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## 金融代写|金融衍生品代写Financial Derivatives代考|Standarded Black-Scholes Market Models

Re-visit to multiperiod market model

Assets: $S_{t+\Delta t}^i=S_t^i\left(1+R_t^i\right)$, or $S_{t+\Delta t}^i-S_t^i=S_t^i R_t^i$.

Randomness: $R_t^i$ are random other than $i=0$.

Assumption: market is arbitrage-free and frictionless.

Information at time $t$ : observed from the history $S_{v: v \leq t}^i$.

Self-financing portfolio $\sum_{j=0}^m \phi_{t-\Delta t}^j S_t^j=\sum_{j=0}^m \phi_t^j S_t^j$.

Equivalently, $X_{t+\Delta t}-X_t=\sum_{j=0}^m \phi_t^j\left(S_{t+\Delta t}^j-S_t^j\right)$.

When $\Delta t \rightarrow 0$, trading time spots go to be continuous.

$S_{t+\Delta t}^i-S_t^i \rightarrow d S_t^i, X_{t+\Delta t}-X_t \rightarrow d X_t$,

$R_t^i \leftarrow$ increment of some Brownian motion (BM).

Information: natural filtration of $S_t^i$, or of the BM.

A standard Black-Scholes market in continuous time

Assets: a risk-free asset (bond), and a risky asset (stock). Transaction is frictionless.

Bond price $S^0(t)$ satisfies $d S^0(t)=r S^0(t) d t, S^0(0)=1$;
Stock price $S(t)$ satisfies $d S(t)=S(t)[\mu d t+\sigma d W(t)], S(0)=s_0$.

Market randomness is described by a standard Brownian motion ${W(t)}_{t \geq 0}$ in a probability space $\left(\Omega, \mathcal{F},\left{\mathcal{F}t\right}{t \geq 0}, \mathbb{P}\right)$.

Market informatio $\left{\mathcal{F}t\right}{t \geq 0}$ is observed from the historical randomeness, i.e. $\mathcal{F}_t=\mathcal{F}_t^W$.

Constant market parameters: interest rate $r$, appreciate rate $\mu$; volatility $\sigma$.

Portfolio and wealth process

Monetary portfolio $\pi^0(t), \pi(t)$

$\pi^0(t)$ : capital in bond; $\pi(t)$ : capital in stock.

$\pi^0(t)$ and $\pi(t)$ must be $\mathcal{F}_t$-measurable.

Total wealth: $X(t)=\pi^0(t)+\pi(t)$.

Self-financing: $d X(t)=\frac{\pi^0(t)}{S^0(t)} d S^0(t)+\frac{\pi(t)}{S(t)} d S(t)$.

Wealth equation:
$$d X(t)=[r X(t)+\pi(t)(\mu-r)] d t+\pi(t) \sigma d W(t), \quad X(0)=x_0 .$$

$\pi^0(\cdot)$ is implied by $\pi(\cdot)$.

To make the equation well-defined, we need $\int_0^T|\sigma \pi(t)|^2 d t<+\infty$ and $\int_0^T|\pi(t)(\mu-r)| d t<+\infty$.

## 金融代写|金融衍生品代写Financial Derivatives代考|General Ito’s formula

We will heavily use the Itô’s formula.

Given standard BM $\left(W_1(t), \cdots, W_m(t)\right)$, define processes $X_1(t), \cdots, X_n(t)$ by $d X_i(t)=a_i(t) d t+\sum_{j=1}^m \sigma_{i, j}(t) d W_j(t)$.

For any deterministic $C^2$ function $f\left(x_1, \cdots, x_n\right)$, denote $A(t)=f\left(X_1(t), \cdots, X_n(t)\right)$. Then
\begin{aligned} d A(t)= & \sum_{i=1}^n \frac{\partial f}{\partial x_i}\left(X_t^1, \cdots, X_t^n\right) d X_t^i \ & +\frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \frac{\partial^2 f}{\partial x_i \partial x_j}\left(X_t^1, \cdots, X_t^n\right)\left[d X_i(t) \times d X_j(t)\right], \end{aligned}
where in $d X_i(t) \times d X_j(t)$, we use the convention
$$d t \times d t=0, d t \times d W_i(t)=0, d W_i(t) \times d W_j(t)=\mathbf{1}_{i=j} d t$$

Often-used applications of Itô’s formula

$W(t)$ is a 1-dimension standard Brownian motion.

Take $d X(t)=\mu(t) d t+\sigma(t) d W(t)$, $d Y(t)=\tilde{\mu}(t) d t+\tilde{\sigma}(t) d W(t)$ and $f(x, y)=x \times y$, then
\begin{aligned} & d[X(t) Y(t)]=X(t) d Y(t)+Y(t) d X(t)+\sigma(t) \tilde{\sigma}(t) d t \ & =[X(t) \tilde{\mu}(t)+Y(t) \mu(t)+\sigma(t) \tilde{\sigma}(t)] d t+[X(t) \tilde{\sigma}(t)+Y(t) \sigma(t)] d W(t) \text {. } \ & \end{aligned}

Take $d X(t)=\mu(t) d t+\sigma(t) d W(t), Y(t)=t$, (i.e. $\tilde{\mu}=1, \tilde{\sigma}=0$ ), then
\begin{aligned} d f(t, X(t)) & =\frac{\partial f}{\partial X}(t, X(t)) d X(t)+\frac{\partial f}{\partial t}(t, X(t)) d t+\frac{1}{2} \frac{\partial^2 f}{\partial X^2}(t, X(t)) \sigma(t)^2 d t \ & =\left[\frac{\partial f}{\partial t}+\frac{\partial f}{\partial X} \mu(t)+\frac{1}{2} \frac{\partial^2 f}{\partial X^2} \sigma^2(t)\right] d t+\frac{\partial f}{\partial X} \sigma(t) d W(t) . \end{aligned}

Replication and pricing
Definition 2 : A contingent claim $\xi$ paid at time $T$ is replicable if it can be replicated by some portfolio with the discounted wealth process $\bar{X}(\cdot)$ being lower bounded.

For any $\xi$ paid at time $T$, if it is replicated by a portfolio with lower bounded discounted wealth process $\bar{X}(\cdot)$, then its price at time $t$ should be $X(t)$.

Is any $\xi$ replicable?
Definition 3 (Completeness): A market is called complete if for any lower bounded $\mathcal{F}_T$-measurable r.v. $\xi$ is replicable.
Theorem 1: A standard Black-Scholes market is complete iff $\sigma \neq 0$.

A proof will be given later.

## 金融代写|金融衍生品代写Financial Derivatives代考|Standarded Black-Scholes Market Models

$S_{t+\Delta t}^i-S_t^i \rightarrow d S_t^i, X_{t+\Delta t}-X_t \rightarrow d X_t$
$R_t^i \leftarrow$ 一些布朗运动 (BM) 的增量。

$\pi^0(t)$ : 债券赕本； $\pi(t)$ : 存货。
$\pi^0(t)$ 和 $\pi(t)$ 一定是 $\mathcal{F}_t$-可衡量的。

$$d X(t)=[r X(t)+\pi(t)(\mu-r)] d t+\pi(t) \sigma d W(t), \quad X(0)=x_0 .$$
$\pi^0(\cdot)$ 隐含于 $\pi(\cdot)$.

## 金融代写|金融行生品代写Financial Derivatives代考|General Ito’s formula

$$d A(t)=\sum_{i=1}^n \frac{\partial f}{\partial x_i}\left(X_t^1, \cdots, X_t^n\right) d X_t^i \quad+\frac{1}{2} \sum_{i=1}^n \sum_{j=1}^n \frac{\partial^2 f}{\partial x_i \partial x_j}\left(X_t^1, \cdots, X_t^n\right)\left[d X_i(t) \times d X_j(t)\right],$$
$$d t \times d t=0, d t \times d W_i(t)=0, d W_i(t) \times d W_j(t)=\mathbf{1}_{i-j} d t$$

$d[X(t) Y(t)]=X(t) d Y(t)+Y(t) d X(t)+\sigma(t) \bar{\sigma}(t) d t \quad=[X(t) \bar{\mu}(t)+Y(t) \mu(t)+\sigma(t) \bar{\sigma}(t)] d t+[X(t) \bar{\sigma}(t)+Y(t) \sigma(t)] d W(t)$.

$d f(t, X(t))=\frac{\partial f}{\partial X}(t, X(t)) d X(t)+\frac{\partial f}{\partial t}(t, X(t)) d t+\frac{1}{2} \frac{\partial^2 f}{\partial X^2}(t, X(t)) \sigma(t)^2 d t \quad=\left[\frac{\partial f}{\partial t}+\frac{\partial f}{\partial X} \mu(t)+\frac{1}{2} \frac{\partial^2 f}{\partial X^2} \sigma^2(t)\right] d t+\frac{\partial f}{\partial X} \sigma(t) d W(t)$.

## 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。

Posted on Categories:数学代写, 衍生品, 金融代写, 金融衍生品

## 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|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。