统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Using the Kelly Criterion for Investing

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随机建模是金融模型的一种形式,用于帮助做出投资决策。这种类型的模型使用随机变量预测不同条件下各种结果的概率。随着现代经济学、金融学实证研究的发展金融中的随机方法Stochastic Methods in Finance作为一种数学工具具有越来越重要的应用价值

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  • Statistical Inference 统计推断
  • Statistical Computing 统计计算
  • Advanced Probability Theory 高等楖率论
  • Advanced Mathematical Statistics 高等数理统计学
  • (Generalized) Linear Models 广义线性模型
  • Statistical Machine Learning 统计机器学习
  • Longitudinal Data Analysis 纵向数据分析
  • Foundations of Data Science 数据科学基础
统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Using the Kelly Criterion for Investing

统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|William T. Ziemba and Leonard C. MacLean

The Kelly capital growth strategy is defined as allocate your current wealth to risky assets so that the expected logarithm of wealth is maximized period by period. So it is a one-period static calculation that can have transaction costs and other market imperfections considered. Log utility dates to Daniel Bernoulli in 1738 who postulated that marginal utility was monotone increasing but declined with wealth and, specifically, is equal to the reciprocal of wealth, $w$, which yields the utility of wealth $u(w)=\log w$. Prior to this it was assumed that decisions were made on an expected value or linear utility basis. This idea ushered in declining marginal utility or risk aversion or concavity which is crucial in investment decision making. In his chapter, in Latin, he also discussed the St. Petersburg paradox and how it might be analyzed using $\log w$.

The St. Petersburg paradox actually originates from Daniel’s cousin, Nicolas Bernoulli, a professor at the University of Basel where Daniel was also a professor of mathematics. In 1708 , Nicolas submitted five important problems to Professor Pierre Montmort. This problem was how much to pay for the following gamble:
A fair coin with $\frac{1}{2}$ probability of heads is repeatedly tossed until heads oecurs, ending the game. The investor pays $c$ dollars and receives in return $2^{k-1}$ with probability $2^{-k}$ for $k=1,2, \ldots$ should a head occur. Thus, after each succeeding loss, assuming a head does not appear, the bet is doubled to $2,4,8, \ldots$ ete. Clearly the expected value is $\frac{1}{2}+\frac{1}{2}+\frac{1}{2}+\ldots$. or infinity with linear utility.
Bell and Cover (1980) argue that the St. Petersburg gamble is attractive at any price $c$, but the investor wants less of it as $c \rightarrow \infty$. The proportion of the investor’s wealth invested in the St. Petersburg gamble is always positive but decreases with the $\cos t c$ as $c$ increases. The rest of the wealth is in cash.

Bernoulli offers two solutions since he felt that this gamble is worth a lot less than infinity. In the first solution, he arbitrarily sets a limit to the utility of very large payoffs. Specifically, any amount over 10 million is assumed to be equal to $2^{24}$. Under that bounded utility assumption, the expected value is
\frac{1}{2}(1)+\frac{1}{4}(2)+\frac{1}{8}(4)+\cdots+\left(\frac{1}{2}\right)^{24}\left(2^{24}\right)+\left(\frac{1}{2}\right)^{25}\left(2^{24}\right)+\left(\frac{1}{2}\right)^{2 h}\left(2^{24}\right)+\ldots=12+\text { the original } 1=13 .

When utility is log the expected value is
\frac{1}{2} \log 1+\frac{1}{4} \log 2+\cdots+\frac{1}{8} \log 4+\cdots=\log 2=0.69315
Use of a concave utility function does not eliminate the paradox.
For example, the utility function $U(x)=x / \log (x+A)$, where $A>2$ is a constant, is strictly concave, strictly increasing, and infinitely differentiable yet the
As Menger (1967) pointed out in 1934 , the log, the square root, and many other, but not all, concave utility functions eliminate the original St. Petersburg paradox but it does not solve one where the payoffs grow faster than $2^{n}$. So if log is the utility function, one creates a new paradox by having the payoffs increase at least as fast as $\log$ reduces them so one still has an infinite sum for the expected utility. With exponentially growing payoffs one has
\frac{1}{2} \log \left(e^{1}\right)+\frac{1}{4} \log \left(e^{2}\right)+\cdots=\infty

统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Risk Aversion

We can break risk aversion, both absolute and relative, into categories of investors as Ziemba (2010) has done in his response to letters he received from Professor Paul A Samuelson $(2006,2007,2008$ ) (Table 1.1).

Ziemba named Ida after Ida May Fuller who paid $\$ 24.75$ into US social security and received her first social security check numbered 00-000-001 on January 31 , 1940 , the actual first such check. She lived in Ludlow, Vermont, to the age of 100 and collected $\$ 22,889$. Such are the benefits and risks of this system; see Bertoccchi, Schwartz, and Ziemba (2010) for more on this. Victor is named for the hedge fund trader Victor Niederhoffer who seems to alternate between very high returns and blowing up; see Ziemba and Ziemba (2007) for some but not all of his episodes. The other three investors are the overbetting Tom who is growth-security dominated in the sense of MacLean, Ziemba, and Blazenko (1992), our $E$ log investor Dick and Harriet, approximately half Kelly, who Samuelson says fits the data well. We agree that in practice, half Kelly is a toned down version of full Kelly that provides a lot more security to compensate for its loss in long-term growth. Figure 1.l shows this behavior in the context of Blackjack where Thorp first used Kelly strategies.
The edge for a successful card counter varies from about $-5$ to $+10 \%$ depending upon the favorability of the deck. By wagering more in favorable situations and less or nothing when the deck is unfavorable, an average weighted edge is about $2 \%$. An approximation to provide insight into the long-run behavior of a player’s fortune is to assume that the game is a Bernoulli trial with a probability of success $=0.51$ and probability of loss $1=0.49$.

Figure $1.1$ shows the relative growth rate $f \ln (1+p)+(1-f) \ln (1-p)$ versus the fraction of the investor’s wealth wagered, $f$. This is maximized by the Kelly $\log$ bet $f^{}-p-q-0.02$. The growth rate is lower for smaller and for larger bets than the Kelly bet. Superimposed on this graph is also the probability that the investor doubles or quadruples the initial wealth before losing half of this initial wealth. Since the growth rate and the security are both decreasing for $f>f^{}$, it follows that it is never advisable to wager more than $f^{*}$.

Observe that the $E$ log investor maximizes long-run growth and that the investor who wagers exactly twice this amount has a growth rate of zero plus the risk-free rate of interest. The fractional Kelly strategies are on the left and correspond to.

统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Understanding the Behavior of E log Strategies

There are many possible investment situations and $E$ log Kelly wagering is useful for some of them. Good uses of the strategy are in situations with many repeated bets that approximate an infinite sequence as in the Breiman, etc., theory. See the papers in MacLean, Thorp, and Ziemba $(2010 b)$ for such extensions; MacLean, Thorp, and Ziemba (2010a) for good and bad Kelly and fractional Kelly properties; and MacLean, Thorp, Zhao, and Ziemba $(2011)$ for simulations of typical behavior. Luenberger (1993) looks at long-run asymptotic behavior. Futures and options trading. sports hetting, including horseracing, are gond examples. The policies tend to non-diversify, plunge on a small number of the best assets, have a lot of volatility,

and produce more total wealth in the end than other strategies. Notable investors who use such strategies are Warren Buffett of Berkshire Hathaway, George Soros of the Quantum funds, and John Maynard Keynes who ran the King’s College Cambridge endowment from 1927 to 1945 . Figure $1.2 \mathrm{a}$, b shows the best and worst months for the Buffett and Soros funds. Observe that Buffett and Soros are asymptotically equivalent in both the left and right tails. Figure $1.3$ shows their wealth graphs. These correspond to typical Kelly behavior. Some Kelly advocates with a gambling background have produced nice smooth graphs such as those of Hong Kong racing guru Bill Benter, famed hedge fund traders Ed Thorp and Jim Simons; seẽ Fig. 1.4ã-d for thẽ vărioùs wealth grạhhs.

According to Ziemba (2005), Keynes was approximately an $80 \%$ Kelly bettor with a utility function of $-w^{-0.25}$. In Ziemba (2005) it is argued that Buffett and Soros are full Kelly bettors. They focus on long run wealth gains, not worrying about short term monthly losses. They tend to have few positions and try not to lose on any of them and not focusing on diversification. Table $1.2$ supports this showing their top 10 equity holdings on September 30,2008 . Soros is even more of a plunger with more than half his equity portfolio in just one position.

统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Using the Kelly Criterion for Investing


统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|William T. Ziemba and Leonard C. MacLean

凯利资本增长策略被定义为将您当前的财富分配给风险资产,从而使财富的预期对数逐期最大化。因此,它是一个单期静态计算,可以考虑交易成本和其他市场不完善性。对数效用可以追溯到 1738 年的丹尼尔·伯努利,他假设边际效用单调增加但随财富下降,具体来说,等于财富的倒数,在, 产生财富的效用你(在)=日志⁡在. 在此之前,假设决策是基于预期价值或线性效用做出的。这个想法导致边际效用下降或风险厌恶或凹度下降,这对投资决策至关重要。在他的拉丁文章节中,他还讨论了圣彼得堡悖论以及如何使用日志⁡在.

圣彼得堡悖论实际上起源于丹尼尔的表弟尼古拉斯·伯努利,他是巴塞尔大学的教授,丹尼尔也是该大学的数学教授。1708 年,尼古拉斯向皮埃尔·蒙莫特教授提交了五个重要问题。这个问题是为以下赌博支付多少钱:
一个公平的硬币12反复投掷正面的概率,直到出现正面,结束游戏。投资者支付C美元并获得回报2到−1有概率2−到为了到=1,2,…如果出现头部。因此,在每次连续输球后,假设没有出现正面,则赌注翻倍至2,4,8,…等。显然期望值是12+12+12+…. 或具有线性效用的无穷大。
Bell 和 Cover (1980) 认为圣彼得堡的赌局不惜任何代价都具有吸引力C,但投资者想要的更少,因为C→∞. 投资于圣彼得堡赌博的投资者财富比例始终为正,但随着某物⁡吨C作为C增加。其余的财富是现金。

伯努利提供了两种解决方案,因为他认为这场赌博的价值远低于无穷大。在第一个解决方案中,他任意设置了非常大收益的效用的限制。具体而言,假设任何超过 1000 万的金额等于224. 在有界效用假设下,期望值为
12(1)+14(2)+18(4)+⋯+(12)24(224)+(12)25(224)+(12)2H(224)+…=12+ 原本的 1=13.

例如,效用函数ü(X)=X/日志⁡(X+一种), 在哪里一种>2是一个常数,是严格凹的,严格递增的,并且是无限可微的,但
正如 Menger (1967) 在 1934 年指出的那样,对数、平方根和许多其他但不是全部的凹效用函数消除了原始的圣彼得堡悖论,但它并不能解决回报增长速度快于2n. 因此,如果 log 是效用函数,则通过让收益增加至少与日志减少它们,因此对于预期效用仍然有无限的总和。随着回报呈指数增长

统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Risk Aversion

正如 Ziemba (2010) 在回复 Paul A Samuelson 教授的来信时所做的那样,我们可以将绝对和相对的风险厌恶分为投资者类别(2006,2007,2008)(表 1.1)。

Ziemba 以 Ida May Fuller 的名字命名 Ida$24.75进入美国社会保障并于 1940 年 1 月 31 日收到了她的第一张社会保障支票,编号为 00-000-001,实际上是第一张这样的支票。她住在佛蒙特州的勒德洛,直到 100 岁$22,889. 这就是这个系统的好处和风险;有关这方面的更多信息,请参见 Bertoccchi、Schwartz 和 Ziemba (2010)。Victor 以对冲基金交易员 Victor Niederhoffer 的名字命名,他似乎在高回报和暴涨之间交替出现;请参阅 Ziemba 和 Ziemba (2007) 了解他的部分但不是全部剧集。其他三个投资者是过度投注的汤姆,他在 MacLean、Ziemba 和 Blazenko (1992) 的意义上是增长安全主导的,我们的和日志投资者迪克和哈里特,大约一半凯利,萨缪尔森说他很适合数据。我们同意,在实践中,半凯利是全凯利的低调版本,它提供了更多的安全性来弥补其在长期增长中的损失。图 1.l 显示了索普首次使用凯利策略的二十一点背景下的这种行为。
成功算牌的优势从大约−5到+10%取决于甲板的好感度。通过在有利的情况下多下注,而在牌组不利时少下注或不下注,平均加权优势约为2%. 深入了解玩家财富的长期行为的近似方法是假设游戏是具有成功概率的伯努利试验=0.51和损失概率1=0.49.

数字1.1显示相对增长率Fln⁡(1+p)+(1−F)ln⁡(1−p)与投资者下注的财富比例相比,F. 这是由凯利最大化日志赌 $f^{ }-pq-0.02.吨H和Gr这在吨Hr一种吨和一世s一世这在和rF这rs米一种一世一世和r一种ndF这r一世一种rG和rb和吨s吨H一种n吨H和到和一世一世是b和吨.小号你p和r一世米p这s和d这n吨H一世sGr一种pH一世s一种一世s这吨H和pr这b一种b一世一世一世吨是吨H一种吨吨H和一世nv和s吨这rd这你b一世和s这rq你一种dr你p一世和s吨H和一世n一世吨一世一种一世在和一种一世吨Hb和F这r和一世这s一世nGH一种一世F这F吨H一世s一世n一世吨一世一种一世在和一种一世吨H.小号一世nC和吨H和Gr这在吨Hr一种吨和一种nd吨H和s和C你r一世吨是一种r和b这吨Hd和Cr和一种s一世nGF这rf>f^{ },一世吨F这一世一世这在s吨H一种吨一世吨一世sn和v和r一种dv一世s一种b一世和吨这在一种G和r米这r和吨H一种nf^{*}$。


统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考|Understanding the Behavior of E log Strategies

有许多可能的投资情况和和log Kelly 投注对他们中的一些人很有用。该策略的良好用途是在许多重复投注的情况下,这些投注近似于 Breiman 等理论中的无限序列。参见 MacLean、Thorp 和 Ziemba 中的论文(2010b)对于此类扩展;MacLean、Thorp 和 Ziemba (2010a) 的好坏凯利属性和分数凯利属性;和 MacLean、Thorp、Zhao 和 Ziemba(2011)用于模拟典型行为。Luenberger (1993) 着眼于长期渐近行为。期货和期权交易。包括赛马在内的体育运动就是例子。政策倾向于非多元化,在少数最好的资产上暴跌,波动性很大,

并最终产生比其他策略更多的总财富。使用这种策略的著名投资者包括伯克希尔哈撒韦公司的沃伦巴菲特、量子基金的乔治索罗斯和 1927 年至 1945 年管理剑桥国王学院捐赠基金的约翰梅纳德凯恩斯。数字1.2一种, b 显示巴菲特和索罗斯基金的最佳和最差月份。观察巴菲特和索罗斯在左尾和右尾上是渐近等价的。数字1.3显示他们的财富图。这些对应于典型的凯利行为。一些具有赌博背景的凯利倡导者制作了漂亮的平滑图,例如香港赛车大师比尔本特、著名的对冲基金交易员埃德索普和吉姆西蒙斯;seẽ 图 1.4ã-d 表示 thẽ vărioùs 财富 grạhhs。

根据 Ziemba (2005),凯恩斯大约是80%具有效用函数的凯利投注者−在−0.25. 在 Ziemba (2005) 中,有人认为巴菲特和索罗斯是完全的凯利投注者。他们专注于长期的财富收益,而不是担心短期的每月损失。他们往往很少有头寸,并尽量不输给任何一个,也不专注于多元化。桌子1.2支持这显示他们在 2008 年 9 月 30 日持有的前 10 名股票。索罗斯甚至更像是一个暴跌者,他一半以上的股票投资组合只在一个头寸上。

统计代写|金融中的随机方法作业代写Stochastic Methods in Finance代考 请认准statistics-lab™

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在概率论概念中,随机过程随机变量的集合。 若一随机系统的样本点是随机函数,则称此函数为样本函数,这一随机系统全部样本函数的集合是一个随机过程。 实际应用中,样本函数的一般定义在时间域或者空间域。 随机过程的实例如股票和汇率的波动、语音信号、视频信号、体温的变化,随机运动如布朗运动、随机徘徊等等。


贝叶斯统计概念及数据分析表示使用概率陈述回答有关未知参数的研究问题以及统计范式。后验分布包括关于参数的先验分布,和基于观测数据提供关于参数的信息似然模型。根据选择的先验分布和似然模型,后验分布可以解析或近似,例如,马尔科夫链蒙特卡罗 (MCMC) 方法之一。贝叶斯统计概念及数据分析使用后验分布来形成模型参数的各种摘要,包括点估计,如后验平均值、中位数、百分位数和称为可信区间的区间估计。此外,所有关于模型参数的统计检验都可以表示为基于估计后验分布的概率报表。





随着AI的大潮到来,Machine Learning逐渐成为一个新的学习热点。同时与传统CS相比,Machine Learning在其他领域也有着广泛的应用,因此这门学科成为不仅折磨CS专业同学的“小恶魔”,也是折磨生物、化学、统计等其他学科留学生的“大魔王”。学习Machine learning的一大绊脚石在于使用语言众多,跨学科范围广,所以学习起来尤其困难。但是不管你在学习Machine Learning时遇到任何难题,StudyGate专业导师团队都能为你轻松解决。


基础数据: $N$ 个样本, $P$ 个变量数的单样本,组成的横列的数据表
变量定性: 分类和顺序;变量定量:数值
数学公式的角度分为: 因变量与自变量


随机过程,是依赖于参数的一组随机变量的全体,参数通常是时间。 随机变量是随机现象的数量表现,其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值(如1秒,5分钟,12小时,7天,1年),因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中,往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录,以得到其自身发展的规律。


多元回归分析渐进(Multiple Regression Analysis Asymptotics)属于计量经济学领域,主要是一种数学上的统计分析方法,可以分析复杂情况下各影响因素的数学关系,在自然科学、社会和经济学等多个领域内应用广泛。


MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中,其中问题和解决方案以熟悉的数学符号表示。典型用途包括:数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发,包括图形用户界面构建MATLAB 是一个交互式系统,其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题,尤其是那些具有矩阵和向量公式的问题,而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问,这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展,得到了许多用户的投入。在大学环境中,它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域,MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要,工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数(M 文件)的综合集合,可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。



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