### 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Optimization

statistics-lab™ 为您的留学生涯保驾护航 在代写风险理论投资组合方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写风险理论投资组合代写方面经验极为丰富，各种代写风险理论投资组合相关的作业也就用不着说。

• 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|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|UNCONSTRAINED OPTIMIZATION

When there are no constraints imposed on the set of feasible solutions, we have an unconstrained optimization problem. Thus, the goal is to maximize or to minimize the objective function with respect to the function arguments without any limits on their values. We consider directly the $n$-dimensional case; that is, the domain of the objective function $f$ is the $n$-dimensional space and the function values are real numbers, $f: \mathbb{R}^{n} \rightarrow \mathbb{R}$. Maximization is denoted by
$$\max f\left(x_{1}, \ldots, x_{n}\right)$$
and minimization by
$$\min f\left(x_{1}, \ldots, x_{n}\right)$$
A more compact form is commonly used, for example
$$\min {x \in \mathbb{R}^{n}} f(x)$$ denotes that we are searching for the minimal value of the function $f(x)$ by varying $x$ in the entire $n$-dimensional space $\mathbb{R}^{n}$. A solution to equation (2.1) is a value of $x=x^{0}$ for which the minimum of $f$ is attained, $$f{0}=f\left(x^{0}\right)=\min {x \in \mathbb{R}^{\pi}} f(x) .$$ Thus, the vector $x{0}$ is such that the function takes a larger value than $f_{0}$ for any other vector $x$,
$$f\left(x^{0}\right) \leq f(x), x \in \mathbb{R}^{n}$$
Note that there may be more than one vector $x^{0}$ satisfying the inequality in equation (2.2) and, therefore, the argument for which $f_{0}$ is achieved may not be unique. If (2.2) holds, then the function is said to attain its global minimum at $x^{0}$. If the inequality in $(2.2)$ holds for $x$ belonging only to a small neighborhood of $x^{0}$ and not to the entire space $\mathbb{R}^{n}$, then the objective function is said to have a local minimum at $x^{0}$. This is usually denoted by
$$f\left(x^{0}\right) \leq f(x)$$

for all $x$ such that $\left|x-x^{0}\right|_{2}<\epsilon$ where $\left|x-x^{0}\right|_{2}$ stands for the Euclidean distance between the vectors $x$ and $x^{0}$,
$$\left|x-x^{0}\right|_{2}=\sqrt{\sum_{i=1}^{n}\left(x_{i}-x_{i}^{0}\right)^{2}}$$
and $\epsilon$ is some positive number. A local minimum may not be global as there may be vectors outside the small neighborhood of $x_{0}$ for which the objective function attains a smaller value than $f\left(x_{0}\right)$. Figure $2.2$ shows the graph of a function with two local maxima, one of which is the global maximum.

There is a connection between minimization and maximization. Maximizing the objective function is the same as minimizing the negative of the objective function and then changing the sign of the minimal value,
$$\max {x \in \mathbb{R}^{n}} f(x)=-\min {x \in \mathbb{R}^{n}}[-f(x)] .$$
This relationship is illustrated in Figure 2.1. As a consequence, problems for maximization can be stated in terms of function minimization and vice versa.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Minima and Maxima of a Differentiable Function

If the second derivatives of the objective function exist, then its local maxima and minima, often called generically local extrema, can be characterized.

Denote by $\nabla f(x)$ the vector of the first partial derivatives of the objective function evaluated at $x$,
$$\nabla f(x)=\left(\frac{\partial f(x)}{\partial x_{1}}, \ldots, \frac{\partial f(x)}{\partial x_{n}}\right) .$$
This vector is called the function gradient. At each point $x$ of the domain of the function, it shows the direction of greatest rate of increase of the function in a small neighborhood of $x$. If for a given $x$, the gradient equals a vector of zeros,
$$\nabla f(x)=(0, \ldots, 0)$$
then the function does not change in a small neighborhood of $x \in \mathbb{R}^{n}$. It turns out that all points of local extrema of the objective function are characterized by a zero gradient. As a result, the points yielding the local extrema of the objective function are among the solutions of the system of equations,
\mid \begin{aligned} &\frac{\partial f(x)}{\partial x_{1}}=0 \ &\cdots \ &\frac{\partial f(x)}{\partial x_{n}}=0 \end{aligned}
The system of equation $(2.3)$ is often referred to as representing the first-order condition for the objective function extrema. However, it is only a necessary condition; that is, if the gradient is zero at a given point in the $n$-dimensional space, then this point may or may not be a point of a local extremum for the function. An illustration is given in Figure 2.2. The top plot shows the graph of a two-dimensional function and the bottom plot contains the contour lines of the function with the gradient calculated at a grid of points. There are three points marked with a black dot that have a zero gradient. The middle point is not a point of a local maximum even though it has a zero gradient. This point is called a saddle point since the graph resembles the shape of a saddle in a neighborhood of it. The left and the right points are where the function has two local maxima corresponding to the two peaks visible on the top plot. The right peak is a local maximum that is not the global one and the left peak represents the global maximum.

This example demonstrates that the first-order conditions are generally insufficient to characterize the points of local extrema. The additional condition that identifies which of the zero-gradient points are points

of local minimum or maximum is given through the matrix of second derivatives,
$$H=\left(\begin{array}{cccc} \frac{\partial^{2} f(x)}{\partial x_{1}^{2}} & \frac{\partial^{2} f(x)}{\partial x_{1} \partial x_{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{1} \partial x_{e}} \ \frac{\partial^{2} f(x)}{\partial x_{2} \partial x_{1}} & \frac{\partial^{2} f(x)}{\partial x_{2}^{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{2} \partial x_{n}} \ \vdots & \vdots & \ddots & \vdots \ \frac{\partial^{2} f(x)}{\partial x_{n} \partial x_{1}} & \frac{\partial^{2} f(x)}{\partial x_{n} \partial x_{2}} & \cdots & \frac{\partial^{2} f(x)}{\partial x_{n}^{2}} \end{array}\right)$$

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Convex Functions

In section 2.2.1, we demonstrated that the first-order conditions are insufficient in the general case to describe the local extrema. However, when certain assumptions are made for the objective function, the first-order conditions can become sufficient. Furthermore, for certain classes of functions, the local extrema are necessarily global. Therefore, solving the first-order conditions, we obtain the global extremum.

A general class of functions with nice optimal properties is the class of convex functions. Not only are the convex functions easy to optimize but they have also important application in risk management. In Chapter 6, we discuss general measures of risk. It turns out that the property which guarantees that diversification is possible appears to be exactly the convexity

property. As a consequence, a measure of risk is necessarily a convex functional. ${ }^{1}$

Precisely, a function $f(x)$ is called a convex function if it satisfies the property: For a given $\alpha \in[0,1]$ and all $x^{1} \in \mathbb{R}^{n}$ and $x^{2} \in \mathbb{R}^{n}$ in the function domain,
$$f\left(\alpha x^{1}+(1-\alpha) x^{2}\right) \leq \alpha f\left(x^{1}\right)+(1-\alpha) f\left(x^{2}\right)$$
The definition is illustrated in Figure 2.3. Basically, if a function is convex, then a straight line connecting any two points on the graph lies “above” the graph of the function.

There is a related term to convex functions. A function $f$ is called concave if the negative of $f$ is convex. In effect, a function is concave if it

satisfies the property: For a given $\alpha \in[0,1]$ and all $x^{1} \in \mathbb{R}^{n}$ and $x^{2} \in \mathbb{R}^{n}$ in the function domain,
$$f\left(\alpha x^{1}+(1-\alpha) x^{2}\right) \geq \alpha f\left(x^{1}\right)+(1-\alpha) f\left(x^{2}\right) .$$
We use convex and concave functions in the discussion of the efficient frontier in Chapter 8 .

If the domain $D$ of a convex function is not the entire space $\mathbb{R}^{n}$, then the set D satisfies the property,
$$\alpha x^{1}+(1-\alpha) x^{2} \in D$$
where $x^{1} \in D, x^{2} \in D$, and $0 \leq \alpha \leq 1$. The sets that satisfy equation (2.6) are called convex sets. Thus, the domains of convex (and concave) functions should be convex sets. Geometrically, a set is convex if it contains the straight line connecting any two points belonging to the set. Rockafellar (1997) provides detailed information on the implications of convexity in optimization theory.
We summarize several important properties of convex functions:

• Not all convex functions are differentiable. If a convex function is two times continuously differentiable, then the corresponding Hessian defined in equation $(2.4)$ is a positive semidefinite matrix. ${ }^{2}$
• All convex functions are continuous if considered in an open set.
• The sublevel sets
$$L_{c}={x: f(x) \leq c}$$
where $c$ is a constant, are convex sets if $f$ is a convex function. The converse is not true in general. Section $2.2 .3$ provides more information about non-convex functions with convex sublevel sets.
• The local minima of a convex function are global. If a convex function $f$ is twice continuously differentiable, then the global minimum is obtained in the points solving the first-order condition,
$$\nabla f(x)=0 .$$
• A sum of convex functions is a convex function:
$$f(x)=f_{1}(x)+f_{2}(x)+\ldots+f_{k}(x)$$
is a convex function if $f_{i}, i=1, \ldots, k$ are convex functions.

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|UNCONSTRAINED OPTIMIZATION

F(X0)≤F(X),X∈Rn

F(X0)≤F(X)

|X−X0|2=∑一世=1n(X一世−X一世0)2

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Minima and Maxima of a Differentiable Function

∇F(X)=(∂F(X)∂X1,…,∂F(X)∂Xn).

∇F(X)=(0,…,0)

∣∂F(X)∂X1=0 ⋯ ∂F(X)∂Xn=0

H=(∂2F(X)∂X12∂2F(X)∂X1∂X2⋯∂2F(X)∂X1∂X和 ∂2F(X)∂X2∂X1∂2F(X)∂X22⋯∂2F(X)∂X2∂Xn ⋮⋮⋱⋮ ∂2F(X)∂Xn∂X1∂2F(X)∂Xn∂X2⋯∂2F(X)∂Xn2)

## 金融代写|风险理论投资组合代写Market Risk, Measures and Portfolio 代考|Convex Functions

F(一种X1+(1−一种)X2)≤一种F(X1)+(1−一种)F(X2)

F(一种X1+(1−一种)X2)≥一种F(X1)+(1−一种)F(X2).

• 并非所有凸函数都是可微的。如果一个凸函数是两次连续可微的，则等式中定义的相应 Hessian(2.4)是一个半正定矩阵。2
• 如果在开集中考虑，所有凸函数都是连续的。
• 子级集
大号C=X:F(X)≤C
在哪里C是一个常数，如果是凸集F是一个凸函数。反之亦然。部分2.2.3提供有关具有凸子水平集的非凸函数的更多信息。
• 凸函数的局部最小值是全局的。如果一个凸函数F是两次连续可微的，则在求解一阶条件的点中获得全局最小值，
∇F(X)=0.
• 凸函数之和是一个凸函数：
F(X)=F1(X)+F2(X)+…+Fķ(X)
是一个凸函数，如果F一世,一世=1,…,ķ是凸函数。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## 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。