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## avatest™帮您通过考试

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Natural FV Processes Are Predictable

The main result of this section is to show that a process $A \in \mathbb{V}$ is natural if and only if it is predictable. To achieve this, we need to consider the right continuous filtration $\left(\mathcal{F}_{.}^{+}\right)$along with the given filtration.

Recall that we had observed in Corollary $4.5$ that a $\left(\mathcal{F}^{+}\right)$predictable process $f$ such that $f_0$ is $\mathcal{F}0$ measurable is $\left(\mathcal{F}{.}\right)$predictable.

In his work on decomposition of submartingales, P. A. Meyer had introduced a notion of natural increasing process. It was an ad hoc definition, given with the aim of showing uniqueness in the Doob-Meyer decomposition.

Definition 8.31 Let $A \in \mathbb{V}0$; i.e. $A$ is an adapted process with finite variation paths and $A_0=0$. Suppose $|A|$ is locally integrable where $|A|_t=\operatorname{Var}{[0, t]}(A) . A$ is said to be natural if for all bounded r.c.l.I. martingales $M$
$[M, A]$ is a local martingale.
Let $\mathbb{W}=\left{V \in \mathbb{V}0:|V|\right.$ is locally integrable where $\left.|V|_t=\operatorname{VaR}{[0, t]}(V)\right}$.
Remark 8.32 Let $A \in \mathbb{V}_0$ be such that $[A, A]$ is locally integrable. Since $\Delta[A, A]$ $=(\Delta A)^2$, it follows that $(\triangle A)$ is locally integrable and as a consequence $A$ is locally integrable and thus $A \in \mathbb{W}$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Decomposition of Semimartingales Revisited

In view of this identification of natural $\mathrm{FV}$ processes as predictable, we can recast Theorem $5.50$ as follows.
Theorem $8.38$ Let $X$ be a stochastic integrator such that
(i) $X_t=X_{t \wedge T}$ for all $t$.
(ii) $\mathrm{E}\left[\sup {s \leq T}\left|X_s\right|\right]<\infty$. (iii) $\mathrm{E}\left[[X, X]_T\right]<\infty$. Then $\mathrm{X}$ admits a decomposition $$X=M+A, M \in \mathbb{M}^2, A \in \mathbb{V}, A_0=0 \text { and } A \text { is predictable } .$$ Further, the decomposition (8.4.1) is unique. Proof Let $X=M+A$ be the decomposition in Theorem 5.50. As seen in Corollary 5.50, the process $A$ satisfies $A_t=A{t \wedge T}$. Since $\mathrm{E}\left[[M, A]_T\right]=0$, we have
$$\mathrm{E}\left[[X, X]_T\right]=\mathrm{E}\left[[M, M]_T\right]+\mathrm{E}\left[[A, A]_T\right]$$
and hence $\mathrm{E}\left[[A, A]_T\right]<\infty$ and so $A \in \mathbb{W}$. Thus $A$ satisfies conditions of Theorem $8.37$ and hence $A$ is predictable. For uniqueness, if $X=N+B$ is another decomposition with $N \in \mathbb{M}^2$ and $B \in \mathbb{V}$ and $B$ being predictable, then $M-N=B-A$ is a predictable process with finite variation paths which is also a martingale and hence by Theorem $8.29, M=N$ and $B=A$.

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Doob-Meyer Decomposition

$$0 \leq C_t \leq A_t \quad \forall t$$

$$U=D+B$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Square Integrable Martingales

$$M_t=\xi 1_{[\tau, \infty)}(t)$$

$$A_t=\mathrm{E}\left[\xi^2 \mid \mathcal{F} \tau-\right] 1[\tau, \infty)(t) .$$

$$M_t^2-A_t=\left(\xi^2-\mathrm{E}\left[\xi^2 \mid \mathcal{F}_\tau-\right]\right) 1[\tau, \infty)(t)$$

## 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。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Stochastic Differential Equations

Let us consider the stochastic differential equation (3.5.1) where instead of a Brownian motion as in Chap. 3, here $W=\left(W^1, W^2, \ldots, W^d\right)$ is a amenable semimartingale. The growth estimate (7.2.5) enables one to conclude that in this case too, Theorem $3.30$ is true and the same proof works essentially-using (7.2.5) instead of (3.4.4). Moreover, using random time change, one can conclude that the same is true even when $W$ is any continuous semimartingale. We will prove this along with some results on approximations to the solution of an SDE.

We are going to consider the following general framework for the SDE driven by continuous semimartingales, where the evolution from a time $t_0$ onwards could depend upon the entire past history of the solution rather than only on its current value as was the case in Eq. (3.5.1) driven by a Brownian motion.

Let $Y^1, Y^2, \ldots Y^m$ be continuous semimartingales w.r.t. the filtration $(\mathcal{F}$.). Let $Y=\left(Y^1, Y^2, \ldots Y^m\right)$. Here we will consider an SDE
$$d U_t=b(t, \cdot, U) d Y_t, \quad t \geq 0, \quad U_0=\xi_0$$
where the functional $b$ is given as follows. Recall that $\mathbb{C}d=\mathbb{C}\left([0, \infty), \mathbb{R}^d\right)$. Let $$a:[0, \infty) \times \Omega \times \mathbb{C}_d \rightarrow \mathrm{L}(d, m)$$ be such that for all $\zeta \in \mathbb{C}_d$, $(t, \omega) \mapsto a(t, \omega, \zeta)$ is an r.c.l.l. $(\mathcal{F}$.$) adapted process$ and there is an increasing r.c.l.l. adapted process $K$ such that for all $\zeta_1, \zeta_2 \in \mathbb{C}_d$, $$\sup {0 \leq s \leq t}\left|a\left(s, \omega, \zeta_2\right)-a\left(s, \omega, \zeta_1\right)\right| \leq K_t(\omega) \sup _{0 \leq s \leq t}\left|\zeta_2(s)-\zeta_1(s)\right| .$$
Finally, $b:[0, \infty) \times \Omega \times \mathbb{C}_d \rightarrow \mathrm{L}(d, m)$ be given by
$$b(s, \omega, \zeta)=a(s-, \omega, \zeta)$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Pathwise Formula for Solution of SDE

In this section, we will consider the SDE
$$d V_t=f(t-, H, V) d X_t$$
for an $\mathbb{R}^d$-valued process $V$ where $f:[0, \infty) \times \mathbb{D}r \times \mathbb{C}_d \mapsto \mathrm{L}(d, m), H$ is an $\mathbb{R}^r$-valued r.c.l.l. adapted process, $X$ is a $\mathbb{R}^m$-valued continuous semimartingale. Here $\mathbb{D}_r=\mathbb{D}\left([0, \infty), \mathbb{R}^r\right), \mathbb{C}_d=\mathbb{C}\left([0, \infty), \mathbb{R}^d\right)$. For $t<\infty, \zeta \in \mathbb{C}_d$ and $\gamma \in \mathbb{D}_r$, let $\gamma^t(s)=\gamma(t \wedge s)$ and $\zeta^t(s)=\zeta(t \wedge s)$. We assume that $f$ satisfies \begin{aligned} f(t, \gamma, \zeta) & =f\left(t, \gamma^t, \zeta^t\right), \quad \forall \gamma \in \mathbb{D}_r, \zeta \in \mathbb{C}_d, 0 \leq t<\infty \ t & \mapsto f(t, \gamma, \zeta) \text { is an r.c.l.l. function } \forall \gamma \in \mathbb{D}_r, \zeta \in \mathbb{C}_d \end{aligned} We also assume that there exists a constant $C_T<\infty$ for each $T<\infty$ such that $\forall \gamma \in \mathbb{D}_r, \zeta_1, \zeta_2 \in \mathbb{C}_d, 0 \leq t \leq T$ $$\left|f\left(t, \gamma, \zeta_1\right)-f\left(t, \gamma, \zeta_2\right)\right| \leq C_T\left(1+\sup {0 \leq s \leq t}|\gamma(s)|\right)\left(\sup _{0 \leq s \leq t}\left|\zeta_1(s)-\zeta_2(s)\right|\right)$$
As in Sect. 6.2, we will now obtain a mapping $\Psi$ that yields a pathwise solution to the SDE (7.4.1).

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Stochastic Differential Equations

$$d U_t=b(t, \cdot, U) d Y_t, \quad t \geq 0, \quad U_0=\xi_0$$

$$a:[0, \infty) \times \Omega \times \mathbb{C}d \rightarrow \mathrm{L}(d, m)$$ 对所有人来说 $\zeta \in \mathbb{C}_d,(t, \omega) \mapsto a(t, \omega, \zeta)$ 是一个 $r c |(\mathcal{F}$. adaptedprocess并且有一个越来越多的 rcll 适应过程K这样对于所 有人 $\zeta_1, \zeta_2 \in \mathbb{C}_d$ $$\sup 0 \leq s \leq t\left|a\left(s, \omega, \zeta_2\right)-a\left(s, \omega, \zeta_1\right)\right| \leq K_t(\omega) \sup {0 \leq s \leq t}\left|\zeta_2(s)-\zeta_1(s)\right| .$$

## 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。

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## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Quadratic Variation of a Square Integrable Martingale

The next lemma connects the quadratic variation map $\Psi$ and r.c.l.l. martingales.
Lemma 5.16 Let $\left(N_t, \mathcal{F}t\right)$ be an r.c.l.l. martingale such that $\mathrm{E}\left(N_t^2\right)<\infty$ for all $t>0$. Suppose there is a constant $C<\infty$ such that with $$\tau=\inf \left{t>0:\left|N_t\right| \geq C \text { or }\left|N{t-}\right| \geq C\right}$$
one has
$$N_t=N_{t \wedge \tau} .$$
Let
$$A_t(\omega)=\Psi(N .(\omega))(t) .$$
Then $\left(A_t\right)$ is an $\left(\mathcal{F}t\right)$ adapted r.c.l.l. increasing process such that $X_t:=N_t^2-A_t$ is also a martingale. Proof Let $\Psi_n(\gamma)$ and $t_i^n(\gamma)$ be as in the previous section. \begin{aligned} A_t^n(\omega) & =\Psi_n(N .(\omega))(t) \ \sigma_i^n(\omega) & =t_i^n(N .(\omega)) \ Y_t^n(\omega) & =N_t^2(\omega)-N_0^2(\omega)-A_t^n(\omega) \end{aligned} It is easy to see that for each $n,\left{\sigma_i^n: i \geq 1\right}$ are stopping times (see Theorem 2.46) and that $$A_t^n=\sum{i=0}^{\infty}\left(N_{\sigma_{i+1}^n \wedge t}-N_{\sigma_i^n \wedge t}\right)^2 .$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Square Integrable Martingales Are Stochastic Integrators

The main aim of this section is to show that square integrable martingales are stochastic integrators.

The treatment is essentially classical, as in Kunita-Watanabe [46], but with an exception. The role of $\langle M, M\rangle$-the predictable quadratic variation in the KunitaWatanabe treatment-is here played by the quadratic variation $[M, M]$.

Recall that $\mathbb{M}^2$ denotes the class of r.c.l.l. martingales $M$ such that $\mathrm{E}\left[M_t^2\right]<\infty$ for all $t<\infty$ with $M_0=0$.

Lemma 5.27 Let $M, N \in \mathbb{M}^2$ and $f, g \in \mathbb{S}$. Let $X=J_M(f)$ and $Y=J_N(g)$. Let $Z_t=X_t Y_t-\int_0^t f_s g_s d[M, N]s^\psi$. Then $X, Y, Z$ are martingales. Proof The proof is almost the same as proof of Lemma 3.10, and it uses $M_t N_t-$ $[M, N]_t^\psi$ is a martingale along with Theorem $2.59$, Corollary $2.60$ and Theorem $2.61$. Corollary 5.28 Let $M \in \mathbb{M}^2$ and $f \in \mathbb{S}$. Then $Y_t=\int_0^t f d M$ and $Z_t=\left(Y_t\right)^2-$ $\int_0^t f_s^2 d[M, M]_s^\psi$ are martingales and $$\mathrm{E}\left[\sup {0 \leq t \leq T}\left|\int_0^t f d M\right|^2\right] \leq 4 \mathrm{E}\left[\int_0^T f_s^2 d[M, M]_s^\nu\right] .$$
Proof Lemma $5.27$ gives $Y, Z$ are martingales. The estimate (5.4.1) now follows from Doob’s inequality.

Theorem 5.29 Let $M \in \mathbb{M}^2$. Then $M$ is a stochastic integrator. Further, for $f \in$ $\mathbb{B}(\widetilde{\Omega}, \mathcal{P})$, the processes $Y_t=\int_0^t f d M$ and $Z_t=Y_t^2-\int_0^t f_s^2 d[M, M]s^*$ are martingales, $[Y, Y]_t^\psi=\int_0^t f_s^2 d[M, M]_s^\psi$ and $$\mathrm{E}\left[\sup {0 \leq t \leq T}\left|\int_0^t f d M\right|^2\right] \leq 4 \mathrm{E}\left[\int_0^T f_s^2 d[M, M]s^\psi\right], \quad \forall T<\infty .$$ Proof Fix $T<\infty$. Suffices to prove the result for the case when $M_t=M{t \wedge T}$. The rest follows by localization. See Theorem $4.49$. Recall that $\widetilde{\Omega}=[0, \infty) \times \Omega$ and $\mathcal{P}$ is the predictable $\sigma$-field on $\widetilde{\Omega}$. Let $\mu$ be the measure on $(\widetilde{\Omega}, \mathcal{P})$ defined for $A \in \mathcal{P}$
$$\mu(A)=\int\left[\int_0^T 1_A(\omega, s) d[M, M]s^\psi(\omega)\right] d \mathrm{P}(\omega) .$$ Note that $$\mu(\widetilde{\Omega})=\mathrm{E}\left[[M, M]_T^{\nsim}\right]=\mathrm{E}\left[\left|M_T\right|^2\right]<\infty$$ and for $f \in \mathbb{B}(\tilde{\Omega}, \mathcal{P})$ the norm on $\mathbb{L}^2(\tilde{\Omega}, \mathcal{P}, \mu)$ is given by $$|f|{2, \mu}=\sqrt{\mathrm{E}\left[\int_0^T f_s^2 d[M, M]_s^\psi\right]}$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Quadratic Variation of a Square Integrable Martingale

\left 缺少或无法识别的分隔符

$$N_t=N_{t \wedge \tau} .$$

$$A_t(\omega)=\Psi(N .(\omega))(t) .$$

$$A_t^n(\omega)=\Psi_n(N .(\omega))(t) \sigma_i^n(\omega) \quad=t_i^n(N .(\omega)) Y_t^n(\omega)=N_t^2(\omega)-N_0^2(\omega)-A_t^n(\omega)$$

$$A_t^n=\sum i=0^{\infty}\left(N_{\sigma_{i+1}^n \wedge t}-N_{\sigma_2^n \wedge t}\right)^2 .$$

## 数学代写|随机微积分代写STOCHASTIC CALCULUS代考|Square Integrable Martingales Are Stochastic Integrators

$$\mu(A)=\int\left[\int_0^T 1_A(\omega, s) d[M, M] s^\psi(\omega)\right] d \mathrm{P}(\omega) .$$

$$\mu(\widetilde{\Omega})=\mathrm{E}\left[[M, M]_T^{\infty}\right]=\mathrm{E}\left[\left|M_T\right|^2\right]<\infty$$

$$|f| 2, \mu=\sqrt{\mathrm{E}\left[\int_0^T f_s^2 d[M, M]_s^\psi\right]}$$

## 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。

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## avatest™帮您通过考试

avatest™的各个学科专家已帮了学生顺利通过达上千场考试。我们保证您快速准时完成各时长和类型的考试，包括in class、take home、online、proctor。写手整理各样的资源来或按照您学校的资料教您，创造模拟试题，提供所有的问题例子，以保证您在真实考试中取得的通过率是85%以上。如果您有即将到来的每周、季考、期中或期末考试，我们都能帮助您！

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## 金融代写|投资组合代写Portfolio Theory代考|Portfolio Selection

This chapter is an introduction to the theory of portfolio selection, which together with capital asset pricing theory provides the foundation and the building blocks for the management of portfolios. The goal of portfolio selection is the construction of portfolios that maximize expected returns consistent with individually acceptable levels of risk. Using both historical data and investor expectations of future returns, portfolio selection uses modeling techniques to quantify “expected portfolio returns” and “acceptable levels of portfolio risk” and provides methods to select an optimal portfolio.

The theory of portfolio selection presented in this chapter, often referred to as mean-variance portfolio analysis or simply mean-variance analysis, is a normative theory. A normative theory is one that describes a standard or norm of behavior that investors should pursue in constructing a portfolio rather than a prediction concerning actual behavior.

Asset pricing theory goes on to formalize the relationship that should exist between asset returns and risk if investors behave in a hypothesized manner. In contrast to a normative theory, asset pricing theory is a positive theory-a theory that hypothesizes how investors behave rather than how investors should behave. Based on that hypothesized behavior of investors, a model that provides the expected return (a key input for constructing portfolios based on mean-variance analysis) is derived and is called an asset pricing model.

Together, portfolio selection theory and asset pricing theory provide a framework to specify and measure investment risk and to develop relationships between expected asset return and risk (and hence between risk and required return on an investment). However, it is critically important to understand that portfolio selection is a theory that is independent of any theories about asset pricing. The validity of portfolio selection theory does not rest on the validity of asset pricing theory.

It would not be an overstatement to say that modern portfolio theory has revolutionized the world of investment management. Allowing managers to quantify the investment risk and expected return of a portfolio has provided the scientific and objective complement to the subjective art of investment management. More importantly, whereas at one time the focus of portfolio management used to be the risk of individual assets, the theory of portfolio selection has shifted the focus to the risk of the entire portfolio. This theory shows that it is possible to combine risky assets and produce a portfolio whose expected return reflects its components, but with considerably lower risk. In other words, it is possible to construct a portfolio whose risk is smaller than the sum of all its individual parts!

## 金融代写|投资组合代写Portfolio Theory代考|Utility Function and Indifference Curves

There are many situations where entities (i.e., individuals and firms) face two or more choices. The economic “theory of choice” uses the concept of a utility function to describe the way entities make decisions when faced with a set of choices. A utility function assigns a (numeric) value to all possible choices faced by the entity. The higher the value of a particular choice, the greater the utility derived from that choice. The choice that is selected is the one that results in the maximum utility given a set of constraints faced by the entity.

In portfolio theory too, entities are faced with a set of choices. Different portfolios have different levels of expected return and risk. Typically, the higher the level of expected return, the larger the risk. Entities are faced with the decision of choosing a portfolio from the set of all possible risk-return combinations, where when they like return, they dislike risk. Therefore, entities obtain different levels of utility from different risk-return combinations. The utility obtained from any possible risk-return combination is expressed by the utility function. Put simply, the utility function expresses the preferences of entities over perceived risk and expected return combinations.
A utility function can be expressed in graphical form by a set of indifference curves. Exhibit $3.1$ shows indifference curves labeled $u_1, u_2$, and $u_3$. By convention, the horizontal axis measures risk and the vertical axis measures expected return. Each curve represents a set of portfolios with different combinations of risk and return. All the points on a given indifference curve indicate combinations of risk and expected return that will give the same level of utility to a given investor. For example, on utility curve $u_1$, there are two points $u$ and $u^{\prime}$, with $u$ having a higher expected return than $u^{\prime}$, but also having a higher risk. Because the two points lie on the same indifference curve, the investor has an equal preference for (or is indifferent to) the two points, or, for that matter, any point on the curve. The (positive) slope of an indifference curve reflects the fact that, to obtain the same level of utility, the investor requires a higher expected return in order to accept higher risk.

## 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。

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## 金融代写|投资组合代写Portfolio Theory代考|CAP-M and diversification

For any portfolio $P$
\begin{aligned} \sigma^2(P) & =w^{\prime} \Sigma w \ & =w^{\prime}\left(\underline{\beta} \underline{\beta^{\prime}} \sigma^2(R)+\Sigma_e\right) w \ & =\left(\sum_{i=1}^n w_i \beta_i\right)^2 \sigma^2(R)+\sum_{i=1}^n w_i^2 \sigma^2\left(e_i\right) \end{aligned}
Now let $w_i=\frac{1}{n}$. Then
$$\sigma^2(P)=\bar{\beta}^2 \sigma^2(R)+\frac{1}{n} \overline{\sigma^2(e .)}$$
and so
$$\sigma(P) \longrightarrow \bar{\beta} \sigma(R) .$$
This reaffirms that $\beta_i$ is a measure of the contribution of the $i^{\text {th }}$ security to the risk of the portfolio. $\beta_i \sigma(R)$ is called the market, or undiversifiable, risk of security $i . \sigma\left(e_i\right)$ is called the non-market risk, unsystematic risk, unique risk or residual risk of equity $i$. This risk is diversifiable.

## 金融代写|投资组合代写Portfolio Theory代考|The Sharpe-Lintner-Mossin CAP-M

The CAP-M is what is known as an equilibrium model. The market participants as a whole act to put the market into equilibrium.

A number of additional simplifying assumptions (over and above those of Markowitz) are made in the CAP-M which are thought to be not too far removed from reality, yet are useful in order to simplify (or even make possible) the derivation of the model. Of course, a set of such assumptions is necessary in any economic model. In this model, they are:

1. Short sales are allowed.
2. There is a risk free rate for lending and borrowing money. The rate is the same for lending and borrowing, and investors have any amount of credit.
3. There are no transaction costs in the buying and selling of capital assets.
4. Similarly, there are no income or capital gains taxes.
5. The market consists of all assets. (No assets are exclusively private property.)

## 金融代写|投资组合代写Portfolio Theory代考|CAP-M and diversification

$$\sigma^2(P)=\bar{\beta}^2 \sigma^2(R)+\frac{1}{n} \overline{\sigma^2(e .)}$$

$$\sigma(P) \rightarrow \bar{\beta} \sigma(R) .$$

## 金融代写|投资组合代写Portfolio Theory代考|The Sharpe-Lintner-Mossin CAPM

1. 允许卖空。
2. 资本咨的买卖没有交易成本。
3. 同样，也设有所得祝或洛本利得㙂。
4. 市场由所有咨产组成。 (没有资旁完全是私有财产）

## 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。

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## 金融代写|投资组合代写Portfolio Theory代考|The expected return and risk of a portfolio of assets

Suppose we have a portfolio with $n$ assets, the $i^{t h}$ of which delivers a return $R_{t, i}$ at time $t$. This return has a mean $\mu_{t, i}$ and a variance $\sigma_{t, i}^2$. Suppose the proportion of the value of the portfolio that asset $i$ makes up is $w_i\left(\right.$ so $\sum_{i=1}^n w_i=1$ ).

What is the mean and standard deviation of the return $R$ of the portfolio? All known values are assumed to be known at time $t$, and the $t$ will be implicit in what follows. We can suppress the subscript $t$ as long as we understand that all of the parameters are dynamic and we need to refresh the estimates on a daily basis.
$$\mu:=\mathbb{E}[R]=\mathbb{E}\left[\sum_{i=1}^n w_i R_i\right]=\sum_{i=1}^n w_i \mathbb{E}\left[R_i\right]=\sum_{i=1}^n w_i \mu_i$$
and
ance matrix. So, the return on the portfolio has
\begin{aligned} & \mathbb{E}[R]=w^{\prime} \mu \ & \sigma(R)=\sqrt{w^{\prime} \Sigma w} \end{aligned}

## 金融代写|投资组合代写Portfolio Theory代考|The benefits of diversification

Let us consider some special cases. Suppose the assets are all independent, in particular, they are uncorrelated, so $\rho_{i j}=\delta_{i j}$. ( $\delta_{i j}$ is the indicator function.) Then $\sigma^2(R)=\sum_{i=1}^n w_i^2 \sigma_i^2$. Suppose further that the portfolio is equally weighted, so $w_i=\frac{1}{n}$ for every $i$. Then
$$\sigma^2(R)=\sum_{i=1}^n \frac{1}{n^2} \sigma_i^2=\frac{1}{n} \sum_{i=1}^n \frac{\sigma_i^2}{n} \longrightarrow 0$$
as $n \longrightarrow \infty$. If we accept that variance is a measure of risk, then the risk goes to 0 as we obtain more and more assets.

Suppose now that the portfolio is equally weighted, but that the assets are not necessarily uncorrelated. Then
\begin{aligned} \sigma^2(R) & =\sum_{i=1}^n \sum_{j=1}^n \frac{1}{n^2} \sigma_{i j} \ & =\frac{1}{n} \sum_{i=1}^n \frac{\sigma_i^2}{n}+\frac{n-1}{n} \sum_{i=1}^n \sum_{j=1, j \neq i}^n \frac{\sigma_{i j}}{n(n-1)} \ & =\frac{1}{n} \overline{\sigma_i^2}+\frac{n-1}{n} \overline{\sigma_{i j, i \neq j}} \ & \longrightarrow \frac{\sigma_{i j, i \neq j}}{\longrightarrow} \text { as } \longrightarrow \infty \end{aligned}
The limit is the average covariance, which is a measure of the undiversifiable market risk.

## 金融代写|投资组合代写Portfolio Theory代考|The expected return and risk of a portfolio of assets

$$\mu:=\mathbb{E}[R]=\mathbb{E}\left[\sum_{i=1}^n w_i R_i\right]=\sum_{i=1}^n w_i \mathbb{E}\left[R_i\right]=\sum_{i=1}^n w_i \mu_i$$

$$\mathbb{E}[R]=w^{\prime} \mu \quad \sigma(R)=\sqrt{w^{\prime} \Sigma w}$$

## 金融代写|投资组合代写Portfolio Theory代考|The benefits of diversification

$$\sigma^2(R)=\sum_{i=1}^n \frac{1}{n^2} \sigma_i^2=\frac{1}{n} \sum_{i=1}^n \frac{\sigma_i^2}{n} \longrightarrow 0$$

$$\sigma^2(R)=\sum_{i=1}^n \sum_{j=1}^n \frac{1}{n^2} \sigma_{i j} \quad=\frac{1}{n} \sum_{i=1}^n \frac{\sigma_i^2}{n}+\frac{n-1}{n} \sum_{i=1}^n \sum_{j=1, j \neq i}^n \frac{\sigma_{i j}}{n(n-1)}=\frac{1}{n} \overline{\sigma_i^2}+\frac{n-1}{n} \overline{\sigma_{i j, i \neq j}} \quad \longrightarrow \frac{\sigma_{i j, i \neq j}}{\longrightarrow} \text { as } \longrightarrow \infty$$

## 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。

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## 金融代写|金融衍生品代写Financial Derivatives代考|CORRELATIONS

The discussion so far has centered on the estimation and forecasting of volatility. correlations also play a key role in the calculation of VaR. In this section, we show how correlation estimates can be updated in a similar way to volatility estimates. The correlation between two variables $X$ and $Y$ can be defined as
$$\frac{\operatorname{cov}(X, Y)}{\sigma_X \sigma_Y}$$
where $a x$ and $a Y$ are the standard deviation of $X$ and $Y$ and $\operatorname{cov}(X, Y)$ is the covariance between $X$ and $Y$. The covariance between $X$ and $Y$ is defined as
$$E\left[\left(X-f i_X\right)\left{Y-f i_Y\right}\right]$$
where $f i_X$ and $f i_Y$ are the means of $X$ and $K$, and $E$ denotes the expected value. Although it is easier to develop intuition about the meaning of a correlation than it is for a covariance, it is covariances that are the fundamental variables of our analysis. Define $x t$ and $y,-$ as the percentage changes in $X$ and $Y$ between the end of day $i-1$ and the end of day $i$ :
$$x_i=\frac{X_i-X_{i-1}}{X_{i-1}}, \quad v_i=\frac{Y_i-Y_{i-1}}{Y_{i-1}}$$
where $X$, and $Y t$ are the values of $X$ and $Y$ at the end of day $i$. We also define:
axn : Daily volatility of variable $X$, estimated for day $n$
ayn : Daily volatility of variable $Y$, estimated for day $n$
covn : Estimate of covariance between daily changes in $X$ and $Y$, calculated on day $n$ Our estimate of the correlation between $X$ and $Y$ on day $n$ is
$$\frac{\operatorname{cov}n}{\sigma{x, n} \sigma_{y, n}}$$

## 金融代写|金融衍生品代写Financial Derivatives代考|Consistency Condition for Covariances

Once all the variances and covariances have been calculated, a variance-covariance matrix can beconstructed. When $/{ }^{\wedge} j$, the $(/, j)$ element of this matrix shows the covariance between variable I and variable $j$. When $;=j$, it shows the variance of variable i. Not all variance-covariance matrices are internally consistent. The condition for an $N$ $x$ Nvariance-covariance matrix, $Q$, to be internally consistent is

$$w^J Q \cdot w>0$$
for all $N \times 1$ vectors $w$, where $w T$ is the transpose of $w$. A matrix that satisfies this property is known as positive semidefinite.

## 金融代写|金融衍生品代写Financial Derivatives代考|CORRELATIONS

$$\frac{\operatorname{cov}(X, Y)}{\sigma_X \sigma_Y}$$

\left 缺少或无法识别的分隔符

$$x_i=\frac{X_i-X_{i-1}}{X_{i-1}}, \quad v_i=\frac{Y_i-Y_{i-1}}{Y_{i-1}}$$

axn : 变量的每日波动率 $X$, 估计一天 $n$
ayn：榇量的每日波动率 $Y$ ，估计一天 $n$
covn：每日变化之间的协方差估计 $X$ 和 $Y$ ，按日计算 $n$ 我们对两者之间相关性的估计 $X$ 和 $Y$ 在一天 $n$ 是
$$\frac{\operatorname{cov} n}{\sigma x, n \sigma_{y, n}}$$
Covariances

## 金融代写|金融衍生品代写Financial Derivatives代考|Consistency Condition for Covariances

$$w^J Q \cdot w>0$$

## 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。

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## 金融代写|金融衍生品代写Financial Derivatives代考|ESTIMATING VOLATILITIES

In this unit we explain how historical data can be used to produce estimates of the current andfuture levels of volatilities and correlations. The unit is relevant both to the calculation of valueat risk using the model-building approach and to the valuation of derivatives. When calculatingvalue at risk, we are most interested in the current levels of volatilities and correlations because weare assessing possible changes in the value of a portfolio over a very short period of time. Whenvaluing derivatives, forecasts of volatilities and correlations over the whole life of the derivative areusually required.The unit considers models with imposing names such as exponentially weighted movingaverage (EWMA), autoregressive conditional heteroscedasticity $(\mathrm{ARCH})$, and generalized autoregressive conditional heteroscedasticity (GARCH). The distinctive feature of the models is thatthey recognize that volatilities and correlations are not constant. During some periods a particularvolatility or correlation may be relatively low, whereas during other periods it may be relativelyhigh. The models attempt to keep track of the variations in the volatility or correlation throughtime. $\sigma^2$
Consider a time series of returns $r t+i, i=1, \cdots, \tau$ and $T=t+\tau$, the sample variance,
$$\hat{\sigma}^2=\frac{1}{\tau-1} \sum_{i=1}^\tau\left(\gamma_{t+i}-\mu\right)^2$$
where $r t$ is the return at time $t$, and $\mu$ is the average return over the $\tau$-period, and $\tau=$ $\sqrt{\sigma^2}$ is the unconditional volatility for the period $t$ to $T$. If $T-t$ is e.g. a ten-year period and $t$ is measured in daily interval, then $b \sigma^2$ in (1) is the daily variance, $b \sigma^2 d$, over the ten-year period. If $t$ is measured in weekly interval, then $b \sigma^2$ in (1) is the weekly variance, $b \sigma^2 w$, over the ten-year period. Since variance is linear in time and can be aggregated but not standard deviation,

$$\hat{\sigma}w^2=5 \times \hat{\sigma}_d^2$$ with a multiplier of 5 since there are 5 trading days in a week. To derive volatility, which is often linked to the standard deviation, we have the weekly volatility \begin{aligned} \hat{\sigma}_w & =\sqrt{5 \times \hat{\sigma}_d^2} \ & =\sqrt{5} \times \hat{\sigma}_d \end{aligned} and daily volatility is simply $b \sigma d$. It is a well known fact that volatility does not remain constant through time, the conditional volatility, $\sigma t$, is a more relevant information for asset pricing and risk management at time $t$. So it is a common practice to break $T-t$ up into smaller superiors such that $$\begin{gathered} T-t=\left(T_n-T{n-1}\right)+\left(T_{n-1}-T_{n-2}\right)+\left(T_{n-3}-T_{n-3}\right)+\ldots+\left(T_1-t\right) \ =T_n+T_{n-1}+T_{n-2}+\ldots+T_1 \end{gathered}$$
and (1) becomes
$$\hat{\sigma}t^2=\frac{1}{T_j-1} \sum{i=1}^{T_j}\left(T_{t+i}-\mu\right)^2, \quad j=1, \ldots, n$$

## 金融代写|金融衍生品代写Financial Derivatives代考|USING SQUARED RETURN AS A PROXY FOR DAILY VOLATILITY

Volatility is a latent variable. Before high frequency data became widely available, many researchers resorted to using daily squared return, calculated from market closing prices, to proxy daily volatility. Lopez (2001) shows that $\varepsilon^{2 t}$ is an unbiased but extremely imprecise estimator of $\sigma^{2 t}$ due to its asymmetric distribution. Let
$$r_t=\mu+\varepsilon_t, \quad \varepsilon_t=\sigma_t z_t$$
If $r_t \sim N\left(0, \sigma_t^2\right)$, then $E\left(\left|r_t\right|\right)=\sigma_t \sqrt{2 / \pi}$. Hence $\hat{\sigma}_t=\frac{\left|r_t\right|}{\sqrt{2 / \pi}}$ if $r_t$ has conditional and $z_t \sim N(0,1)$. Then

$$E\left[\varepsilon_t^2 \mid \Phi_{t-1}\right]=\sigma_t^2 E\left[z_t^2 \mid \Phi_{t-1}\right]=\sigma_t^2$$
since $z_t^2 \sim \chi_{(1)}^2$. However, since the median of a $\chi_{(1)}^2$ distribution is $0.455$, is $\varepsilon_t^2$ less than $\frac{1}{2} \sigma_t^2$ more than $50 \%$ of the time. In fact
$$P_r\left(\varepsilon_t^2 \in\left[\frac{1}{2} \sigma_t^2, \frac{3}{2} \sigma_t^2\right]\right)=P_r\left(z_t^2 \in\left[\frac{1}{2}, \frac{3}{2}\right]\right)=0.2588,$$
which means that $\varepsilon^{2 t}$ is $50 \%$ greater or smaller than $\sigma^{2 t}$ nearly $75 \%$ of the time! Under the null hypothesis that rt in (4) is generated by a GARCH(1,1) process, Andersen and Bollerslev (1998) show that the population $R^2$ for the regression
$$\varepsilon_t^2=\alpha+\beta \hat{\sigma}_t^2+v_t$$
is equal to $k^{-1}$ where $k$ is the kurtosis of the standardized residuals, $z t$, and $k$ is finite. For conditional Gaussian error, the $R^2$ from a correctly specified GARCH $(1,1)$ model is bounded from above by 1/3. Christodoulakis and Satchell (1998) extend the results to include compound normals and the Gram-Charlier class of distributions and show that the mis-estimation offorecast performance is likely to be worsened by non-normality.

## 金融代写|金融衍生品代写Financial Derivatives代考|ESTIMATING VOLATILITIES

$$\hat{\sigma}^2=\frac{1}{\tau-1} \sum_{i=1}^\tau\left(\gamma_{t+i}-\mu\right)^2$$
$$\hat{\sigma} w^2=5 \times \hat{\sigma}d^2$$ $$\hat{\sigma}_w=\sqrt{5 \times \hat{\sigma}_d^2} \quad=\sqrt{5} \times \hat{\sigma}_d$$ 的信息.. 所以打破是一种常见的俼法 $T-t$ 上䍧到更小的上级，这样 $$T-t=\left(T_n-T n-1\right)+\left(T{n-1}-T_{n-2}\right)+\left(T_{n-3}-T_{n-3}\right)+\ldots+\left(T_1-t\right)=T_n+T_{n-1}+T_{n-2}+\ldots+T_1$$
(1) 变成
$$\hat{\sigma} t^2=\frac{1}{T_j-1} \sum i=1^{T_j\left(T_{t+i}-\mu\right)^2, \quad j=1, \ldots, n}$$

## 金融代写|金融衍生品代写Financial Derivatives代考|USING SQUARED RETURN AS A PROXY FOR DAILY VOLATILITY

$$r_t=\mu+\varepsilon_t, \quad \varepsilon_t=\sigma_t z_t$$

$$P_r\left(\varepsilon_t^2 \in\left[\frac{1}{2} \sigma_t^2, \frac{3}{2} \sigma_t^2\right]\right)=P_r\left(z_t^2 \in\left[\frac{1}{2}, \frac{3}{2}\right]\right)=0.2588,$$
Bollerslev (1998) 表朋总体 $R^2$ 对扣回忉
$$\varepsilon_t^2=\alpha+\beta \hat{\sigma}_t^2+v_t$$

## 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。

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## 金融代写|金融衍生品代写Financial Derivatives代考|Applications

Monte Carlo simulation tends to be numerically more efficient than other procedures when there are three or more stochastic variables. This is because the time taken to carry out a Monte Carlo simulation increases approximately linearly with the number of variables, whereas the time taken for most other procedures increases exponentially with the number of variables. One advantage of Monte Carlo simulation is that it can provide a standard error for the estimates that it makes. Another is that it is an approach that can accommodate complex payoffs and complex stochastic processes. It can also be used when the payoff depends on some function of the whole pathfollowed by a variable, not just its terminal value. As already noted, a limitation of the Monte Carlo simulation approach is that it is difficult to use it for non-European-style derivatives.

## 金融代写|金融衍生品代写Financial Derivatives代考|Calculating the Greek Letters

The Greek letters can be calculated using Monte Carlo simulation. Suppose that we are interested in the partial derivative of / with $x$, where / is the value of the derivative and $x$ is the value of an underlying variable or a parameter. First, Monte Carlo simulation is used in the usual way to calculate an estimate, $/$, for the value of the derivative. A small increase $S x$ is then made in the value of $x$, and a new value for the derivative, $/^$, is calculated in the same way as /. An estimate for the hedge parameter is given by $$\frac{\hat{\jmath}^-\hat{\jmath}}{S x}$$

In order to minimize the standard error of the estimate, the number of time intervals $N$, the random number streams, and the number of trials $M$ should be the same for calculating both / and /*.
Sampling through a Tree
Instead of implementing Monte Carlo simulation by randomly sampling from the stochastic process for an underlying variable, we can sample paths for the underlying variable using a binomial tree. Suppose we have a binomial tree where the probability of an “up” movement is 0.6. The procedure for sampling a random path through the tree is as follows. At each node, we sample a random number between zero and one. If the number is less than $0.4$, we take the down path. If it is greater than $0.4$, we take the up path. Once we have a complete path from the initial node to the end of the tree, we can calculate a payoff. This completes the first trial. A similar procedure is used to complete more trials. The mean of the payoffs is discounted at the risk-free rate to get an estimate of the value of the derivative.

## 金融代写|金融行生品代写Financial Derivatives代考|Calculating the Greek Letters

$$\frac{\hat{\jmath}^{-} \hat{\jmath}}{S x}$$

## 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。

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## 融代写|量化决策模型代写Quantitative Decision Models 代考| Pritchett’s Precious Time Pieces

Pritchett’s Precious Time Pieces
The company buys, sells, and repairs old clocks and clock parts. Rebuilt springs sell for $\$ 10$per unit. Fixed cost of equipment to build springs is$\$1,000$.
Variable cost for spring material is $\$ 5$per unit. $$s=10 \quad f=1,000 \quad v=5$$ Number of spring sets sold$=X$$$\text { Profits }=s X-f-v X=10 X-1000-5 X$$ If sales$=0$, profits$=-\$1,000$
If sales $=1,000$, profits $=[(10)(1,000)-1,000-(5)(1,000)]$ $=\$ 4,000$Pritchett’s Precious Time Pieces Companies are often interested in their break-even point (BEP). The BEP is the number of units sold that will result in$\$0$ profit.
$$0=s X-f-v X, \text { or } 0=(s-v) X-f$$
Solving for $X$, we have
$$\begin{gathered} f=(s-v) X \ X=\frac{f}{s-v} \end{gathered}$$
$$\text { BEP }=\frac{\text { Fixed cost }}{(\text { Selling price per unit })-(\text { Variable cost per unit) }}$$

## 金融代写|量化决策模型代写Quantitative Decision Models 代考|Pritchett’s Precious Time Pieces

Pritchett’s Precious Time Pieces
BEP for Pritchett’s Precious Time Pieces
$$B E P=\ 1,000 /(\ 10-\ 5)=200 \text { units }$$
Sales of less than 200 units of rebuilt springs will result in a loss

Sales of over 200 units of rebuilt springs will result in a profit
$$\text { BEP }=\frac{\text { Fixed cost }}{(\text { Selling price per unit })-(\text { Variable cost per unit })}$$

Bagels ‘R Us
Assume you are the new owner of Bagels R Us and you want to develop a mathematical model for your daily profits and breakeven point. Your fixed overhead is $\$ 100$per day and your variable costs are$0.50$per bagel (these are GREAT bagels). You charge$\$1$ per bagel.
\begin{tabular}{|c}
Profits = Revenue – Expenses \
\begin{tabular}{|l}
(Price per Unit) $\times$ \
(Number Sold)
\end{tabular} \
$-$ Fixed Cost $-($ Variable Cost/Unit) $\times$ (Number Sold) \
Profits $=\$ \mathbf{1}^*$Number Sold$-\mathbf{\$1 0 0}-\mathbf{\$ . 5 0} *$Number Sold \end{tabular} Bagels ‘$RUs \begin{aligned} &f=\ 100, s=\ 1, v=\ .50 \ &\text { BEP }=f I(s-v) \ &\text { BEP }=\ 100 /(\ 1-\ .50) \ &\text { BEP }=200 \text { units } \end{aligned} When the number of units sold is equal to 200 , the profit is 0 . ## 量化决策模型代写 ## 金融代写|量化决策模型代写Quantitative Decision Models 代考|Pritchett’s Precious Time Pieces Pritchett’s Precious Time Pieces 该公司购买、销售和修理日钟表和钟表雾件。重建的弹簧售价为\$10$ 每单位。

$$s=10 \quad f=1,000 \quad v=5$$

$$\text { Profits }=s X-f-v X=10 X-1000-5 X$$

Pritchett’s Precious Time Pieces

1
$-$ 固定成本 $($ 可变成本/单位 $) \times($ 售出数量 $) ।$

$$f=\ 100, s=\ 1, v=\ .50 \quad \mathrm{BEP}=f I(s-v) \mathrm{BEP}=\ 100 /(\ 1-\ .50) \quad \mathrm{BEP}=200 \text { units }$$

## 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。