Efficient Computation of Hedging Portfolios for Options with Discontinuous Payoffs

We consider the problem of computing hedging portfolios for options that may have discontinuous payoffs, in the framework of diffusion models in which the number of factors may be larger than the number of Brownian motions driving the model. Extending the work of Fournié et al. (1999) , as well as Ma and Zhang (2000) , using integration by parts of Malliavin calculus, we find two representations of the hedging portfolio in terms of expected values of random variables that do not involve differentiating the payoff function. Once this has been accomplished, the hedging portfolio can be computed by simple Monte Carlo. We find the theoretical bound for the error of the two methods. We also perform numerical experiments in order to compare these methods to two existing methods, and find that no method is clearly superior to others.     

Explicit Representation of the Minimal Variance Portfolio in Markets Driven by Lévy Processes

In a market driven by a Lévy martingale, we consider a claim ξ. We study the problem of minimal variance hedging and we give an explicit formula for the minimal variance portfolio in terms of Malliavin derivatives. We discuss two types of stochastic (Malliavin) derivatives for ξ: one based on the chaos expansion in terms of iterated integrals with respect to the power jump processes and one based on the chaos expansion in terms of iterated integrals with respect to the Wiener process and the Poisson random measure components. We study the relation between these two expansions, the corresponding two derivatives, and the corresponding versions of the Clark-Haussmann-Ocone theorem.     

Quantiles of the Euler Scheme for Diffusion Processes and Financial Applications

In this paper we briefly present the results obtained in our paper ( Talay and Zheng 2002a ) on the convergence rate of the approximation of quantiles of the law of one component of  ( X_t )  , where  ( X_t )  is a diffusion process, when one uses a Monte Carlo method combined with the Euler discretization scheme. We consider the case where  ( X_t )  is uniformly hypoelliptic (in the sense of Condition (UH) below), or the inverse of the Malliavin covariance of the component under consideration satisfies the condition (M) below. We then show that Condition (M) seems widely satisfied in applied contexts. We particularly study financial applications: the computation of quantiles of models with stochastic volatility, the computation of the VaR of a portfolio, and the computation of a model risk measurement for the profit and loss of a misspecified hedging strategy.     

Portfolio Value-at-Risk with Heavy-Tailed Risk Factors

This paper develops efficient methods for computing portfolio value-at-risk (VAR) when the underlying risk factors have a heavy-tailed distribution. In modeling heavy tails, we focus on multivariate t distributions and some extensions thereof. We develop two methods for VAR calculation that exploit a quadratic approximation to the portfolio loss, such as the delta-gamma approximation. In the first method, we derive the characteristic function of the quadratic approximation and then use numerical transform inversion to approximate the portfolio loss distribution. Because the quadratic approximation may not always yield accurate VAR estimates, we also develop a low variance Monte Carlo method. This method uses the quadratic approximation to guide the selection of an effective importance sampling distribution that samples risk factors so that large losses occur more often. Variance is further reduced by combining the importance sampling with stratified sampling. Numerical results on a variety of test portfolios indicate that large variance reductions are typically obtained. Both methods developed in this paper overcome difficulties associated with VAR calculation with heavy-tailed risk factors. The Monte Carlo method also extends to the problem of estimating the conditional excess, sometimes known as the conditional VAR.     

用于正弦波频率估计的修正Kay算法

Kay算法能够估计出采样点较少的正弦波频率，但低信噪比下估计性能不佳。针对此问题，提出了修正Kay算法。首先基于最大似然估计准则，推导了观测信号模值与相位的条件概率密度函数，进而重建了Kay算法的相位差噪声矢量协方差矩阵与权值矩阵。实验结果表明，修正算法能够有效估计正弦波信号频率，与Kay算法相比，抗噪性更强。     

GREEKS FORMULAS FOR AN ASSET PRICE MODEL WITH GAMMA PROCESSES

Greeks formulas of Delta, Rho, Vega, and Gamma are derived in closed form for asset price dynamics described by gamma processes and Brownian motions time‐changed by a gamma process. The model considered here includes many well‐known models of practical interest, such as the variance gamma model and the Black–Scholes model. Our approach is based upon the Malliavin calculus for jump processes by making full use of a scaling property of gamma processes with respect to the Girsanov transform. The existence of their variance is investigated. Numerical results are provided to illustrate that the derived Greeks formulas have faster rate of convergence relative to the finite difference method.     

Asymptotically Optimal Importance Sampling and Stratification for Pricing Path-Dependent Options

This paper develops a variance reduction technique for Monte Carlo simulations of path-dependent options driven by high-dimensional Gaussian vectors. The method combines importance sampling based on a change of drift with stratified sampling along a small number of key dimensions. The change of drift is selected through a large deviations analysis and is shown to be optimal in an asymptotic sense. The drift selected has an interpretation as the path of the underlying state variables which maximizes the product of probability and payoff—the most important path. The directions used for stratified sampling are optimal for a quadratic approximation to the integrand or payoff function. Indeed, under differentiability assumptions our importance sampling method eliminates variability due to the linear part of the payoff function, and stratification eliminates much of the variability due to the quadratic part of the payoff. The two parts of the method are linked because the asymptotically optimal drift vector frequently provides a particularly effective direction for stratification. We illustrate the use of the method with path-dependent options, a stochastic volatility model, and interest rate derivatives. The method reveals novel features of the structure of their payoffs.     

基于遗传算法的抽指加权叉指换能器进化设计

提出了极性抽指加权叉指换能器设计的新方法，克服了传统设计方法的繁杂性。将极性抽指加权叉指换能器作为染色体，通过独特的（-1，1）的二值编码，以目标频率响应曲线和待进化的叉指换能器频率响应曲线在考虑的频率范围内的1601个采样点的误差值为进化目标，对种群中的染色体进行选择、交叉和变异等遗传操作，自动进化出符合目标要求的极性抽指加权叉指换能器极性加权状况。进化实验结果表明，应用本文提出的进化方法设计出的极性抽指加权叉指换能器的频响曲线与目标频响曲线基本重合，达到设计要求，进化设计方法效率高，实用性强。     

PORTFOLIO OPTIMIZATION WITH DOWNSIDE CONSTRAINTS

We consider the portfolio optimization problem for an investor whose consumption rate process and terminal wealth are subject to downside constraints. In the standard financial market model that consists of d risky assets and one riskless asset, we assume that the riskless asset earns a constant instantaneous rate of interest,   r > 0  , and that the risky assets are geometric Brownian motions. The optimal portfolio policy for a wide scale of utility functions is derived explicitly. The gradient operator and the Clark–Ocone formula in Malliavin calculus are used in the derivation of this policy. We show how Malliavin calculus approach can help us get around certain difficulties that arise in using the classical "delta hedging" approach.     

Analytical approximations of local‐Heston volatility model and error analysis

This paper studies the expansion of an option price (with bounded Lipschitz payoff) in a stochastic volatility model including a local volatility component. The stochastic volatility is a square root process, which is widely used for modeling the behavior of the variance process (Heston model). The local volatility part is of general form, requiring only appropriate growth and boundedness assumptions. We rigorously establish tight error estimates of our expansions, using Malliavin calculus. The error analysis, which requires a careful treatment because of the lack of weak differentiability of the model, is interesting on its own. Moreover, in the particular case of call–put options, we also provide expansions of the Black–Scholes implied volatility that allow to obtain very simple formulas that are fast to compute compared to the Monte Carlo approach and maintain a very competitive accuracy.     

卫星入轨段多源测轨数据融合定轨的加权方法

针对卫星发射入轨段遥外测多种不等精度、不同类型的测轨数据放在一起定轨时的合理加权问题,采用基于微分轨道改进基本原理,推导得到适用于不同类测轨数据联合定轨的动力学条件方程,提出了依据不同测量数据均方差确定该测量元素对应的动力学条件方程权重的方法。仿真验算和实战数据检验表明,该方法充分发挥了分布于各种测轨数据中的高精度测元的作用,融合后的定轨精度得到显著提高。     

利用主客观集成赋权法的多目标威胁评估

在利用逼近于理想解的排序法(Technique for Order Preference by Similarity to an Ideal Solution,TOPSIS)进行多目标威胁评估时，针对如何获取合理的目标威胁评估因子的权重比，提出了一种主客观集成赋权方法。该方法首先利用主观赋权法和客观赋权法获取两组权重值；然后，通过构造多目标规划模型，将两组权重值进行综合处理，得到更合理的权重值。仿真结果表明，相较于主观赋权法和客观赋权法，所提方法计算出的威胁评估因子的权重值，在用TOPSIS法计算目标威胁评估时，能够得到更加合理、有效的评估结果。     

Is the online-offline buy-online-pickup-in-store retail strategy best among other product delivery strategies under variable lead time?

Every industry improves customer satisfaction by either preventing stockout situations or fulfilling the customer's demand by reducing lead time and variance of the lead time with several services offered. This study considers online selling, offline selling, and buy-online-pickup-in-store policies for selling products from retail stores. This study builds a characteristic relation between buy-online-pickup-in-store, backorder, and on-hand inventory by giving an exact total profit function instead of the expected profit function. The marginal values on lead time and variance of the lead time earn more exact outcomes for any retail strategy. The specific transportation cost for different retail strategies is considered and tested. This paper shows the analytical results which prove that the total profit of the sustainable supply chain is a convex function of both lead time and variance of the lead time. As different channel selling is crucial to any industry for increasing customer demand, the retailer sells their products by hybrid channel and advertises their products online to make them more famous. The model is solved through a classical optimization technique to get a semi-closed-form solution. In this method, it is possible to find out the optimum results for the decision variables separately by equating the first order partial derivative of the total profit with respect to the decision variables to zero and the global optimality by the Hessian matrix. A numerical experiment is conducted here, and the result shows that the supply chain system gives 14.48% more profit if advertisement policy is applied. The supply chain system gives 57.31% less profit and 72.72% less profit when only an online or offline channel is adopted, respectively. A sensitivity analysis is done for the managerial effects of cost parameters on the total profit.     

A cohort analysis of household vehicle expenditure in the U.S. and Japan: A possibility of generational marketing

This paper shows the usefulness of cohort analysis for generational marketing. Aggregate data classified by age and period are decomposed into age, period, and generational cohort effects. We compare two cohort-analysis models, the constrained multiple regression model and the Bayesian cohort model. The empirical results that are common to the household vehicle expenditure ratio in the U.S. and Japan are as follows: (1) among a total of three effects, the period effect is the smallest; (2) with the exception of the latest birth cohort, the cohort effect shows a clear upward trend; (3) the age effect decreases in the 20s and 30s, and next increases with a peak detected in the late 50s, and finally decreases. We provide marketing implications for cohort segmentation and forecasting.     

Volatility components: The term structure dynamics of VIX futures

In this study we empirically study the variance term structure using volatility index (VIX) futures market. We first derive a new pricing framework for VIX futures, which is convenient to study variance term structure dynamics. We construct five models and use Kalman filter and maximum likelihood method for model estimations and comparisons. We provide evidence that a third factor is statistically significant for variance term structure dynamics. We find that our parameter estimates are robust and helpful to shed light on economic significance of variance factor model. © 2009 Wiley Periodicals, Inc. Jrl Fut Mark 30:230–256, 2010     

一种高性能内插滤波器的设计

在全数字接收机系统中，随着高阶调制解调技术的应用，传统内插滤波器的性能已不能满足要求。为此，通过研究一种多项式函数的频率响应，提出了一种高性能内插滤波器的设计方法。该方法在频域逼近的基础上，以线性加权的最小均方误差（MMSE）为优化准则，利用Matlab系统函数进行线性约束条件下的最优化迭代，设计非常灵活。仿真结果表明，该方法设计的内插滤波器性能明显优于常用的内插滤波器，尤其适合于高阶正交幅度调制（QAM）信号。     

基于粒子群优化的数字波束形成算法

波束形成的加权系数求解是一个优化过程,现有算法大多经过多次迭代,计算量大,实现复杂。为降低波束形成算法复杂度,将粒子群优化原理应用于数字波束形成中,提出了基于粒子群优化的自适应数字波束形成算法。该算法将每一组权值作为一个粒子,将阵列加权和的输出信号与干扰噪声比(SINR)作为适应度函数,通过比较各个粒子的适应度值,进行迭代搜索得到最优解。该算法可使天线阵在主波束对准有用信号,同时能有效抑制两个以上的干扰,且对阵列通道误差有较好的稳健性。计算机仿真验证了算法的有效性。     

一种能耗均衡的无线传感器网络路由算法

为了延长采用电池供电的无线传感器网络的生命周期，提出了一种综合考虑单个节点能耗和节点传输信息至汇集节点所需总能耗的路由算法。该算法首先根据网络中节点到汇集节点从小到大的距离顺序选择待规划节点，然后计算各对应候选节点的评价参数，该参数由单节点能耗和节点传输信息至汇集节点所需总能耗加权得到，最后选择评价参数最小的候选节点作为待规划节点的中继节点。仿真结果表明，该算法的生命周期明显长于LEACH(Low Energy Adaptive Clustering Hierarchy)算法。     

Arrow–Debreu equilibria for rank‐dependent utilities with heterogeneous probability weighting

We study Arrow–Debreu equilibria for a one‐period‐two‐date pure exchange economy with rank‐dependent utility agents having heterogeneous probability weighting and outcome utility functions. In particular, we allow the economy to have a mix of expected utility agents and rank‐dependent utility ones, with nonconvex probability weighting functions. The standard approach for convex economy equilibria fails due to the incompatibility with second‐order stochastic dominance. The representative agent approach devised in Xia and Zhou (2016) does not work either due to the heterogeneity of the weighting functions. We overcome these difficulties by considering the comonotone allocations, on which the rank‐dependent utilities become concave. Accordingly, we introduce the notion of comonotone Pareto optima, and derive their characterizing conditions. With the aid of the auxiliary problem of price equilibria with transfers, we provide a sufficient condition in terms of the model primitives under which an Arrow–Debreu equilibrium exists, along with the explicit expression of the state‐price density in equilibrium. This new, general sufficient condition distinguishes the paper from previous related studies with homogeneous and/or convex probability weightings.     

Robust estimation of the optimal hedge ratio

When using derivative instruments such as futures to hedge a portfolio of risky assets, the primary objective is to estimate the optimal hedge ratio (OHR). When agents have mean‐variance utility and the futures price follows a martingale, the OHR is equivalent to the minimum variance hedge ratio,which can be estimated by regressing the spot market return on the futures market return using ordinary least squares. To accommodate time‐varying volatility in asset returns, estimators based on rolling windows, GARCH, or EWMA models are commonly employed. However, all of these approaches are based on the sample variance and covariance estimators of returns, which, while consistent irrespective of the underlying distribution of the data, are not in general efficient. In particular, when the distribution of the data is leptokurtic, as is commonly found for short horizon asset returns, these estimators will attach too much weight to extreme observations. This article proposes an alternative to the standard approach to the estimation of the OHR that is robust to the leptokurtosis of returns. We use the robust OHR to construct a dynamic hedging strategy for daily returns on the FTSE100 index using index futures. We estimate the robust OHR using both the rolling window approach and the EWMA approach, and compare our results to those based on the standard rolling window and EWMA estimators. It is shown that the robust OHR yields a hedged portfolio variance that is marginally lower than that based on the standard estimator. Moreover, the variance of the robust OHR is as much as 70% lower than the variance of the standard OHR, substantially reducing the transaction costs that are associated with dynamic hedging strategies. © 2003 Wiley Periodicals, Inc. Jrl Fut Mark 23:799–816, 2003