### 金融代写|利率建模代写Interest Rate Modeling代考|ACTL40004

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling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。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling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|利率建模代写Interest Rate Modeling代考|Arbitrage Pricing

Let $\tau>0$ be a time horizon, and let $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ be a probability space, where $\mathcal{F}{t \in[0, \tau]}$ is the augmented filtration and $\mathbf{P}$ is a real-world measure.

Let $\mathcal{M}=\left{S_{i t}\right}_{i \in \Lambda}$ denote a family of price processes of assets, where $\mathcal{M}$ may contain very wide classes of asset, for example, a stock market or a bonds market. We remark that each $S_{i t}$ is not assumed to be an Ito process. We denote by $\Lambda$ a set of indices of assets. For example, a finite set of numbers ${1, \cdots, n}$ can be used for a finite asset market, and a time interval $[0, \tau]$ can be used for the family of bonds with maturity $T \leq \tau$. In this context, $\mathcal{M}$ is referred to as a market.
For simplicity, we work with a finite asset market $\mathcal{M}=\left{S_{i t}\right}_{i=1, \cdots, n} .$ A trading strategy is an $\mathbf{R}^{n}$-valued process $\theta_{t}=\left(\theta_{1 t}, \cdots, \theta_{n t}\right)$, where each $\theta_{i t}$ represents a volume of asset $i$. Then, the value of the portfolio $p(t)$ at $t$ is given by
$$p(t)=\theta_{t} S_{t}=\sum_{i=1}^{n} \theta_{i t} S_{i t}$$
$\theta$ is said to be a self-financing trading strategy if
$$d p(t)=\theta_{t} d S_{t}$$
In the self-financing trading portfolio, we neither add nor withdraw money. The next example shows the meaning of (3.2).

## 金融代写|利率建模代写Interest Rate Modeling代考|Change of Num´eraire

This section presents the most basic argument in arbitrage theory. We show that the arbitrage price is not affected by choice of numéraire, and make an arbitrage pricing system that is more practical. First, we present Theorem $3.2 .1$, which gives an arbitrage-free condition in a martingale approach. Theorem 3.2.1(2) will be applied to constructing the HJM model in Chapter $4 .$ Additionally, Theorem 3.2.1(2) is generalized to Theorem 3.2.2, which will be used in constructing the LIBOR market model in Chapter $5 .$
Numéraire and numéraire measure
A numéraire process is a price process of a traded asset that takes positive value almost surely. For example, a risk-free bond can be used as a numéraire.
Consider the market $\mathcal{M}=\left{S_{i t}\right}_{i \in \Lambda}$, which appeared in the previous section. Let an arbitrary $i \in \Lambda$ be fixed, and define $z_{t}$ by
$$z_{t}=\xi_{t} \frac{S_{i t}}{S_{i 0}} \text {. }$$
From the assumption of an arbitrage-free market, $z_{t}$ is a P-martingale with $z_{0}=1$ and $z_{t}>0$ a.s. Let $\mathbf{P}{\mathbf{i}}$ denote a measure defined by the Radon-Nikodym derivative such that $$\frac{d \mathbf{P}{\mathbf{i}}}{d \mathbf{P}}=z_{\tau}\left(=\xi_{\tau} \frac{S_{i \tau}}{S_{i 0}}\right)$$
From (3.7), we have $z_{t} / S_{i t}=\xi_{t} / S_{i 0}$. For any arbitrary $j$, multiplying both sides of the above by $S_{j t}$ gives
$$z_{t} \frac{S_{j t}}{S_{i t}}=\frac{\xi_{t} S_{j t}}{S_{i 0}} \text {. }$$
The arbitrage-free assumption implies that the right side is a P-martingale. From Proposition 2.4.2, we see that $S_{j t} / S_{i t}$ is a $\mathbf{P}{\mathrm{i}}$-martingale for all $j$. Obviously, $\mathbf{P}{\mathbf{i}}$ is equivalent to $\mathbf{P}$. This is called the $S_{i}$ numéraire measure.
Accordingly, we have the following theorem.

## 金融代写|利率建模代写Interest Rate Modeling代考|Market Price of Risk

The previous section presented a general framework of arbitrage pricing without use of a Brownian motion. Next, we introduce another framework for a bond market where bond prices are represented by Ito processes. In this setting, we consider the market price of risk to define the arbitrage-free market.
Let $W_{t}$ be a $d$-dimensional $\mathbf{P}$-Brownian motion defined on $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$, and let $\left{B{i t}\right}_{i \in \Lambda}$ be a family of price processes of zero-coupon bonds. We denote by $\mathcal{B}=\left{B_{i t}\right}_{i \in \Lambda}$ the bond market. In this section, we assume that values $B_{i t}$ are represented by Ito processes such that
$$\frac{d B_{i t}}{B_{i}}=\mu_{i t} d t+v_{i t} d W_{t}, \quad i=1, \cdots, n,$$
where $\mu_{i t}$ is a one-dimensional adapted process, and $v_{i t}$ is a $d$-dimensional adapted process satisfying
$$\int_{0}^{T}\left|\mu_{i t}\right| d s<\infty, \quad \int_{0}^{T}\left|v_{i t}\right|^{2} d s<\infty$$ respectively. We sometimes denote these as $\mu_{i}$ and $v_{i}$, omitting the subscript t. From the Ito formula (2.66), we have
$$B_{i t}=B_{i 0} \exp \left{\int_{0}^{t}\left(\mu_{i}-\frac{\left|v_{i}\right|^{2}}{2}\right) d s+\int_{0}^{t} v_{i} d W_{s}\right}$$
for each $i$. Also, $\xi_{t}$ denotes an adapted process defined by
$$\frac{d \xi_{t}}{\xi_{t}}=-r_{t} d t-\varphi_{t} d W_{t}$$
with $\xi_{0}=1$, where $r_{t}$ is a one-dimensional adapted process, and $\varphi_{t}$ is a $d-$ dimensional adapted process such that $\int_{0}^{T}|r| d s<\infty$ and $\int_{0}^{T}|\varphi|^{2} d s<\infty$. We sometimes denote these as $r$ and $\varphi$, omitting the subscript $t$. From $(2.66), \xi_{t}$ is represented by
$$\xi_{t}=\exp \left{\int_{0}^{t}\left(-r-\frac{|\varphi|^{2}}{2}\right) d s-\int_{0}^{t} \varphi d W_{s}\right}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Arbitrage Pricing

p(吨)=θ吨小号吨=∑一世=1nθ一世吨小号一世吨
θ据说是一种自筹资金的交易策略，如果

dp(吨)=θ吨d小号吨

## 金融代写|利率建模代写Interest Rate Modeling代考|Change of Num´eraire

Numéraire 和 numéraire measure

d磷一世d磷=和τ(=Xτ小号一世τ小号一世0)

## 金融代写|利率建模代写Interest Rate Modeling代考|Market Price of Risk

d乙一世吨乙一世=μ一世吨d吨+在一世吨d在吨,一世=1,⋯,n,

∫0吨|μ一世吨|ds<∞,∫0吨|在一世吨|2ds<∞分别。我们有时将这些表示为μ一世和在一世, 省略下标 t。根据 Ito 公式 (2.66)，我们有

B_{i t}=B_{i 0} \exp \left{\int_{0}^{t}\left(\mu_{i}-\frac{\left|v_{i}\right|^{2} }{2}\right) d s+\int_{0}^{t} v_{i} d W_{s}\right}B_{i t}=B_{i 0} \exp \left{\int_{0}^{t}\left(\mu_{i}-\frac{\left|v_{i}\right|^{2} }{2}\right) d s+\int_{0}^{t} v_{i} d W_{s}\right}

dX吨X吨=−r吨d吨−披吨d在吨

\xi_{t}=\exp \left{\int_{0}^{t}\left(-r-\frac{|\varphi|^{2}}{2}\right) d s-\int_{ 0}^{t} \varphi d W_{s}\right}

## 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。