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A summary about realized volatility in high freqeuncy financial data

A summary about realized volatility in high freqeuncy financial data
A summary about realized volatility in high freqeuncy financial data

A summary about realized volatility in high

freqeuncy financial data

Abundant trading data is created every day in nowaday's financial world, and it offers plenty of information. After Torben G. Andersen and Tim Bollerslev(1998) proposed interdaily volatility model, more and more researchers focus on stochastic volatility research about high frequency data. Such as C du Toit WJ Conradie(2006) studied realized volatility measurement with microstructure effects,Ole E.Barndorff-Nielsen(2002), who write many papers from 2001 to 2011 in this field, use realized variance and realized volatility to estimate quadratic variation and stochastic volatility respectively, in 2001, Ole E.Barndorff-Nielsen and Neil Shephard studied higher order variation and stochastic volatility models, they proposed a notation “higher order variations ”that defined as

[]1[]()|()(1)|n r r

j x t x j x j δδδ==--∑

when r>2. In a study by Barndor-Nielsen and Shephard (2001d) of the properties of realised volatility, that is the sum of squares of intra-day returns on speculative assets, it became

necessary in addition to quadratic variation of the stochastic processes to consider also aspects of higher order variation. The requisite mathematical results on higher order variation seem of some independent interest.

Now the following is some intruduction of “realized volatility ”.

A triple (Ω;F;P) is called a probability space, if Ω is a given set, F is a σ-algebra on Ω,and P is a probability measure on (Ω;F).

A sequence {Xn} of random variables defined on a probability space (Ω;F;P) is said to converge to X in

probability, denoted by lim m p Xn X →∞

=, if lim (||)0,0n P Xn X εε→∞

-≥=?> {Xn} is said to converge to X in mean square if

2lim (||)0n E Xn X →∞

-= Markov's Inequality guarantees that convergence in mean square is stronger than convergence in probability. It's well know that convergence in probability implies convergence in distribution.

A stochastic process 0{}t t

B ≥ is said to be a standard

Brownian motion if

(1) B 0= 0;

(2) 0{}t t B ≥ has stationary independent increments;

(3) for every t > 0, Bt is normally distributed with mean 0 and variance t.

Denote by Ft the σ-algebra generated by the random

variable 0{}t s t B ≤≤, and by F the σ-algebra generated by 0{}s s B ≥.

A process g (t;w) : [0,)n R ∞?Ω→ is called Ft-adapted if for each t ≥ 0 the function (,)g t ωω→ is Ft-measurable.

Suppose that Xt is a real-valued Ft-adapted stochastic process defined on a probability space (Ω;F;P). The quantity

11

20()j j n t t j X X +-=-∑ is called the realized volatility of the stochastic process Xt,where {}010n t t t t =≤≤≤= is any partition of the interval

[0,t].The quadratic variation of Xt,if exists,is defined by the following

112||||00[,]lim

()j j n t t j X X t p X X +-∏→==-∑

where 1max{||:1,,}j j t t j n -∏=-=

is the length of the longest subinterval in the partition

{}010n t t t t =≤≤≤=

For the Itoprocess

t t t t dX dt dB μσ=+

where Bt is a standard Brownian motion, t μis the drift coefficient and 2t σis the instantaneous variance of the return

process Xt,the following result is well known.

Theorem A.Let 0{}

t t B ≥ be the Brownian motion, and ,t t μσ be

Ft-adapted stochastic process. Then the quadratic variation of the Ito process is

20

[,]t

t s X X ds σ=?.

Let (,)S T νν= be the class of functions

(,):[0,)f t R ω∞?Ω→

such that

(1)(,)(,)t f t ωω→ is B F ?-measurable,where B denotes the Borel σ-algebra on [0,)∞.

(2) (,)f t ω is Ft-adapted;

(3)2[(,)]T

S

E f t dt ω<∞?.

The new econometrics is motivated by the advent of complete records of quotes or transaction prices for many ?nancial assets. Although market microstructure e ?ects (e.g. discreteness of prices, bid/ask bounce, irregular trading etc.)means that there is a mismatch between asset pricing theory based on semimartingales and the data at very ?ne time intervals (see, for example,Bai, Russell, and Tiao (2000)) it does suggest the desirability of establishing an asymptotic distribution theory for estimators as we use more and more highly frequent observations.Realized variance, being the summation of squared intra-day returns, has quickly gained

popularity as a measure of daily volatility. Following Parkinson(1980) we replace each squared intra-day return by the high-low range for that period to create a novel and more e?cient estimator called the realized range. In theory, the realized variance is an unbiased and highly e?cient estimator, as illustrated in Andersen et al.(2001b), and converges to the true underlying integrated variance when the length of the intra-day intervals goes to zero, see Barndor?-Nielsen and Shephard (2002).In practice, market microstructure e?ects such as bid-ask bounce pose limitations to the choice of sampling frequency. Returns at very high frequencies are distorted such that the realized variance becomes biased and inconsistent, see Bandi and Russell (2005a,b), and Hansen and Lunde (2006b). Popular choices in empirical applications are the ?ve- and thirty-minute intervals, which are believed to strike a balance between the increasing accuracy of higher frequencies and the adverse e?ects of market microstructure frictions, see e.g. Andersen and Bollerslev (1998), Andersen et al. (2001a), Andersen et al. (2003), and Fleming etal. (2003).An alternative way of measuring volatility is based on the di?erence between the maximum and minimum prices observed during a certain period. Parkinson (1980) shows that the daily

(log) high-low range, properly scaled, not only is an unbiased estimator of daily volatility but is ?ve times more e ?cient than the squared daily close-to-close return. Correspondingly, Andersen and Bollerslev (1998) and Brandt and Diebold (2006) ?nd that the e ?ciency of the daily high-low range is between that of the realized variance computed using 3-hour and 6-hour returns.

In the paper “Estimating quadratic variation using realised variance ”written by Ole E. Barndorff-Nielsen(2002), he proposed two questions about “realised variance ”, that is the

sum of M squared returns.And the main result of this paper is

22,1()(0,1)

hn M j n j y s ds N σ=-

→∑? This is a mixed Gaussian limit theory, that is the denominator is itself random. Of course this theory can be used to provide approximations for realised volatility as well as realised variance. The distribution of realised volatilies can also be approximated indirectly via

42,1()(0,1)hn M j n j y s ds N σ=-

→∑?

using the delta method which gives

(0,1)N →

The log-based approximation 2[2]

,log()log()(0,1)M

j n n y N σ-→∑ is likely to be preferred in practice when we construct con ?dence intervals for realised volatility.

In a study “A realised volatility measurement using

quadratic variation and dealing with microstructure e?ects”by C du Toit? and WJ Conradie?, they add microstructure effects into realised volatility research.In Anderson, et al. (2001a, 2001b), Barndor?-Nielsen and Shepard (2001, 2002a, 2002b,2002c) and Comte and Renault (1998), a modelfree (non-parametric) volatility measurement is speci?ed and studied.

The main results of this paper is

And

Building on realized variance and bipower variationmeasures constructed fromhigh-frequency financial prices, Torben G. Andersena, Tim Bollerslev, Xin Huang (2011) propose a simple reduced form framework for effectively incorporating intraday data into the modeling of daily return volatility. They decompose the total daily return variability into the continuous sample path variance, the variation arising from discontinuous jumps that occur during the trading day, as well as the overnight return variance. Their empirical results, based on long samples of high-frequency equity and bond futures returns, suggest that the dynamic dependencies in the daily

continuous sample path variability arewell described by an approximate long-memoryHAR–GARCHmodel,while the overnight returnsmay bemodeled by an augmented GARCH type structure. The dynamic dependencies in the non-parametrically identified significant jumps appear to be well described by the combination of an ACHmodel for the time-varying jump intensities coupled with a relatively simple log-linear structure for the jump sizes.

Analysis of high frequency finacial data poses interest to econometric modeling and statistical analysis. Modeling and measuring financial volatility are key steps for derivative pricing, portfolio allocation and risk management.How can we model and measure high frequency data effectively? In theory, the sum of squares of log returns sampled at high frequency estimates their variance(see,for example in the paper “Andersen,T.G,Bollerslev,T.,Diebold,F.X.and Labys,P,Modeling and forecasting realized volatility,Econometrica,71,579-625,2003”and “Barndorff-Nielsen,Ole E. and Shephard,N,Econometric analysis of realized volatility and its use in estimating stochastic volatility models,Journal of royal statistical society,series B,64,253-280,2002”). For example, for log

price data following a diffusion process without noise, the realized volatility converges to its quadratic variation(Barndorff-Nielsen,Ole E. and Shephard,N,Econometric analysis of realized volatility and its use in estimating stochastic volatility models,Journal of royal statistical society,series B,64,253-280,2002). Much research has been done recently.

So much for the review about riealised volatility, my major study direction is the higher order realised volatility in high freqeuncy finanacial data. Now, I have aready had a result in this field, and the main idea is all of the risk function can be represented by a polynomial. In my working paper, we calculate the expectation and the variance of the polynomial and we find some interesting results.The main result of our working paper is the constant-matrix in the variance matrix. That is to say any risk variance can be represents by a vector multiple the constant-matrix and multiple the transpose fo the vector above.This is a very interesting result which shocked me at first. I do not know whether this is a surprising result, and I do not know whether effect in the real financial world, but I know its an interesting result in mathematics at least, especially the constand-matrix. For my own reason, the working

paper have not done, the paper is not excellent now, there still much word for me to do. I hope I can finish my paper in six month and have a very good result in the end. I still hope there are more experts to work with me together, so that I can improve myself much.

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英文summary写作范例

Article Children Must be Taught to Tell Right from Wrong William Kilpatrick Many of today 's young people have a difficult time seeing any moral dimension ( 道德层 面 ) to their actions. There are a number of reasons why that 's true, but none more prominent than a failed system of education that eschews ( 回避 ) teaching children the traditional moral values that bind Americans together as a society and a culture. That failed approach, called “decision - making, ” was introduced in schools 25 years ago. It tells children to decide for themselves what is right and what is wrong. It replaced “character education. ( 品格教 育 )” Character education didn 't ask children to reinvent the moral wheel ( 浪费时间重新发明早已存 在的道德标准); instead, it encouraged them to practice habits of courage, justice and self-control. In the 1940s, when a character education approach prevailed, chewing gum; today they worry about robbery and rape. Decision-making curriculums pose thorny ( 棘手的 ) ethical dilemmas to students, with the impression that all morality is problematic and that all questions of right and wrong are in dispute. Youngsters are forced to question values and virtues they 've never acquired in the first place or upon which they have only a tenuous ( 薄弱的 ) hold. The assumption behind this method is that students will arrive at good moral conclusions if only they are given the chance. But the actual result is moral confusion. For example, a recent national study of 1,700 sixth- to ninth-graders revealed that a majority of boys considered rape to be acceptable under certain conditions. Astoundingly, many of the girls agreed. This kind of moral illiteracy is further encouraged by values-education (价值观教育 ) programs that are little more than courses in self-esteem ( 自尊 ). These programs are based on the questionable assumption that a child who feels good about himself or herself won 't want to do anything wrong. But it is just as reasonable to make an opposite assumption: namely, that a child who has uncritical self-regard w ill conclude that he or she can 't do anything bad. Such naive self-acceptance results in large part from the non-directive ( 无指导性的 ), non-judgmental ( 无是非观的 ), as-long-as-you-feel-comfortable-with-your-choices mentality ( 思 想) that has pervaded ( 渗透) public education for the last two and one-half decades. Many of today 's drug education, sex education and values -education courses are based on the same 1960s philosophy that helped fuel the explosion in teen drug use and sexual activity in the first place. Meanwhile, while educators are still fiddling with ( 胡乱摆弄 ) outdated “feel - good ” approaches, New York, Washington, and Los Angeles are burning. Youngsters are leaving school believing that matters of right and wrong are always merely subjective. If you pass a stranger on the street and decide to murder him because you need money —if it feels right —you go with that feeling. Clearly, murder is not taught in our schools, but such a conclusion —just about any conclusion —can be reached and justified using the decision-making method. It is time to consign ( 寄出 ) the fads (风尚 ) of “decision - making ” and “non- judgmentalism ” to the ash heap of failed policies, and return to a proved method. Character education provides a much more realistic approach to moral formation. It is built on an understanding that we learn morality not by debating it but by practicing it. Sample teachers worried about students leaving them

英语必修二知识点整理

英语必修二知识点整理 Unit1 Cultural relics 1.基础梳理 rare valuable survive vase dynasty amaze honey design fancy style decorate jewel artist belong to remove troop reception doubt former worth local apartment paint castle trail envidence entrance sink sailor maid in formal debate take apart keep…in one’s heart 2.词语归纳 1)state 指“国家”时,常表示“政权,国体”等政治性概念,首字母常常大写。表示“状态,情况”时,为可数名词,常作单数;in a state表示“处于混乱或者是不整洁的状态”;get into a state变得十分紧张。 in state 庄严堂皇的,隆重的 state也可以作动词,表示“陈述,阐明,声明”,多用于正式场合或者是公文,商务信函,日常用语中应该避免。 表示“据说,据称”常用于it或者sb/sth作主语的被动句中。 2)rare 作形容词,表示“罕见的,稀少的,稀有的,难得的”,可形容人或者是物。rare也可以指肉,表示“未熟的,半熟的”。 rare也是作副词,相当于rarely,意思是“很,非常”。 rare和and连用,相当于一个副词,意思是“很,极,非常”。 3)belong 不能用于被动语态和进行时。 belong in 适宜于,用利于,应该用在……。 4)gift

summary 范文

Original: My neighbor's children love playing hide-and-seek as all children do, but no one imagine that a game they played last week would be reported in the local newspaper. One afternoon, they were playing in the vacant lot down the corner. Young Paul, who is only five years old, found the perfect place to hide. His sister, Natalie, had shut her eyes and was counting to ten when Paul noticed the storage mail box at the corner and saw that the metal door was standing open. The mailman had just taken out several sacks of mail and had carried them to his truck which was standing at the curb a few feet away. Paul climbed into the storage box and pulled the door closed so hard that it locked. Soon realizing what he had done, he became frightened and started crying. Meanwhile, Natalie was looking for him everywhere but could not find him. It was lucky that she happened to pause at the corner for a minute and heard her brother's cries. She immediately ran to tell the mailman who hurried back from his truck to unlock the metal door. Paul was now free, but he had had such a bad scare that he could not stop crying. The mailman, however, soon found a way of making him laugh again. He told him that the next time he wanted to hide in a mail box, he should remember to put a stamp on himself! Summary: The children were playing hide-and-seek in a vacant lot one afternoon. Finding that the storage mailbox had been left open, Paul hid and locked himself in it accidentally. His sister, Natalie, heard his cries and realized where he was hiding, so she immediately told the mailman to unlock the metal door. After letting him out, the mailman made him stop crying by telling him to put a stamp on himself the next he wanted to hide in a mailbox. Original: Why do some animals die out? In the past two hundred years people have caused many kinds of animals to die out--to become extinct. People keep building houses and factories in fields and woods. As they spread over the land, they destroy animals' homes. If the animals can't find a place to live, they die out. Sixteen kinds of Hawaiian birds have become extinct for this reason. Other animals, such as the Florida Key deer, may soon die out because they are losing their homes.Hunters have caused some animals to become extinct, too. In the last century, hunters killed all the passenger pigeons in North America and most of the buffalos. Today they are fast killing off hawks and wolves. Pollution is killing many animals today, too. As rivers become polluted, fish are poisoned. Many die. Birds that eat the poisoned fish can't lay strong, healthy eggs. New birds aren’t born. So far, no animals have become extinct because of pollution. But some, such as the bald eagle and the brown pelican, have become rare and may die out. Scientists think that some animals become extinct because of changes in climate. The places where they live become hotter or cooler, drier or wetter. The food that they eat cannot grow there any more. If the animals can't learn to eat something else, they die. Dinosaurs may have died out for this reason. Summary:

人教版高中英语必修二知识点归纳总结

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危险!书可能会改变你的人生 1 刘易斯?卡罗尔书中的爱丽丝不小心掉进了兔子洞里,但她在那里发现了一个神奇的仙境。当我们打开一本书时,我们也会像爱丽丝那样走进一个全新的世界。我们能从一个年长者的角度,或通过一个孩子的眼睛来观察生活;我们可以周游世界,遍访现实生活中从没想过要访问的国家和文化;我们可以体验未曾经历过的事情,这些事情也许令人困惑,也许引人入胜;可能是不愉快的,也可能是令人痛苦的,但无论如何都至少能把我们从现实世界中解放出来。 2 英国诗人威廉?柯珀(1731—1800)说:“变化是生活的调味品,它让生活变得有滋有味。”虽然他没有说在什么地方以及怎样才能找到变化,但我们知道他说得对。我们知道我们生活在一个充满变化与差异的世界里,我们知道人们的生活各不相同,过日子的方式也不尽相同,人们做不同的工作,有不同的信仰,持不同的观点,有不同的风俗习惯,操不同的语言。通常,我们不知道这些差异的大小,但一旦发生了不平常的事情并引起了我们的注意,这种变化或差异与其说是机会,毋宁说是威胁。 3 读书让我们能够安全地享受和庆贺这种变化与差异,并为我们提供成长的机会。在家里安详平和的环境中与他人的生活互动,这是阅读小说才享有的特权。我们甚至感觉到——哪怕只是在一瞬间——我们和其他文化读者的共同点或许要多于我们和家门口 随便碰到的一个人的共同点。我们学会把目光移出我们周围的环境,投向天边,去领略一下异域风光。 4 如果我们怀疑读书是否能给我们力量的话,我们就应该自己去一趟当地的图书馆或书店,或者,如果我们足够幸运的话,可以读一读家里书架上的书。我们会惊奇于古今小说的标题所创造出来的壮观景象:约翰?斯坦贝克的《愤怒的葡萄》、约翰?欧文的《第四只手》、亚历山大?索尔仁尼琴的《癌病房》、欧内斯特?海明威的《丧钟为谁而鸣》、格雷厄姆?格林的《哈瓦那特派员》、奥黛丽?尼芬格的《时间旅行者的妻子》、保罗? 托迪的《到也门钓鲑鱼》。一旦开始阅读,我们就应该思考一下我们在书中读到的别样人生。 5 每一本书都有自己的语言、方言、词汇和语法。我们不见得总能理解其中的每一个字、每一句话,但不管我们是痴迷其中,还是觉得被排斥在外,我们的情感被调动起来了。尽管在地理上有一定的距离,但其他民族、其他文化未必就离我们那么遥远。在书里我们可能遇见生活在不同气候、有不同信仰、属于不同种族的人。即便是住在同一条街上的邻居,我们也可能对其一无所知,而只能通过阅读结识。

英文Summary写作方法、范例及常用句式

摘要是对一篇文章的主题思想的简单陈述。它用最简洁的语言概括了原文的主题。写摘要主要包括三个步骤:(1)阅读;(2)写作;(3)修改成文。 第一步:阅读 A.认真阅读给定的原文材料。如果一遍不能理解,就多读两遍。阅读次数越多,你对原文的理解就越深刻。 B.给摘要起一个标题。用那些能概括文章主题思想的单词、短语或短句子作为标题。也可以采用文中的主题句作为标题。主题句往往出现在文章的开头或结尾。一个好标题有助于确定文章的中心思想。C.现在,就该决定原文中哪些部分重要,哪些部分次重要了。对重要部分的主要观点进行概括。 D.简要地记下主要观点——主题、标题、细节等你认为对概括摘要重要的东西。 第二步:动手写作 A. 摘要应该只有原文的三分之一或四分之一长。因此首先数一下原文的字数,然后除以三,得到一个数字。摘要的字数可以少于这个数字,但是千万不能超过这个数字。 B. 摘要应全部用自己的话完成。不要引用原文的句子。 C. 应该遵循原文的逻辑顺序。这样你就不必重新组织观点、事实。 D. 摘要必须全面、清晰地表明原文所载的信息,以便你的读者不需翻阅原文就可以完全掌握材料的原意。 1 / 19

E. 写摘要时可以采用下列几种小技巧: 1) 删除细节。只保留主要观点。 2) 选择一至两个例子。原文中可能包括5个或更多的例子,你只需从中筛选一至二个例子。 3) 把长段的描述变成短小、简单的句子。如果材料中描述某人或某事用了十个句子,那么你只要把它们变成一两句即可。 4) 避免重复。在原文中,为了强调某个主题,可能会重复论证说明。但是这在摘要中是不能使用的。应该删除那些突出强调的重述句。 5) 压缩长的句子。如下列两例: “His courage in battle might without exaggeration be called lion-like.” 可以概括为:”He was very brave in battle.” “He was hard up for money and was being pressed by his creditor.” 可以概括为:“He was in financial difficulties.” 6) 你还可以使用词组代替整句或者从句。请看下面的例子:“Beautiful mountains like Mount Tai, Lushan Mountain, and Mount Huang, were visited by only a few people in the past. Today, better wages, holidays with pay, new hotels on these mountains, and better train and bus services, have brought them within reach of many who never thought of visiting them ten years ago.” 2 / 19

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