sectors from the point of view of a non-satiable risk averse investor. In particular, we assume that the vector of returns of a given market is in the domain of attraction of a symmetric stable Paretian law in order to take into account the asymptotic behaviour of the financial returns. We determine the stochastic dominance rule for stable symmetric distributions, where the stability parameter plays a crucial role. Consequently, the multivariate rule for ordering markets is based on a comparison between i) location parameters, ii) dispersion parameters, and iii) stability indices. Finally, we apply the method to the equity markets of the four countries with the highest gross domestic product in 2013, namely, the US, China, Japan and Germany. In this empirical comparison we examine the ex ante and ex post dominance between stock markets, either assuming that the returns are jointly (or conditionally, for a robust approach) Gaussian distributed, or in the domain of attraction of a stable sub-Gaussian law.     

General linear formulations of stochastic dominance criteria

We develop and implement linear formulations of general Nth order stochastic dominance criteria for discrete probability distributions. Our approach is based on a piece-wise polynomial representation of utility and its derivatives and can be implemented by solving a relatively small system of linear inequalities. This approach allows for comparing a given prospect with a discrete set of alternative prospects as well as for comparison with a polyhedral set of linear combinations of prospects. We also derive a linear dual formulation in terms of lower partial moments and co-lower partial moments. An empirical application to historical stock market data suggests that the passive stock market portfolio is highly inefficient relative to actively managed portfolios for all investment horizons and for nearly all investors. The results also illustrate that the mean–variance rule and second-order stochastic dominance rule may not detect market portfolio inefficiency because of non-trivial violations of non-satiation and prudence.     

Profit criteria involving risk in price setting of virtual products

This work deals with pricing of “virtual” products, i.e., products that a retailer can supply after demand has been realized. Such products allow the retailer to avoid holding costs and ensure timely fulfillment of demand with no risk of shortage. Demand is commonly price-dependent and uncertain, and we seek to maximize each of three criteria: expected profit, the likelihood of achieving a profit target, and the profit for a given percentile. Simultaneous multiple criteria are also explored. Two forms of demand uncertainty are considered in the analysis: the multiplicative form, where, due to stochastic dominance, all the investigated profit criteria—and, in fact, any utility function of the profit—can be optimized simultaneously; and the additive form, where stochastic dominance cannot occur. Under the multiplicative form of demand, the property of stochastic dominance is shown to hold in a two-echelon supply chain (comprising both the supplier and the retailer) and in a centralized system.     

Stochastic dominance: A bibliographical rectification and a restatement of whitmore's theorem

The aim of this note is to make clear that the fundamental theorem on stochastic dominance of degree two was discovered in 1932 by Karamata. Apart from this historical rectification, it is interesting to remark that the proof is simple and elegant and can be used to characterize stochastic dominance of any order. We give a simple proof along these lines of Whitmore's theorem about third degree stochastic dominance.     

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.     

Relaxations of linear programming problems with first order stochastic dominance constraints

Linear stochastic programming problems with first order stochastic dominance (FSD) constraints are non-convex. For their mixed 0-1 linear programming formulation we present two convex relaxations based on second order stochastic dominance (SSD). We develop necessary and sufficient conditions for FSD, used to obtain a disjunctive programming formulation and to strengthen one of the SSD-based relaxations.     

INSDECM—an interactive procedure for stochastic multicriteria decision problems

A new interactive technique for a discrete stochastic multiattribute decision making problem is proposed in this paper. It is assumed that performance probability distribution for each action on each attribute is known. Two concepts are combined in the procedure: stochastic dominance and interactive approach. The first one is employed for generating efficient actions and constructing rankings of actions with respect to attributes. The second concept is used when the communication between the DM and the model is conducted. It is assumed that decision maker’s restrictions are defined by specifying minimal or maximal values of scalar criteria measuring either expected outcome or variability of outcomes. As such restrictions are, in general, not consistent with stochastic dominance rules, we suggest verifying this consistency and asking the decision maker to redefine inconsistent restrictions.     

Optimization with multivariate stochastic dominance constraints

We consider stochastic optimization problems where risk-aversion is expressed by a stochastic ordering constraint. The constraint requires that a random vector depending on our decisions stochastically dominates a given benchmark random vector. We identify a suitable multivariate stochastic order and describe its generator in terms of von Neumann–Morgenstern utility functions. We develop necessary and sufficient conditions of optimality and duality relations for optimization problems with this constraint. Assuming convexity we show that the Lagrange multipliers corresponding to dominance constraints are elements of the generator of this order, thus refining and generalizing earlier results for optimization under univariate stochastic dominance constraints. Furthermore, we obtain necessary conditions of optimality for non-convex problems under additional smoothness assumptions.     

From stochastic dominance to mean-risk models: Semideviations as risk measures

Two methods are frequently used for modeling the choice among uncertain outcomes: stochastic dominance and mean-risk approaches. The former is based on an axiomatic model of risk-averse preferences but does not provide a convenient computational recipe. The latter quantifies the problem in a lucid form of two criteria with possible trade-off analysis, but cannot model all risk-averse preferences. In particular, if variance is used as a measure of risk, the resulting mean–variance (Markowitz) model is, in general, not consistent with stochastic dominance rules. This paper shows that the standard semideviation (square root of the semivariance) as the risk measure makes the mean-risk model consistent with the second degree stochastic dominance, provided that the trade-off coefficient is bounded by a certain constant. Similar results are obtained for the absolute semideviation, and for the absolute and standard deviations in the case of symmetric or bounded distributions. In the analysis we use a new tool, the Outcome–Risk (O–R) diagram, which appears to be particularly useful for comparing uncertain outcomes.     

不确定性对混合CVaR约束库存系统的影响

运用应用概率中的随机占优研究需求不确定性对混合CVaR约束库存系统最优订购量和最优利润的影响。引入刻画决策者风险态度的“风险偏好系数”,得到系统最优订购量和最优利润关于风险偏好系数的单调性。研究表明随机大需求总会导致系统较高的最优订购量和最优利润;在割准则序意义下,最优订购量可能随需求可变性的增加而增加也可能随需求可变性的增加而减少;在二阶随机占优且风险偏好系数大于等于1的情况下系统最优利润具有随机单调性,然而当风险偏好系数小于1时最优利润在二阶随机占优意义下的结论不一定成立,我们通过一个数值例子来说明。     

Valid inequalities and restrictions for stochastic programming problems with first order stochastic dominance constraints

Stochastic dominance relations are well studied in statistics, decision theory and economics. Recently, there has been significant interest in introducing dominance relations into stochastic optimization problems as constraints. In the discrete case, stochastic optimization models involving second order stochastic dominance constraints can be solved by linear programming. However, problems involving first order stochastic dominance constraints are potentially hard due to the non-convexity of the associated feasible regions. In this paper we consider a mixed 0–1 linear programming formulation of a discrete first order constrained optimization model and present a relaxation based on second order constraints. We derive some valid inequalities and restrictions by employing the probabilistic structure of the problem. We also generate cuts that are valid inequalities for the disjunctive relaxations arising from the underlying combinatorial structure of the problem by applying the lift-and-project procedure. We describe three heuristic algorithms to construct feasible solutions, based on conditional second order constraints, variable fixing, and conditional value at risk. Finally, we present numerical results for several instances of a real world portfolio optimization problem. This research was supported by the NSF awards DMS-0603728 and DMI-0354678.     

Risk aversion for an electricity retailer with second-order stochastic dominance constraints

In this paper we present the problem faced by an electricity retailer which searches to determine its forward contracting portfolio and the selling prices for its potential clients. This problem is formulated as a two-stage stochastic program including second-order stochastic dominance constraints. The stochastic dominance theory is used in order to reduce the risk suffering from low profits. The resulting deterministic equivalent problem is a mixed-integer linear program which is solved using commercial branch-and-cut software. Numerical results for a realistic case study are reported and relevant conclusions are drawn.     

Portfolio construction based on stochastic dominance and target return distributions

Mean-risk models have been widely used in portfolio optimization. However, such models may produce portfolios that are dominated with respect to second order stochastic dominance and therefore not optimal for rational and risk-averse investors. This paper considers the problem of constructing a portfolio which is non-dominated with respect to second order stochastic dominance and whose return distribution has specified desirable properties. The problem is multi-objective and is transformed into a single objective problem by using the reference point method, in which target levels, known as aspiration points, are specified for the objective functions. A model is proposed in which the aspiration points relate to ordered outcomes for the portfolio return. This concept is extended by additionally specifying reservation points, which act pre-emptively in the optimization model. The theoretical properties of the models are studied. The performance of the models on real data drawn from the Hang Seng index is also investigated.