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A note on calculating expected shortfall for discrete time stochastic volatility models

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Financial Innovation 2021年第1期7卷 926-941页

作者： Michael Grabchak Eliana Christou Department of Mathematics and Statistics University of North Carolina at Charlotte9201 University City BlvdCharlotteNC 28223USA

In this paper we consider the problem of estimating expected shortfall(ES)for discrete time stochastic volatility(SV)***,we develop Monte Carlo methods to evaluate ES for a variety of commonly used SV *** includes both models where the innovations are independent of the volatility and where there is *** dependence aims to capture the well-known leverage *** performance of our Monte Carlo methods is analyzed through simulations and empirical analyses of four major US indices.

关键词： expected shortfall Stochastic volatility Value-at-risk

Change-point detection for expected shortfall in time series

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Journal of Management Science and Engineering 2021年第3期6卷 324-335页

作者： Lingyu Sun Dong Li Department of Mathematical Sciences Tsinghua UniversityBeijing 100084China Center for Statistical Science Department of Industrial EngineeringTsinghua UniversityBejing 100084China

expected shortfall(ES)is a popular risk measure and plays an important role in risk and portfolio ***,change-point detection of risk measures has been attracting much attention in *** on the self-normalized CUSUM statistic in Fan,Glynn and Pelger(2018)and the Wild Binary Segmentation(WBS)algorithm in Fryzlewicz(2014),this paper proposes a variant WBS procedure to detect and estimate change points of ES in time *** strengthened Schwarz information criterion is also introduced to determine the number of change *** Carlo simulation studies are conducted to assess the finite-sample performance of our variant WBS procedure about ES in time *** empirical application is given to illustrate the usefulness of our procedure.

关键词： Change point expected shortfall Risk measure

Uncertainty Comparison Between Value-at-Risk and expected shortfall

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Communications in Mathematical Research 2024年第1期40卷 102-124页

作者： Qing Liu Weimin Liu Liang Peng Gengsheng Qin School of Statistics Jjiangxi University of Finance and EconomicsNanchangJiangxi 330013China Department of Mathematics and Statistics Georgia State UniversityAtlantaGA 30303USA Maurice R.Greenberg School of Risk Science Georgia State UniversityAtlantaGA 20303USA

Value-at-Risk(VaR)and expected shortfall(ES)are two key risk measures in financial risk *** these two measures has been a hot debate,and most discussions focus on risk measure *** paper uses independent data and autoregressive models with normal or t-distribution to examine the effect of the heavy tail and dependence on comparing the nonparametric inference uncertainty of these two risk *** and numerical analyses suggest that VaR at 99%level is better than ES at 97.5%level for distributions with heavier tails.

关键词： α-mixing asymptotic variance expected shortfall Value-at-Risk

Three Methods to Calculate the Financial Risk Measurement: Value- At-Risk and expected shortfall

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Journal of Finance Research 2020年第2期4卷 145-150页

作者： Yulin Liu University of Leeds LeedsWest YorkshireLS28EYUK

This paper analyzes the relationship between the risk factor of each stock and the portfolio’s risk based on a small portfolio with four ***,and the reason why these risk factors can be regarded as a market ***,it evaluates the properties of the convex and coherent risk indicators of the capital requirement index composed of VaR and ES,and use three methods(the historical estimation method,boudoukh’s mixed method and Monte Carlo method)to estimate the risk measurement indicators VaR and ES respectively based on the assumption of multivariate normal distribution’risk factors and multivariate student t-copula distribution’s one,finally it figures out that these three calculation results are very close.

关键词： Value at risk expected shortfall Risk factors Student’s t-copula

The sum of two independent polynomially-modified hyperbolic secant random variables with application in computational finance

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International Journal of Modeling, Simulation, and Scientific Computing 2021年第4期12卷 201-214页

作者： A.A.L.Zadeh Hojatollah Zakerzadeh Hamzeh Torabi Department of Statistics Yazd University YazdIran

In this paper,by reshaping the hyperbolic secant distribution using Hermite polynomial,we devise a polynomially-modified hyperbolic secant distribution which is more flexible than secant distribution to capture the skewness,heavy-tailedness and kurtosis of *** a portfolio possibly consists of multiple assets,the distribution of the sum of independent polynomially-modified hyperbolic secant random variables is *** exceptional cases,we evaluate risk measures such as value at risk and expected shortfall(ES)for the sum of two independent polynomially-modified hyperbolic secant random ***,using real datasets from four international computers stocks,such as Adobe Systems,Microsoft,Nvidia and Symantec Corporations,the effectiveness of the proposed model is shown by the goodness of Gram–Charlier-like expansion of hyperbolic secant law,for performance of value at risk and ES estimation,both in and out of the sample period.

关键词： expected shortfall Gram–Charlier-like expansions heavy tailed distributions value at risk

STUDY ON THE INTERRELATION OF EFFICIENT PORTFOLIOS AND THEIR FRONTIER UNDER t DISTRIBUTION AND VARIOUS RISK MEASURES

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Applied Mathematics(A Journal of Chinese Universities) 2006年第4期21卷 369-382页

作者： Wang Yi Chen Zhiping Zhang Kecun Department of Scientific Computing and Applied Softwares Faculty of Science Xi＇an Jiaotong University Xi＇an 710049 China.

In order to study the effect of different risk measures on the efficient portfolios （fron- tier） while properly describing the characteristic of return distributions in the stock market, it is assumed in this paper that the joint return distribution of risky assets obeys the multivariate t-distribution. Under the mean-risk analysis framework, the interrelationship of efficient portfolios （frontier） based on risk measures such as variance, value at risk （VaR）, and expected shortfall （ES） is analyzed and compared. It is proved that, when there is no riskless asset in the market, the efficient frontier under VaR or ES is a subset of the mean-variance （MV） efficient frontier, and the efficient portfolios under VaR or ES are also MV efficient; when there exists a riskless asset in the market, a portfolio is MV efficient if and only if it is a VaR or ES efficient portfolio. The obtained results generalize relevant conclusions about investment theory, and can better guide investors to make their investment decision.

关键词： mean-risk model portfolio optimization value at risk expected shortfall efficient frontier.

Management of a Complex Portfolio of Assets with Stochastic Drifts and Volatilities

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Open Journal of Statistics 2022年第6期12卷 827-838页

作者： Wendkouni Yaméogo Korotimi Ouédraogo Diakarya Barro LANIBIO Université Joseph KI-ZERBO Ouagadougou Burkina Faso Université Thomas SANKARA Ouagadougou Burkina Faso

In financial analysis risk quantification is essential for efficient portfolio management in a stochastic framework. In this paper we study the value at risk, the expected shortfall, marginal expected shortfall and value at risk, incremental value at risk and expected shortfall, the marginal and discrete marginal contributions of a portfolio. Each asset in the portfolio is characterized by a trend, a volatility and a price following a three-dimensional diffusion process. The interest rate of each asset evolves according to the Hull and White model. Furthermore, we propose the optimization of this portfolio according to the value at risk model.

关键词： Value at Risk expected shortfall Stochastic Process Interest Rate

基于ES模型的沪深300股指期货基差风险研究

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基于ES模型的沪深300股指期货基差风险研究

作者：王梦琨浙江财经大学

学位级别：硕士

股指期货具有价格发现,规避风险以及提高资金配置效率等功能。我国于2010年4月16日正式推出沪深300股指期货,这是我国资本市场发展的一大里程碑。股指期货的跨期性、杠杆性、联动性和多样性等特点很好地弥补金融市场的缺陷,不仅可以健... 详细信息

股指期货具有价格发现,规避风险以及提高资金配置效率等功能。我国于2010年4月16日正式推出沪深300股指期货,这是我国资本市场发展的一大里程碑。股指期货的跨期性、杠杆性、联动性和多样性等特点很好地弥补金融市场的缺陷,不仅可以健全股票市场的价格机制还可以大大降低交易成本,并且为广大参与者提供了一个可以避险的投资工具。沪深300股指期货的推出彻底改变了我国金融市场风险管理工具缺乏的现状,但同时也面临股指期货高杠杆交易巨大风险的挑战。基差是股指价格与股指期货价格之间的差值,在股指期货交易中,由于现货价格与期货价格波动不一致、从而基差波动不确定而导致的风险被称为基差风险。如果基差朝着有利方向变化,则不仅可以取得较好的套期保值效果,而且还可以获得额外的盈利;反之,不仅会影响套保效果,甚至还会蒙受损失。所以对于套期保值者和套利者而言,股指期货的风险主要体现在基差波动的不确定性上。因此,研究沪深300股指期货的基差风险对于我国金融市场而言具有很强的现实意义。本文将对我国沪深300股指期货进行较为详细的阐述,然后运用expected shortfall模型度量股指期货的基差风险,最后实证研究预测沪深300股指期货的基差风险。由于金融时间序列往往具有尖峰后尾的特征,运用正态分布进行计算会带来很大的误差,而非对称Laplace分布能够很好的拟合金融时间序列的尖峰后尾的特征。因此本文运用非对称Laplace分布来拟合沪深300股指期货的基差,建立非对称Laplace分布下的ES模型来分析沪深300股指期货的波动情况,计算出该模型下的ES值和VaR值,分别将模型得出的VaR和ES值与实际的基差波动进行比较,考察ES和VaR对实际基差波动的拟合情况,利用Kupiec在1995年提出的似然比检验方法建立LR和LE统计量,对非对称Laplace分布下的ES模型的计算结果进行有效性检验,检验结果显示非对称Laplace分布的ES模型能够有效的拟合沪深300股指期货基差风险。

关键词：基差风险非对称Laplace分布 VaR expected shortfall 风险度量

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基于尾部风险度量的资金分配的研究

基于尾部风险度量的资金分配的研究

作者：周杨志长春工业大学

学位级别：硕士

在效用理论中,expected shortfall(ES)的风险态度是基于线性效用函数表达的,而ES和基于ES的资金分配难以表达现实中复杂的非线性关系,因此,我们受到ES和基于ES的资金分配的关联性的启发,将基于ES的资金分配扩展到一般的情况。由非线性的... 详细信息

在效用理论中,expected shortfall(ES)的风险态度是基于线性效用函数表达的,而ES和基于ES的资金分配难以表达现实中复杂的非线性关系,因此,我们受到ES和基于ES的资金分配的关联性的启发,将基于ES的资金分配扩展到一般的情况。由非线性的Young函数族来表达非中立的风险态度,根据不同的风险态度评估风险对总风险事件的非线性贡献,来取代基于ES的资金分配处理非线性的情形。本文一方面,基于ES的资金分配和Orlicz范数的定义,研究了基于ES的资金分配的推广,提出了基于Orlicz风险度量的资金分配模型,用估计方程方法解决了基于Orlicz风险度量的资金分配的非参数估计问题。首先我们从ES的资金分配的定义出发,结合Orlicz范数的定义,建立了基于Orlicz风险度量的资金分配模型。其次对基于Orlicz风险度量的资金分配进行了估计方程估计,得到估计的强相合性和渐近正态性,进而得到了渐近正态置信区间,最后用R软件模拟考察了估计方法在不同置信水平和样本容量下的样本表现。另一方面,用估计方程方法解决Haezendonck–Goovaerts风险度量(HGRM)资金分配的非参数估计问题,得到基于HGRM的资金分配的经验估计,估计方程估计以及各自的强相合性和渐近正态性。本文给出HGRM的资金分配的定义。对基于HGRM的资金分配进行经验估计,证得经验估计的相合性和渐近正态性,由此得到渐近正态置信区间。在某些具体问题中,也关心对门限值的估计。为了同时给出基于HGRM的资金分配和门限值的估计量。从基于HGRM的资金分配的定义出发寻找估计方程,建立了基于HGRM的资金分配和门限值的联合估计方程,得到了联合估计量,同样证得了估计量的大样本性质。数值模拟分析了样本容量对基于HGRM的资金分配的估计的影响。

关键词： expected shortfall Orlicz风险度量资金分配 Haezendonck–Goovaerts风险度量