### 金融代写|量化风险管理代写Quantitative Risk Management代考|FINC6023

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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (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

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Examples of Parametric Distributions

Elliptical distributions: this class of distributions includes the Gaussian distribution and the $t$-distribution.

• The random variable $X$ is said to have a Gaussian distribution if its density (with mean $\mu$ and variance $\sigma^{2}$ ) is such that
$$f_{X}(x)=\frac{1}{(2 \pi)^{1 / 2} \sigma} \exp \left(-\frac{1}{2 \sigma^{2}}(x-\mu)^{-2}\right) .$$
This distribution is symmetrical and decreases very quickly towards zero. When $\mu=0$ and $\sigma=1$, then its kurtosis is equal to 3 . This value is used as a benchmark to decide if any distribution has low tail behaviour (kurtosis less than 3 ) or high tail behaviour (kurtosis bigger than 3 ).
• The $t$-distribution density is proportional to:
$$f_{X}(x)=\frac{1}{1+\left(x^{2} / \nu\right)^{(v+1) / 2}} .$$
The $t$-distribution’s tail becomes heavier as $v$ increases ( $v$ is the number of degrees of freedom). This distribution is not easy to use in finance because the $v$ parameter is an integer, rather than a continuous parameter and limits the flexibility of its using.
• Other distributions can be used whose properties are complementary to the two previous distributions because they have more parameters. In finance, we are interested in heavy tailed distribution functions with high kurtosis because it is more prone to extreme values. For instance, the GED distribution can be considered. The density of a GED random variable normalised to have a mean of zero and a variance of one is given by:
$$f(z)=\frac{\operatorname{vexp}\left[-1 /\left.2||_{\lambda}\right|^{v}\right]}{\lambda 2^{1+1 / v} \Gamma(1 / v)},-\infty2, the distribution of z has thinner tails than the normal. This density appears more preferable to the Student- t distribution because v \in R^{+}. ## 金融代写|量化风险管理代写Quantitative Risk Management代考|Non-parametric Modelling for a Distribution The non-parametric setting avoids the uncertainty on the choice of parametric densities and permits limiting the errors due to estimation procedure applied to estimate the parameters of the densities (Silverman 2018). Suppose we are given independent identically distributed real-valued observations \left(X_{1}, \cdots, X_{n}\right) with density f. We estimate f at a grid of points x_{1}, \cdots, x_{M}, for any arbitrary M fixed, and in particular here, we focus on estimation at a single point x. If f is smooth in a small neighbourhood [x-h, x+h] of x, the following approximation can be obtained:$$
2 h \cdot f(x) \approx \int_{x-h}^{x+h} f(u) d u=P(X \in[x-h, x+h])
$$by the mean value theorem. The right-hand side of (4.1.14) can be approximated by counting the number of X_{i} ‘s in the small interval of length 2 h, and then dividing by n. This is an histogram estimator with bincentre x and bandwidth 2 h. Let K(u)= \frac{1}{2} I(|u| \leq 1), where I(.) is the indicator function taking the value 1 when the event is true and zero otherwise. Then, the histogram estimator can be written:$$
\hat{f}{h}(x)=n^{-1} \sum{i=1}^{n} K_{h}\left(x-X_{i}\right)
$$where K_{h}(.)=h^{-1} K_{h}(. / h). The expression (4.1.15) is called the kernel density estimator of f(x) with kernel K(u)=\frac{1}{2} I(|u| \leq 1) and bandwidth h. The step function kernel weights each observation inside the window equally, even though observations closer to x should possess better information than more distant ones. In addition the step function estimator is discontinuous in x, which is unattractive given the smoothness assumption on f. Both objectives can be satisfied by choosing a smoother “window function” K as kernel, i.e., one for which K(u) \rightarrow 0 as |u| \rightarrow 1. A kernel is a piecewise continuous function, symmetric around zero, integrating to one:$$
K(u)=K(-u) ; \int K(u) d u=1 .
$$## 金融代写|量化风险管理代写Quantitative Risk Management代考|Shift of a Distribution In order to shift the distribution F_{X} of a random variable X a distortion function g is applied to the cumulative distribution function F_{X}. Such function g is defined on [0,1] \rightarrow[0 ; 1] such that g(0)=0 and g(1)=1, and is a continuous increasing function (Wang and Young 1998; Wang 2000). Distortion functions arose from empirical observations that people do not evaluate risk as a linear function of the actual probabilities for different outcomes but rather as a nonlinear distortion function. It is used to transform the probabilities of the loss distribution to another probability distribution by re-weighting the original distribution. This transformation increases the weight given to desirable events and deflates others. Different distortions g have been proposed in the literature. Some functions are provided in Table 4.1, where the parameters k and \gamma represent the confidence level and the level of risk aversion. When g is a concave function, its first derivative g^{\prime} is an increasing function, g^{\prime}\left(S_{X}(x)\right) where S_{X}=P[X>x] is a decreasing function { }^{1} in x and g^{\prime}\left(S_{X}(x)\right) represents a weighted coefficient which discounts the probability of desirable events while loading the probability of adverse events. A first approach for the distortion operator is g_{\alpha}(u)=\Phi\left[\Phi^{-1}(u)+\alpha\right], where \Phi is the Gaussian cumulative distribution. In terms of risk measure this last function applies the same perspective of preference to quantify the risk associated with gain and risk. Thus, a risk manager evaluates the risk associated with the upside and downside risks with the same function g implying a symmetric consideration for the two effects due to the distortion. Moreover it induces the same confidence level for the losses and the gain which implies the same level of risk aversion associated with the losses and the gains. In Fig. 4.3 the impact on the logistic distribution of the previous distortion function introduced is presented. We remark that the distorted distribution is always symmetrical under this kind of distortion function, and we observe a shift in the mode of the initial distribution towards the left. To avoid the problem of symmetry, the following distortion function can be used: g_{i}(u)=u+k_{i}\left(u-u^{2}\right) for \left.\left.k \in\right] 0,1\right] et \forall i \in{1,2}. In a risk measure perspective one models losses and gains differently relatively to the values of the parameters k_{i}, i= 1,2 . Thus upside and downside risks are modelled in different ways. Nevertheless the calibration of the parameters k_{i}, i=1,2 is not easy. ## 量化风险管理代考 ## 金融代写|量化风险管理代写Quantitative Risk Management代考|Examples of Parametric Distributions 椭圆分布：这类分布包括高斯分布和吨-分配。 • 随机变量X如果它的密度（平均μ和方差σ2) 是这样的 FX(X)=1(2圆周率)1/2σ经验⁡(−12σ2(X−μ)−2). 这种分布是对称的，并且非常迅速地向零下降。什么时候μ=0和σ=1，则其峰度等于 3 。该值用作基准来确定任何分布是否具有低尾行为（峰度小于 3 ）或高尾行为（峰度大于 3 ）。 • 这吨-分布密度与： FX(X)=11+(X2/ν)(在+1)/2. 这吨-分布的尾巴变得更重在增加（在是自由度数）。这种分布在金融中并不容易使用，因为在parameter 是一个整数，而不是一个连续的参数，限制了它的使用灵活性。 • 可以使用其他分布，其性质与前两个分布互补，因为它们具有更多参数。在金融领域，我们对具有高峰度的重尾分布函数感兴趣，因为它更容易出现极端值。例如，可以考虑 GED 分布。归一化为均值为零且方差为 1 的 GED 随机变量的密度由下式给出：$$
f(z)=\frac{\operatorname{vexp}\left[-1 /\left.2|| _{\lambda}\right|^{v}\right]}{\lambda 2^{1+1 / v} \Gamma(1 / v)},-\infty2,吨H和d一世s吨r一世b在吨一世○n○F和H一个s吨H一世nn和r吨一个一世ls吨H一个n吨H和n○r米一个l.吨H一世sd和ns一世吨是一个pp和一个rs米○r和pr和F和r一个bl和吨○吨H和小号吨在d和n吨−吨d一世s吨r一世b在吨一世○nb和C一个在s和v \in R^{+}\$。

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Non-parametric Modelling for a Distribution

2H⋅F(X)≈∫X−HX+HF(在)d在=磷(X∈[X−H,X+H])

F^H(X)=n−1∑一世=1nķH(X−X一世)

ķ(在)=ķ(−在);∫ķ(在)d在=1.

## 有限元方法代写

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## MATLAB代写

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。