Multi-period minimax hedging strategies

Option pricing and hedging under transaction costs are of major importance to marketmakers and investors. In this paper we present the basic minimax strategy which determines the optimum number of shares that minimizes the worst-case potential hedging error under transaction costs for the next period. We present two extensions of this strategy. The first extension is the two-period minimax where the worst case is defined over a two-period setting. The objective function of the basic minimax strategy is augmented to include the hedging error for the second period. The second extension is the variable minimax strategy where early rebalancing is triggered by the minimax hedging error. Simulation results suggest that the basic minimax strategy and its two extensions are superior in performance to delta hedging and that the variable minimax strategy is superior to both the basic and the two-period strategies. This result is due to the opportunity provided by the variable minimax strategy to rebalance early. The greatest amount of business for traded options is done for at-the-money options; in this paper, we have concluded that the minimax strategies are particularly suitable for managing the risk of such options. In the Appendix, we present the minimax algorithm used for the implementation of these strategies.     

Risk management strategies via minimax portfolio optimization

Recent extreme economic developments nearing a worst-case scenario motivate further examination of minimax linear programming approaches for portfolio optimization. Risk measured as the worst-case return is employed and a portfolio from maximizing returns subject to a risk threshold is constructed. Minimax model properties are developed and parametric analysis of the risk threshold connects this model to expected value along a continuum, revealing an efficient frontier segmenting investors by risk preference. Divergence of minimax model results from expected value is quantified and a set of possible prior distributions expressing a degree of Knightian uncertainty corresponding to risk preference determined. The minimax model will maximize return with respect to one of these prior distributions providing valuable insight regarding an investor’s risk attitude and decision behavior. Linear programming models for financial firms to assist individual investors to hedge against losses by buying insurance and a model for designing variable annuities are proposed.     

一类无约束离散Minimax问题的区间调节熵算法

In this paper,a class of unconstrained discrete minimax problems is described,in which the objective functions are in C^1. The paper deals with this problem by means of taking the place of maximum-entropy function with adjustable entropy function. By constructing an interval extension of adjustable entropy function and some region deletion test rules, a new interval algorithm is presented. The relevant properties are proven, The minimax value and the localization of the minimax points of the problem can be obtained by this method. This method can overcome the flow problem in the maximum-entropy algorithm. Both theoretical and numerical results show that the method is reliable and efficient.     

定数截尾试验下两参数Weibull分布尺度参数的最优稳健Bayes估计

以Г-后验期望损失作为标准,研究了定数截尾试验下两参数W e ibu ll分布尺度参数θ的最优稳健Bayes估计问题.假设尺度参数θ的先验分布在分布族Г上变化,形状参数β已知时,在0-1损失下,得到了θ的最优稳健区间估计,在均方损失下得到θ的最优稳健点估计及区间估计;β未知时,得到了θ的最优稳健点估计及区间估计.最后给出了数值例子,说明了方法的有效性.     

Stein's phenomenon in estimation of means restricted to a polyhedral convex cone

This paper treats the problem of estimating the restricted means of normal distributions with a known variance, where the means are restricted to a polyhedral convex cone which includes various restrictions such as positive orthant, simple order, tree order and umbrella order restrictions. In the context of the simultaneous estimation of the restricted means, it is of great interest to investigate decision-theoretic properties of the generalized Bayes estimator against the uniform prior distribution over the polyhedral convex cone. In this paper, the generalized Bayes estimator is shown to be minimax. It is also proved that it is admissible in the one- or two-dimensional case, but is improved on by a shrinkage estimator in the three- or more-dimensional case. This means that the so-called Stein phenomenon on the minimax generalized Bayes estimator can be extended to the case where the means are restricted to the polyhedral convex cone. The risk behaviors of the estimators are investigated through Monte Carlo simulation, and it is revealed that the shrinkage estimator has a substantial risk reduction.     

Properties of the optimal trajectories for coplanar,aeroassisted orbital transfer

We show that the sequences of function values constructed by two versions of a minimax algorithm converge linearly to the minimum values. Both versions use the Pshenichnyi-Pironneau-Polak search direction subprocedure; the first uses an exact line search to determine the stepsize, while the second one uses an Armijo-type stepsize rule. The proofs depend on a second-order sufficiency condition, but not on strict complementary slackness. Minimax problems in which each function appearing in the max is a composition of a twice continuously differentiable function with a linear function typically do not satisfy second-order sufficiency conditions. Nevertheless, we show that, on such minimax problems, the two algorithms do converge linearly when the outer functions are convex and strict complementary slackness holds at the solutions.The research reported herein was sponsored in part by the National Science Foundation Grant ECS-87-13334, the Air Force Office of Scientific Research Contract AFOSR-86-0116, the State of California MICRO Program Grant 532410-19900, and a Howard Hughes Doctoral Fellowship (Hughes Aircraft Company). The authors would like to thank the referees for their helpful suggestions.     

Some topological properties of the solution sets of parametrized minimax problems

Sufficient conditions are given to guarantee the closedness and uniform boundedness of the solution sets corresponding to a pair of dual parametrized minimax problems. The parameter of the minimax problems belongs to a metrizable space and affects not only the objective function, but also the feasible sets.This research was supported in part by Fondo Nacional de Ciencias, Proyecto No. 01273.     

基于贝叶斯信息更新的失事飞机发现概率模型

了目标搜索区域的确定方法以及失事飞机在目标搜索区域的初始概率分布,得到发现概率的计算公式。以发现概率为目标,构造了一个求解最优搜寻策略的Max Max化规划模型,模型可以动基于贝叶斯方法,提出了一个失事飞机的发现概率模型,利用飞机失联前后的信息数据,给出态地对坠机点的概率分布进行更新,使下一步搜寻任务得到及时的修正和调整。考虑到洋流对坠机点的影响,本文还提出了一个关于基点先验概率分布的重构策略。此外,对任务搜索区域最优路径的选取问题做了进一步探讨,给出了一个任务搜索区域上搜寻路径的选取方法     

线性等式约束连续型minimax问题的区间算法

研究线性等式约束连续型minimax问题,其中目标函数为Lipschitz连续函数,基于线性约束函数的区间迭代运算、区域二分原则和无解区域删除原则,建立了求解线性等式约束连续型minimax问题的区间算法,证明了算法的相关定理,给出了数值算例,该算法保证求出问题的整体解,且是可靠和有效的.     

A gamma-minimax result in credibility theory

The statistical decision problem of experience ratemaking has been described by Bühlmann (1975) as a two-person game of the actuary (player 2) against (malevolent) nature (player 1). Within this framework premium calculation procedures are strategies of player 2 and some types of credibility formulae can be regarded as Bayesian strategies or in terms of statistical decision theory as (linear) Bayes estimators. Since the application of the Bayes principle of game theory to insurance ratemaking is not appropriate — usually the actuary has not enough information for identifying one single prior — it is quite natural to select a premium calculation procedure which is optimal according to the minimax principle of game theory or according to the so-called gamma-minimax principle. This principle is more adaptable than the minimax principle since it allows to take into account vague prior information. In this paper credibility formulae are derived which are gamma-minimax for special types of prior information similar to those in Bühlmann's paper.     

LINEX损失函数下位置参数函数的极小极大估计

主要研究了在LINEX损失函数下位置参数函数的极小极大估计,为了给出它的极小极大估计存在的一个充分条件,将位置参数θ限定在一个有界区间上,并且当其函数h(θ)满足一定条件时,h(θ)的极小极大估计是存在的,并给出了证明.     

A Kalman filter as a minimax estimator

A minimax terminal state estimation problem is posed for a linear plant and a generalized quadratic loss function. Sufficient conditions are developed to insure that a Kalman filter will provide a minimax estimate for the terminal state of the plant. It is further shown that this Kalman filter will not generally be a minimax estimate for the terminal state if the observation interval is arbitrarily long. Consequently, a subminimax estimate is defined, subject to a particular existence condition. This subminimax estimate is related to the Kalman filter, and it may provide a useful estimate for the terminal state when the performance of the Kalman filter is no longer satisfactory.     

限制参数空间时参数的 minimax 估计

一、引言在实际问题中,我们有时事先就对参数有一些认识,因此有必要考虑参数空间被限制时参数的估计问题.但令人吃惊的是,这给理论上的讨论带来了许多困难,尤其是求参数的 minimax 估计等问题,迄今结果不多.     

Asymptotic minimax estimation in semiparametric models

Summary We give several conditions on the estimator of efficient score function for estimating the parametric component of semiparametric models. A semiparametric version of the one-step MLE using an estimator of efficient score function which fulfills the conditions is shown to converge to the normal distribution with minimum variance locally uniformly over a fairly large neighborhood around the assumed semiparametric model. Consequently, it is shown to be asymptotically minimax with bounded subconvex loss functions. A few examples are also considered.     

A minimax search plan for constrained optimization problems

A minimax search strategy is described for locating the boundary point of a region on a line joining a feasible point to an infeasible point. Asymptotic strategies, useful when the number of experiments to be used in the search is not predetermined, are also given. These strategies are useful subroutines for many multidimensional optimization algorithms.The authors thank G. V. Reklaitis for initial discussions concerning this problem. John H. Beamer was an NSF Graduate Fellow at the time when this research was conducted.     

Estimation, prediction and the Stein phenomenon under divergence loss

We consider two problems: (1) estimate a normal mean under a general divergence loss introduced in [S. Amari, Differential geometry of curved exponential families — curvatures and information loss, Ann. Statist. 10 (1982) 357-387] and [N. Cressie, T.R.C. Read, Multinomial goodness-of-fit tests, J. Roy. Statist. Soc. Ser. B. 46 (1984) 440-464] and (2) find a predictive density of a new observation drawn independently of observations sampled from a normal distribution with the same mean but possibly with a different variance under the same loss. The general divergence loss includes as special cases both the Kullback-Leibler and Bhattacharyya-Hellinger losses. The sample mean, which is a Bayes estimator of the population mean under this loss and the improper uniform prior, is shown to be minimax in any arbitrary dimension. A counterpart of this result for predictive density is also proved in any arbitrary dimension. The admissibility of these rules holds in one dimension, and we conjecture that the result is true in two dimensions as well. However, the general Baranchick [A.J. Baranchick, a family of minimax estimators of the mean of a multivariate normal distribution, Ann. Math. Statist. 41 (1970) 642-645] class of estimators, which includes the James-Stein estimator and the Strawderman [W.E. Strawderman, Proper Bayes minimax estimators of the multivariate normal mean, Ann. Math. Statist. 42 (1971) 385-388] class of estimators, dominates the sample mean in three or higher dimensions for the estimation problem. An analogous class of predictive densities is defined and any member of this class is shown to dominate the predictive density corresponding to a uniform prior in three or higher dimensions. For the prediction problem, in the special case of Kullback-Leibler loss, our results complement to a certain extent some of the recent important work of Komaki [F. Komaki, A shrinkage predictive distribution for multivariate normal observations, Biometrika 88 (2001) 859-864] and George, Liang and Xu [E.I. George, F. Liang, X. Xu, Improved minimax predictive densities under Kullbak-Leibler loss, Ann. Statist. 34 (2006) 78-92]. While our proposed approach produces a general class of predictive densities (not necessarily Bayes, but not excluding Bayes predictors) dominating the predictive density under a uniform prior. We show also that various modifications of the James-Stein estimator continue to dominate the sample mean, and by the duality of estimation and predictive density results which we will show, similar results continue to hold for the prediction problem as well.     

Pattern search methods for finite minimax problems

In this paper, we propose pattern search methods for finite minimax problems. Due to the nonsmoothness of this class of problems, we convert the original problem into a smooth one by using a smoothing technique based on the exponential penalty function of Kort and Bertsekas, which technique depends on a smoothing parameter that control the approximation to the finite minimax problems. The proposed methods are based on a sampling of the smooth function along a set of suitable search directions and on an updating rule for the step-control parameter. Under suitable conditions, we get the global convergence results despite the fact that pattern search methods do not have explicit information concerning the gradient and consequently are unable to enforce explicitly a notion of sufficient feasible decrease.     

A multi-product continuous review inventory system in stochastic environment with budget constraint

Common characteristics of inventory systems include uncertain demand and restrictions such as budgetary and storage space constraints. Several authors have examined budget constrained multi-item stochastic inventory systems controlled by continuous review policies without considering marginal review shortage costs. Existing models assume that purchasing costs are paid at the time an order is placed, which is not always the case since in some systems purchasing costs are paid when order arrive. In the latter case the maximum investment in inventory is random since the inventory level when an order arrives is a random variable. Hence payment of purchasing costs on delivery yields a stochastic budget constraint for inventory. In this paper with mixture of back orders and lost sales, we assume that mean and variance of lead time demand are known but their probability distributions are unknown. After that, we apply the minimax distribution free procedure to find the minimum expected value of the random objective function with budget constraint. The random budget constraint is transformed to crisp budget constraint by chance-constraint technique. Finally, the model is illustrated by a numerical example.     

Minimax estimator of regression coefficient in normal distribution under balanced loss function

This article investigates linear minimax estimators of regression coefficient in a linear model with an assumption that the underlying distribution is a normal one with a nonnegative definite covariance matrix under a balanced loss function. Some linear minimax estimators of regression coefficient in the class of all estimators are obtained. The result shows that the linear minimax estimators are unique under some conditions.     

A new expected-improvement algorithm for continuous minimax optimization

Worst-case design is important whenever robustness to adverse environmental conditions must be ensured regardless of their probability. It leads to minimax optimization, which is most often treated assuming that prior knowledge makes the worst environmental conditions obvious, or that a closed-form expression for the performance index is available. This paper considers the important situation where none of these assumptions is true and where the performance index must be evaluated via costly numerical simulations. Strategies to limit the number of these evaluations are then of paramount importance. One such strategy is proposed here, which further improves the performance of an algorithm recently presented that combines a relaxation procedure for minimax search with the well-known Kriging-based EGO algorithm. Expected Improvement is computed in the minimax optimization context, which allows to further reduce the number of costly evaluations of the performance index. The interest of the approach is demonstrated on test cases and a simple engineering problem from the literature, which facilitates comparison with alternative approaches.