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金融代写|固定收益与信贷代写Fixed Income and Credit代考|Self-financing Portfolios in Discrete Time

We consider a financial market living in discrete time on a filtered probability space $(\Omega, \mathcal{F}, P, \mathbf{F})$, so we are allowed trade at discrete points in time $t=0,1,2, \ldots$.

Basic Definitions
On the market we can trade in $N$ different assets with (adapted) price processes $S^1, \ldots, S^N$. We will use the following notation.
Definition $6.1$
\begin{aligned} S_n^i & =\text { the price of one unit of asset No. } i \text { at time } n, \ h_n^i & =\text { number of units of asset No. } i \text { bought at time } n, \ d_n^i & =\text { dividends from asset No. } i \text { at time } n, \ h_n & =\text { the portfolio }\left[h_n^1, \ldots, h_n^N\right], \ c_n & =\text { consumption at time } n, \ V_n & =\text { the value of the portfolio } h_{n-1} \text { at time } n .\end{aligned}
The interpretation of the dividend process $d$ is that if you are holding one unit of asset No. $i$ during the interval $[n-1, n]$, then you obtain the amount $d_n$ at time $n$.

$$d_{n+1}^i=\Delta D_n^i . \quad i=1, \ldots, N$$
The point of all this is that the concept of a cumulative dividend process can easily be extended to continuous time, and we now reformulate Proposition $6.5$ in the new terms as follows.

金融代写|固定收益与信贷代写Fixed Income and Credit代考|Self-financing Portfolios in Continuous Time

We now move to continuous time and consider a financial market with $N$ assets.
Definition $6.9$
$S_t^i=$ the price of one unit of asset No. $i$ at time $t$,
$h_t^i=$ number of units of asset No. $i$ held at time $t$,
$h_t=$ the portfolio $\left[h_t^1, \ldots, h_t^N\right]$,
$D_t^i=$ the cumulative dividend process for asset No. $i$,
$c_t=$ consumption rate at time $n$,
$V_t=$ the value of the portfolio $h_t$ at time tn.
Remark 6.3.1 There are two important differences from the discrete time definitions, and these concern consumption and dividends:

In discrete time, $c_n$ denotes consumption at time $n$, so it is measured in dollars. The consumption rate $c_t$ above, on the other hand, is measured in dollars per unit time, so in continuous time we have a continuous flow of consumption, where the dollar value of consumption over an infinitesimal interval $[t, t+d t]$ is given by $c_t d t$. It is possible to include “impulse consumption” where you get a lump sum of money at a specific point in time, but this would require a more general stochastic integration theory, so we do not include this.

We assume that the cumulative dividend process $D^i$ has a stochastic Itô differential, so in particular it has continuous trajectories. We thus exclude discrete dividends. To include discrete dividends would require a more general stochastic integration theory involving jump processes, and we would also have to change some of the definitions above.

金融代写|固定收益与信贷代写Fixed Income and Credit代考|Self-financing Portfolios in Discrete Time

$S_n^i=$ the price of one unit of asset No. $i$ at time $n, h_n^i \quad=$ number of units of asset No. $i$ bought at time $n, d_n^i=$ dividends from asset No. $i$ at ti:

$$d_{n+1}^i=\Delta D_n^i . \quad i=1, \ldots, N$$

金融代写|固定收益与信贷代写Fixed Income and Credit代考|Self-financing Portfolios in Continuous Time

$S_t^i=$ 一单位资产编号的价格 $i$ 在时间 $t$,
$h_t^i=$ 资产编号单位数 $i$ 在时间举行 $t$,
$h_t=$ 投资组合 $\left[h_t^1, \ldots, h_t^N\right]$ ，
$D_t^i=$ 痹产昊甸计分红过程 $i$,
$c_t=$ 时间消岪率 $n$,
$V_t=$ 投资组合的价值 $h_t$ 在时间 $\mathrm{tn}$ 。

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|固定收益与信贷代写Fixed Income and Credit代考|Stochastic Calculus and the Itô Formula

Let $X$ be a stochastic process and suppose that there exists a real number $a$ and two adapted processes $\mu$ and $\sigma$ such that the following relation holds for all $t \geq 0$.
$$X_t=a+\int_0^t \mu_s d s+\int_0^t \sigma_s d W_s$$
where $a$ is some given real number. As usual $W$ is a Wiener process. To use a less cumbersome notation we will often write eqn (4.16) in the following form:
\begin{aligned} d X_t & =\mu_t d t+\sigma_t d W_t, \ X(0) & =a . \end{aligned}
In this case we say that $X$ has a stochastic differential given by (4.17) with an initial condition given by (4.18). It is important to note that the formal string $d X_t=\mu_t d t+\sigma_t d W_t$ has no independent meaning. It is simply a shorthand version of the expression (4.16) above. From an intuitive point of view the stochastic differential is, however, a much more natural object to consider than the corresponding integral expression. This is because the stochastic differential gives us the “infinitesimal dynamics” of the $X$-process, and as we have seen in Section $4.1$ both the drift term $\mu_t$ and the diffusion term $\sigma_t$ have a natural intuitive interpretation.

金融代写|固定收益与信贷代写Fixed Income and Credit代考|The Multidimensional Itô Formula

Let us now consider a vector process $X=\left(X^1, \ldots, X^n\right)^{\star}$, where the component $X^i$ has a stochastic differential of the form
$$d X_t^i=\mu_t^i d t+\sum_{j=1}^d \sigma_t^{i j} d W_t^j$$
and $W^1, \ldots, W^d$ are $d$ independent Wiener processes.
Defining the drift vector process $\mu$ by
$$\mu_t=\left[\begin{array}{c} \mu^1 \ \vdots \ \mu^n \end{array}\right],$$
the $d$-dimensional vector Wiener process $W$ by
$$W=\left[\begin{array}{c} W^1 \ \vdots \ W^d \end{array}\right]$$
and the $n \times d$-dimensional diffusion matrix process $\sigma_t$ by
$$\sigma=\left[\begin{array}{cccc} \sigma^{11} & \sigma^{12} & \ldots & \sigma^{1 d} \ \sigma^{21} & \sigma^{22} & \ldots & \sigma^{2 d} \ \vdots & \vdots & \ddots & \vdots \ \sigma^{n 1} & \sigma^{n 2} & \ldots & \sigma^{n d} \end{array}\right],$$
we may write the $X$-dynamics as
$$d X_t=\mu_t d t+\sigma_t d W_t$$

金融代写|固定收益与信贷代写Fixed Income and Credit代考|Stochastic Calculus and the Itô Formula

$$X_t=a+\int_0^t \mu_s d s+\int_0^t \sigma_s d W_s$$

$$d X_t=\mu_t d t+\sigma_t d W_t, X(0)=a .$$

金融代写|固定收益与信贷代写Fixed Income and Credit代考|The Multidimensional Itô Formula

$$d X_t^i=\mu_t^i d t+\sum_{j=1}^d \sigma_t^{i j} d W_t^j$$

$$\mu_t=\left[\mu^1 \vdots \mu^n\right]$$

$$W=\left[W^1: W^d\right]$$

$$d X_t=\mu_t d t+\sigma_t d W_t$$

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:固定收益与信贷, 数学代写, 金融代写

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|固定收益与信贷代写Fixed Income and Credit代考|The Model

We consider a financial market with $N$ different financial assets. These assets could in principle be almost anything, like bonds, stocks, options, or whatever financial instrument that is traded on a liquid market. The market only exists at the two points in time $t=0$ and $t=1$, and the price per unit of asset No. $i$ at time $t$ will be denoted by $S_t^i$. We thus have a price vector process $S_t, t=0,1$ and we will view the price vector as a column vector, i.e.
$$S_t=\left[\begin{array}{c} S_t^1 \ \vdots \ S_t^N \end{array}\right]$$
The randomness in the system is modeled by assuming that we have a finite sample space $\Omega=\left{\omega_1, \ldots, \omega_M\right}$ and that the probabilities $p_j=P\left(\omega_j\right), j=1, \ldots, M$ are all strictly positive. The price vector $S_0$ is assumed to be deterministic and known to us, but the price vector at time $t=1$ depends upon the outcome $\omega \in \Omega$, and $S_1^i\left(\omega_j\right)$ denotes the price per unit of asset No. $i$ at time $t=1$ if $\omega_j$ has occured.

金融代写|固定收益与信贷代写Fixed Income and Credit代考|Absence of Arbitrage

We now define a portfolio as an $N$ dimensional row vector $h=\left[h^1, \ldots, h^N\right]$ with the interpretation that $h^i$ is the number of units of asset No. $i$ that we buy at time $t=0$ and keep until time $t=1$.

Since we are buying the assets with deterministic prices at time $t=0$ and selling them at time $t=1$ at stochastic prices, the value process of our portfolio will be a stochastic process $V_t^h$ defined by
$$V_t^h=\sum_{i=1}^N h^i S_t^i=h S_t, \quad t=0,1$$
and in more detail we can write this as
$$V_t^h\left(\omega_i\right)=h S_t\left(\omega_i\right)=h d_i=(h D)_i .$$
There are various similar, but not equivalent, variations of the concept of an arbitrage portfolio. The standard one is the following.

Definition 3.1 The portfolio $h$ is an arbitrage portfolio if it satisfies the conditions
\begin{aligned} V_0^h & =0 \ P\left(V_1^h \geq 0\right) & =1 \ P\left(V_1^h>0\right) & >0 . \end{aligned}
In more detail we can write this as
$$\begin{gathered} V_0^h<0, \ V_1^h\left(\omega_i\right) \geq 0, \quad \text { for all } i=1, \ldots, M \ V_1^h\left(\omega_i\right)>0, \quad \text { for some } i=1, \ldots, M \end{gathered}$$

金融代写|固定收益与信贷代写Fixed Income and Credit代考|The Model

$$S_t=\left[S_t^1: S_t^N\right]$$

金融代与写固定收益与信贷代写Fixed Income and Credit代考|Absence of Arbitrage

$$V_t^h=\sum_{i=1}^N h^i S_t^i=h S_t, \quad t=0,1$$

$$V_t^h\left(\omega_i\right)=h S_t\left(\omega_i\right)=h d_i=(h D)_i .$$

$$V_0^h=0 P\left(V_1^h \geq 0\right) \quad=1 P\left(V_1^h>0\right)>0 .$$

$$V_0^h<0, V_1^h\left(\omega_i\right) \geq 0, \quad \text { for all } i=1, \ldots, M V_1^h\left(\omega_i\right)>0, \quad \text { for some } i=1, \ldots, M$$

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。