ACTL90003 - 统计代写答疑辅导

标签： ACTL90003

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

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写利率建模Interest Rate Modeling这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

利率模型是指一种对利率的运动和演变进行建模的数学方法。它是一种基于市场风险的单因素短利率模型。瓦西克利率模型常用于经济学中，以确定利率在未来的移动方向。

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。

我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|ACTL40004

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.
Self-financing trading strategy
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
$$
\xi_{t}=\exp \left{\int_{0}^{t}\left(-r-\frac{|\varphi|^{2}}{2}\right) d s-\int_{0}^{t} \varphi d W_{s}\right}
$$

利率建模代考

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

让τ>0是一个时间范围，让(Ω,F吨∈[0,τ],磷)是一个概率空间，其中F吨∈[0,τ]是增强过滤和磷是现实世界的衡量标准。

让\mathcal{M}=\left{S_{it}\right}_{i \in \Lambda}\mathcal{M}=\left{S_{it}\right}_{i \in \Lambda}表示资产的价格过程族，其中米可能包含非常广泛的资产类别，例如股票市场或债券市场。我们注意到每个小号一世吨不假定为 Ito 过程。我们表示Λ一组资产指数。例如，一组有限的数字1,⋯,n可用于有限资产市场和时间间隔[0,τ]可用于到期债券家族吨≤τ. 在这种情况下，米被称为市场。
自筹资金交易策略
为简单起见，我们使用有限的资产市场\mathcal{M}=\left{S_{it}\right}_{i=1, \cdots, n} 。\mathcal{M}=\left{S_{it}\right}_{i=1, \cdots, n} 。交易策略是一种Rn价值过程θ吨=(θ1吨,⋯,θn吨), 其中每个θ一世吨代表资产数量一世. 然后，投资组合的价值p(吨)在吨是（谁）给的

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
在自筹资金的交易组合中，我们既不加钱也不提钱。下一个例子显示了 (3.2) 的含义。

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

本节介绍套利理论中最基本的论点。我们证明了套利价格不受计价选择的影响，并提出了一个更实用的套利定价体系。首先，我们提出定理3.2.1，这给出了鞅方法中的无套利条件。定理 3.2.1(2) 将应用于本章中 HJM 模型的构建4.此外，定理 3.2.1(2) 被推广到定理 3.2.2，将用于构建 LIBOR 市场模型5.
Numéraire 和 numéraire measure
计价过程是交易资产的价格过程，几乎可以肯定地取正值。例如，无风险债券可以用作计价债券。
考虑市场\mathcal{M}=\left{S_{it}\right}_{i \in \Lambda}\mathcal{M}=\left{S_{it}\right}_{i \in \Lambda}，出现在上一节中。让任意一个一世∈Λ固定，并定义和吨经过

和吨=X吨小号一世吨小号一世0.
根据无套利市场的假设，和吨是一个 P-鞅和0=1和和吨>0作为让磷一世表示由 Radon-Nikodym 导数定义的度量，使得

d磷一世d磷=和τ(=Xτ小号一世τ小号一世0)
从（3.7），我们有和吨/小号一世吨=X吨/小号一世0. 对于任何任意j，将上面的两边乘以小号j吨给

和吨小号j吨小号一世吨=X吨小号j吨小号一世0.
无套利假设意味着右侧是 P-鞅。从命题 2.4.2，我们看到小号j吨/小号一世吨是一个磷一世-所有人的马丁格尔j. 明显地，磷一世相当于磷. 这被称为小号一世numéraire 量度。
因此，我们有以下定理。

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

上一节介绍了不使用布朗运动的套利定价的一般框架。接下来，我们介绍另一个债券市场框架，其中债券价格由 Ito 过程表示。在这种情况下，我们考虑风险的市场价格来定义无套利市场。
让在吨做一个d维磷-布朗运动定义在(Ω,F吨∈[0,τ],磷)，然后让\left{B{it}\right}_{i \in \Lambda}\left{B{it}\right}_{i \in \Lambda}是零息债券的价格过程族。我们表示\mathcal{B}=\left{B_{it}\right}_{i \in \Lambda}\mathcal{B}=\left{B_{it}\right}_{i \in \Lambda}债券市场。在本节中，我们假设值乙一世吨由 Ito 过程表示，使得

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,
在哪里μ一世吨是一维适应过程，并且在一世吨是一个d维适应过程满足

∫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}
对于每个一世. 还，X吨表示由下式定义的适应过程

dX吨X吨=−r吨d吨−披吨d在吨
和X0=1，在哪里r吨是一维适应过程，并且披吨是一个d−尺寸适应过程，使得∫0吨|r|ds<∞和∫0吨|披|2ds<∞. 我们有时将这些表示为r和披, 省略下标吨. 从(2.66),X吨表示为

\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代考请认准statistics-lab™

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

金融工程是使用数学技术来解决金融问题。金融工程使用计算机科学、统计学、经济学和应用数学领域的工具和知识来解决当前的金融问题，以及设计新的和创新的金融产品。

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

术语广义线性模型（GLM）通常是指给定连续和/或分类预测因素的连续响应变量的常规线性回归模型。它包括多元线性回归，以及方差分析和方差分析（仅含固定效应）。

有限元方法代写

有限元方法（FEM）是一种流行的方法，用于数值解决工程和数学建模中出现的微分方程。典型的问题领域包括结构分析、传热、流体流动、质量运输和电磁势等传统领域。

有限元是一种通用的数值方法，用于解决两个或三个空间变量的偏微分方程（即一些边界值问题）。为了解决一个问题，有限元将一个大系统细分为更小、更简单的部分，称为有限元。这是通过在空间维度上的特定空间离散化来实现的，它是通过构建对象的网格来实现的：用于求解的数值域，它有有限数量的点。边界值问题的有限元方法表述最终导致一个代数方程组。该方法在域上对未知函数进行逼近。[1] 然后将模拟这些有限元的简单方程组合成一个更大的方程系统，以模拟整个问题。然后，有限元通过变化微积分使相关的误差函数最小化来逼近一个解决方案。

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。

随机分析代写

随机微积分是数学的一个分支，对随机过程进行操作。它允许为随机过程的积分定义一个关于随机过程的一致的积分理论。这个领域是由日本数学家伊藤清在第二次世界大战期间创建并开始的。

时间序列分析代写

随机过程，是依赖于参数的一组随机变量的全体，参数通常是时间。随机变量是随机现象的数量表现，其时间序列是一组按照时间发生先后顺序进行排列的数据点序列。通常一组时间序列的时间间隔为一恒定值（如1秒，5分钟，12小时，7天，1年），因此时间序列可以作为离散时间数据进行分析处理。研究时间序列数据的意义在于现实中，往往需要研究某个事物其随时间发展变化的规律。这就需要通过研究该事物过去发展的历史记录，以得到其自身发展的规律。

回归分析代写

多元回归分析渐进（Multiple Regression Analysis Asymptotics）属于计量经济学领域，主要是一种数学上的统计分析方法，可以分析复杂情况下各影响因素的数学关系，在自然科学、社会和经济学等多个领域内应用广泛。

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，工具箱允许您学习和应用专业技术。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。

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

金融代写|利率建模代写Interest Rate Modeling代考|ACTL90003

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写利率建模Interest Rate Modeling这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|ACTL90003

金融代写|利率建模代写Interest Rate Modeling代考|Random Variables

A random variable $X$ is defined as a real-valued measurable function on a sample space $\Omega$. We regard interest rates, bond prices, and stock prices on future days as random variables throughout this book.
The expectation of a random variable $X$ under measure $\mathbf{P}$ is defined as
$$
E[X]=\int_{w \in \Omega} X(w) \mathbf{P}(d w)
$$
The expectation is sometimes referred to as the mean. The variance of $X$ is defined as
$$
v(x)=\int_{w \in \Omega} X(w)^{2} \mathbf{P}(d w)
$$
The variance $V(X)$ is a measure of how widely $X$ varies from $E[X]$.
Expectation is linear on linear combinations of random variables. Let $Y$ be another random variable. As an example, for constants $a, b$, and $c$, the expectation of the linear combination $a+b X+c Y$ is given by
$$
E[a+b X+c Y]=a+b E[X]+c[Y]
$$
The variance of $a+b X+c Y$ is given by
$$
V(a+b X+c Y)=b^{2} V(X)+c^{2} V(Y)+2 b c \operatorname{Cov}(X, Y)
$$

where Cov is the covariance between $X$ and $Y$ and is defined as
$$
\operatorname{Cov}(X, Y)=\int_{w \in \Omega}(X(w)-E[X])(Y(w)-E[Y]) P(d w)
$$
The correlation coefficient $\rho(X, Y)$ is defined as
$$
\rho(X, Y)=\frac{\operatorname{Cov}(X, Y)}{\sqrt{V(X) V(Y)}}
$$
For any constants $a$ and $b$, it holds that
$$
\begin{aligned}
\rho(a X, b Y) &=\frac{\operatorname{Cov}(a X, b Y)}{\sqrt{V(a X) V(b Y)}}=\frac{a b \operatorname{Cov}(X, Y)}{\sqrt{a^{2} b^{2} V(X) V(Y)}} \
&=\rho(X, Y) .
\end{aligned}
$$
Hence, the correlation coefficient is invariant to the magnitudes of $a$ and $b$. The correlation coefficient can be used as a measure of the strength of the linear relationship between $X$ and $Y$. Naturally, it holds that $-1 \leq \rho \leq 1$.

金融代写|利率建模代写Interest Rate Modeling代考|Stochastic Process

For arbitrage pricing, the dynamics of market price change is represented by a stochastic process. To introduce the concept of a stochastic process, we develop a probability space such that stochastic processes are well defined in the space, where the key is a filtration used to define a stochastic process.
Filtration
For a fixed time $\tau>0$, let $\mathcal{F}{t \in[0, \tau]}$ denote an increasing subset of $\mathcal{F}$; that is, $\mathcal{F}{s} \subset \mathcal{F}{t}$ for $0 \leq s{t}$ is a $\sigma$-algebra. Under this definition, $\mathcal{F}{t \in[0, \tau]}$ is called an augmented filtration, or a filtration for short. Along these lines, $\left(\Omega, \mathcal{F}, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ is called a filtered probability space, or a probability space for short.
Without loss of generality, we may set $\mathcal{F}=\mathcal{F}{\tau}$, which means that $\left(\Omega, \mathcal{F}, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ can be represented by a triplet $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ for convenience. In particular, we use a filtration chosen such that it represents a flow of information; namely, $\mathcal{F}{t}$ comprises the information about all events observed up to time $t$. An example of such a filtration is given in Example 2.3.1.
Stochastic process
Recall the probability space introduced in Example 2.1.1, that is, the up and down moves of stock price. Stock price movement between days is uncertain and unpredictable, making it an example of a discrete-time stochastic process.
In a continuous-time setting, a stochastic process $X_{t}$ is defined as a set of random variables indexed by time $t, t \geq 0$ and defined on a probability space $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$. The process $X{t}$ is said to be continuous if almost all sample paths are continuous in $t$. Additionally, $X_{t}$ is said to be adapted if $X_{t}$ is $\mathcal{F}{t^{-}}$ measurable at an arbitrary time $t \in[0, \tau]$. This is called $\mathcal{F}{t}$-adapted for short. In this book, we work mostly with adapted processes.

金融代写|利率建模代写Interest Rate Modeling代考|Stochastic Integral

This section summarizes basic definitions and results for stochastic differential equations but omits proofs. For details and a rigorous description see any of Chung and Williams (1990), Karatzas and Shreve (1998), Öksendal (2003), and Shreve (2004).
Stochastic integral
Let $W_{t}$ be a Brownian motion under a measure $\mathbf{P}$. For a stochastic process $X_{t}$ satisfying $\int_{0}^{t}\left|X_{s}\right|^{2} d s<\infty$ a.s., the stochastic integral of $X_{t}$ with respect to $W_{t}$ is defined by
$$
\int_{0}^{t} X_{s} d W_{s}=\lim {\Delta \rightarrow 0} \sum{i=1}^{n-1} X_{t_{i}}\left{W_{t_{i+1}}-W_{t_{i}}\right}
$$
where $0=t_{0}<\cdots<t_{n}=t$ and $\Delta t=\max {0 \leq i \leq n-1}\left(t{i+1}-t_{i}\right)$.
The following properties are well known.
1) If $X_{t}$ is a deterministic process, that is, generated by a deterministic function on $t$, then it holds that
$$
E\left[\int_{0}^{t} X_{s} d W_{s}\right]=0
$$
2) $\int_{0}^{t} X_{s} d W_{s}$ has normal distribution with mean 0 and variance
$$
E\left[\left(\int_{0}^{t} X_{s} d W_{s}\right)^{2}\right]=\int_{0}^{t}\left|X_{s}\right|^{2} d s
$$
Generally, for a function $\sigma\left(t, X_{t}\right)$ with inputs $t$ and $X_{t}$, and regarding $\sigma\left(t, X_{t}\right)$ as a stochastic process, the stochastic integral is defined by
$$
\int_{0}^{t} \sigma\left(s, X_{s}\right) d W_{s}
$$

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|Random Variables

随机变量X定义为样本空间上的实值可测函数Ω. 在本书中，我们将未来几天的利率、债券价格和股票价格视为随机变量。
随机变量的期望X测量中磷定义为

和[X]=∫在∈ΩX(在)磷(d在)
期望有时被称为均值。的方差X定义为

在(X)=∫在∈ΩX(在)2磷(d在)
方差在(X)是衡量范围有多广的指标X变化于和[X].
期望在随机变量的线性组合上是线性的。让是是另一个随机变量。例如，对于常量一个,b，和C, 线性组合的期望一个+bX+C是是（谁）给的

和[一个+bX+C是]=一个+b和[X]+C[是]
的方差一个+bX+C是是（谁）给的

在(一个+bX+C是)=b2在(X)+C2在(是)+2bC这⁡(X,是)

其中 Cov 是之间的协方差X和是并定义为

这⁡(X,是)=∫在∈Ω(X(在)−和[X])(是(在)−和[是])磷(d在)
相关系数ρ(X,是)定义为

ρ(X,是)=这⁡(X,是)在(X)在(是)
对于任何常数一个和b, 它认为

ρ(一个X,b是)=这⁡(一个X,b是)在(一个X)在(b是)=一个b这⁡(X,是)一个2b2在(X)在(是) =ρ(X,是).
因此，相关系数对于一个和b. 相关系数可以用来衡量两者之间的线性关系的强度X和是. 自然地，它认为−1≤ρ≤1.

金融代写|利率建模代写Interest Rate Modeling代考|Stochastic Process

对于套利定价，市场价格变化的动态由一个随机过程表示。为了引入随机过程的概念，我们开发了一个概率空间，以便在空间中很好地定义随机过程，其中关键是用于定义随机过程的过滤。
过滤
固定时间τ>0，让F吨∈[0,τ]表示增加的子集F; 那是，Fs⊂F吨为了0≤s吨是一个σ-代数。在这个定义下，F吨∈[0,τ]被称为增强过滤，或简称过滤。沿着这些思路，(Ω,F,F吨∈[0,τ],磷)称为过滤概率空间，简称概率空间。
不失一般性，我们可以设F=Fτ，意思就是(Ω,F,F吨∈[0,τ],磷)可以用三元组表示(Ω,F吨∈[0,τ],磷)为了方便。特别是，我们使用选择的过滤器，使其代表信息流；即，F吨包含有关截至时间观察到的所有事件的信息吨. 例 2.3.1 给出了这种过滤的一个例子。
随机过程
回想示例 2.1.1 中介绍的概率空间，即股票价格的上下波动。几天之间的股票价格变动是不确定和不可预测的，这使其成为离散时间随机过程的一个例子。
在连续时间设置中，随机过程X吨被定义为一组按时间索引的随机变量吨,吨≥0并在概率空间上定义(Ω,F吨∈[0,τ],磷). 过程X吨如果几乎所有样本路径在吨. 此外，X吨据说是适应的，如果X吨是F吨−可在任意时间测量吨∈[0,τ]. 这就是所谓的F吨- 简称改编。在本书中，我们主要使用经过调整的流程。

金融代写|利率建模代写Interest Rate Modeling代考|Stochastic Integral

本节总结了随机微分方程的基本定义和结果，但省略了证明。有关详细信息和严格的描述，请参见 Chung 和 Williams (1990)、Karatzas 和 Shreve (1998)、Öksendal (2003) 和 Shreve (2004)。
随机积分
Let在吨是一个测度下的布朗运动磷. 对于随机过程X吨令人满意的∫0吨|Xs|2ds<∞作为，随机积分X吨关于在吨定义为

\int_{0}^{t} X_{s} d W_{s}=\lim {\Delta \rightarrow 0} \sum{i=1}^{n-1} X_{t_{i}}\left {W_{t_{i+1}}-W_{t_{i}}\right}\int_{0}^{t} X_{s} d W_{s}=\lim {\Delta \rightarrow 0} \sum{i=1}^{n-1} X_{t_{i}}\left {W_{t_{i+1}}-W_{t_{i}}\right}
在哪里0=吨0<⋯<吨n=吨和Δ吨=最大限度0≤一世≤n−1(吨一世+1−吨一世).
以下性质是众所周知的。
1) 如果X吨是一个确定性过程，即由一个确定性函数在吨，那么它认为

和[∫0吨Xsd在s]=0
2) ∫0吨Xsd在s具有均值为 0 且方差为正态分布

和[(∫0吨Xsd在s)2]=∫0吨|Xs|2ds
一般来说，对于一个函数σ(吨,X吨)带输入吨和X吨，并且关于σ(吨,X吨)作为一个随机过程，随机积分定义为

∫0吨σ(s,Xs)d在s

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金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

金融代写|利率建模代写Interest Rate Modeling代考|Nested Simulation

Posted on 2022年6月21日2022年6月21日 by statistics-lab

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我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|Nested Simulation

There are many exotic options whose price depends on the path of an underlying asset. If a structured product involves an exotic derivative, then the price of the product is also path-dependent. Since the pricing of complex products is not trivial, recombining trees and Monte Carlo simulation are employed for pricing.

When a portfolio contains exotic products such as those described above, it is difficult to price them for a future date, which is necessary for measuring VaR. Nested simulation is a computing technique for measuring VaR of a derivatives portfolio, which we briefly introduce below.

Let $T$ be a holding period for measuring VaR. We denote by $x_{t}$ the value of the risk factor at time $t$, and by $p_{T}$ the price of a derivative asset at $T$. As illustrated in Fig. $1.8$, the distribution of the risk factor $x_{T}$ at $T$ is simulated by a Monte Carlo method in the outer step. Then, we consider $x_{T}$ to represent a scenario in which VaR is to be measured. At each scenario $x_{T}$, the derivative is priced as $p_{T} \mid x_{T}$ by another Monte Carlo simulation starting at time $T$. This is the inner step. Once we obtain the distribution of $p_{T} \mid x_{T}$, we can compute the VaR in the manner described in Section 1.6.3 of this chapter.
A problem intrinsic to nested simulation is that the number of simulation becomes enormous. There are many studies on algorithms for accelerating nested simulation. As examples, see Devineau and Loisel (2009) and Gordy and Juneja (2010), among others.

金融代写|利率建模代写Interest Rate Modeling代考|Validity of VaR

VaR is widely used as a risk measure in regulatory guidelines in the banking sectors. However it is known that VaR at a particular confidence level does not tell us the loss beyond that level, although the validity of VaR has been studied mathematically and financially in many papers. In this section, we make some brief remarks about the properties of $\mathrm{VaR}$ as a downside risk measure and define expected shortfall as a coherent risk measure. We address these issues basically according to the Bank for International Settlement (BIS, 2004b) for tail risk and according to Artzner et al. (1999) for coherent risk measures.
For this purpose, we denote by $X$ the value of a portfolio at time $T$, and by $\rho$ a risk measure for $X$. Then, $\rho(X)$ represents the size of the risk of $X$ measured by $\rho$. Naturally, the portfolio is allowed to sell some options, which creates the possibility that $X$ will take a negative value. To avoid confusion, we note that a large positive $\rho(X)$ represents a larger risk than a small positive or negative $\rho(X)$. We remark that up to the previous section, risk measures have been defined in terms of the profit-loss measures of a portfolio, such as sensitivity, convexity, and VaR. In this section, we work with a risk measure for $X$ that does not represent profit or loss. In this context, we define a VaR of $V_{\alpha}$ at confidence level $1-\alpha$ over a holding period $T$ as
$$
V_{\alpha}(X)=-\inf {x \mid \mathbf{P}(X \leq x)>\alpha}
$$
Tail risk
To explain the tail risk, we consider two portfolios, whose prices at time $T$ are denoted by $A$ for one portfolio and $B$ for the other. We use $A$ and $B$ as the names of these portfolios. We assume that $A$ and $B$ have the same VaR at confidence level $1-\alpha$, so that $V_{\alpha}(A)=V_{\alpha}(B)$. In Fig. $1.9$, the solid line shows

the distribution of future value of $A$, and the dashed line shows that of $B$. We see that the downside risk of portfolio $B$ is larger than that of $A$. This shows that $V a R_{\alpha}$ does not capture the difference in downside risk between portfolios, as exemplified by $A$ and $B$; this risk is noted in BIS (2004b) as tail risk.

Under VaR regulations, risk managers will choose portfolio $B$ in preference to A because there is a larger opportunity for profit in B. Thus, VaR might mislead risk managers who optimally control their portfolios. For further argument along these lines, see Basak and Shapiro (2001) and Yamai and Yoshiba $(2002)$, among others.

金融代写|利率建模代写Interest Rate Modeling代考|Probability Space

Probability space
We denote by $\Omega$ a set of all possible outcomes, called the sample space, and an element $w \in \Omega$, called a sample. The family $\mathcal{F}$ is a collection of subsets of $\Omega$ and is assumed to be a $\sigma$-algebra. A stochastic event $A$ is represented as an element in $\mathcal{F}$.

The function $\mathbf{P}$ is a probability function defined on $\mathcal{F}$ such that $\mathbf{P}(\Omega)=1$ and $\mathbf{P}(\phi)=0 ;$ this is also called a probability measure. For $A \in \mathcal{F}, \mathbf{P}(\mathbf{A})$ indicates probability of event $A$ occurring. From these, we characterize a probability space by a triplet $(\Omega, \mathcal{F}, \mathbf{P})$.
Example 2.1: Sample space
In financial engineering, a sample $w \in \Omega$ is regarded as a history of up and down moves of a stock price observed in a fixed period. This makes $\Omega$ the set of up and down sequences of the stock price. If a period consists of two days $\left{t_{1}, t_{2}\right}$, then $\Omega$ will contain four samples and be given by
$$
\Omega={u u, u d, d u, d d}
$$
where we represent by $u$ and $d$ the up and down movements, respectively, of the stock price. Accordingly, $\mathcal{F}$ consists of all subsets of $\Omega$,
$$
\mathcal{F}={\phi, \Omega, u u, u d, d u, d d,{u u, u d},{u u, d d,}, \cdots}
$$

which means that $\mathcal{F}$ has $16\left(=2^{4}\right)$ events.
Real-world measure
In financial market analysis, the probability measure derived from observation of market prices is called the market measure, with many synonyms, such as historical measure, physical measure, actual measure, and real-world measure. We regard historical data as being observed under the real-world measure.
Recall the previous example. If our observation estimates that the probability of an up move is constant at $0.5$ and the down-move probability is $0.5$, then we have
$$
\mathbf{P}(u u)=\mathbf{P}(u d)=\mathbf{P}(d u)=\mathbf{P}(d d)=0.25
$$
Accordingly, for an arbitrary event $A \in \mathcal{F}$, it is possible to give $\mathbf{P}(A)$. This probability measure $\mathbf{P}$ is an example of the real-world measure.

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|Nested Simulation

有许多奇异期权的价格取决于标的资产的路径。如果结构性产品涉及奇异衍生品，那么产品的价格也是路径相关的。由于复杂产品的定价并非易事，因此采用重组树和蒙特卡罗模拟进行定价。

当投资组合包含如上所述的奇特产品时，很难为将来的日期定价，这对于衡量 VaR 是必要的。嵌套模拟是一种用于衡量衍生品投资组合 VaR 的计算技术，我们将在下面简要介绍。

让吨是衡量 VaR 的持有期。我们表示X吨风险因素在时间的价值吨，并由p吨衍生资产的价格吨. 如图所示。1.8, 风险因素的分布X吨在吨在外部步骤中通过蒙特卡洛方法进行模拟。那么，我们考虑X吨表示要测量 VaR 的情景。在每个场景X吨，衍生品定价为p吨∣X吨由另一个从时间开始的蒙特卡洛模拟吨. 这是内在的步骤。一旦我们得到分布p吨∣X吨，我们可以按照本章 1.6.3 节中描述的方式计算 VaR。
嵌套模拟的一个内在问题是模拟的数量变得巨大。关于加速嵌套仿真的算法有很多研究。例如，参见 Devineau 和 Loisel (2009) 以及 Gordy 和 Juneja (2010) 等。

金融代写|利率建模代写Interest Rate Modeling代考|Validity of VaR

VaR 被广泛用作银行业监管指南中的风险度量。然而，众所周知，特定置信水平的 VaR 并不能告诉我们超出该水平的损失，尽管在许多论文中已经对 VaR 的有效性进行了数学和财务研究。在本节中，我们对属性的一些简要说明在一个R作为下行风险衡量标准，并将预期短缺定义为连贯的风险衡量标准。我们基本上根据国际清算银行 (BIS, 2004b) 的尾部风险和 Artzner 等人的方法来解决这些问题。(1999) 用于连贯的风险措施。
为此，我们记为X投资组合当时的价值吨，并由ρ风险度量X. 然后，ρ(X)代表风险的大小X通过测量ρ. 自然地，投资组合被允许出售一些期权，这创造了以下可能性：X将取负值。为避免混淆，我们注意到一个大的正ρ(X)代表比小的正面或负面更大的风险ρ(X). 我们注意到，直到上一节，风险度量都是根据投资组合的损益度量来定义的，例如敏感性、凸性和 VaR。在本节中，我们使用风险度量X不代表利润或损失。在这种情况下，我们定义 VaR 为在一个在置信水平1−一个持有期吨作为

在一个(X)=−信息X∣磷(X≤X)>一个
尾部风险
为了解释尾部风险，我们考虑两个投资组合，它们的价格吨表示为一个对于一个投资组合和乙对于另一个。我们用一个和乙作为这些投资组合的名称。我们假设一个和乙在置信水平上具有相同的 VaR1−一个，以便在一个(一个)=在一个(乙). 在图。1.9，实线表示

未来价值的分布一个, 虚线表示乙. 我们看到投资组合的下行风险乙大于一个. 这表明在一个R一个没有捕捉投资组合之间下行风险的差异，例如一个和乙; 这种风险在 BIS (2004b) 中被称为尾部风险。

在 VaR 规定下，风险管理者将选择投资组合乙优先于 A，因为 B 有更大的获利机会。因此，VaR 可能会误导最佳控制其投资组合的风险经理。有关这些方面的进一步论点，请参见 Basak 和 Shapiro (2001) 以及 Yamai 和 Yoshiba(2002)，等等。

金融代写|利率建模代写Interest Rate Modeling代考|Probability Space

概率空间
我们表示为Ω一组所有可能的结果，称为样本空间和一个元素在∈Ω，称为样本。家庭F是子集的集合Ω并假定为σ-代数。随机事件一个表示为一个元素F.

功能磷是一个概率函数定义在F这样磷(Ω)=1和磷(φ)=0;这也称为概率测度。为了一个∈F,磷(一个)表示事件的概率一个发生。根据这些，我们用三元组来表征概率空间(Ω,F,磷).
例 2.1：样本空间
在金融工程中，样本在∈Ω被认为是在固定时期内观察到的股票价格上下波动的历史。这使得Ω股票价格的上涨和下跌序列的集合。如果一个时期由两天组成\left{t_{1}, t_{2}\right}\left{t_{1}, t_{2}\right}，然后Ω将包含四个样本并由下式给出

Ω=在在,在d,d在,dd
我们代表的地方在和d分别是股价的上涨和下跌。因此，F由所有子集组成Ω,

F=φ,Ω,在在,在d,d在,dd,在在,在d,在在,dd,,⋯

意思就是F有16(=24)事件。
真实世界测度
在金融市场分析中，通过观察市场价格得出的概率测度称为市场测度，有许多同义词，如历史测度、物理测度、实际测度和真实世界测度。我们认为历史数据是在现实世界的衡量标准下观察到的。
回想一下前面的例子。如果我们的观察估计上涨的概率是恒定的0.5向下移动的概率是0.5，那么我们有

磷(在在)=磷(在d)=磷(d在)=磷(dd)=0.25
因此，对于任意事件一个∈F, 可以给磷(一个). 这种概率测度磷是真实世界测量的一个例子。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

金融代写|利率建模代写Interest Rate Modeling代考|Covariance VaR Models

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写利率建模Interest Rate Modeling这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|Covariance VaR Models

金融代写|利率建模代写Interest Rate Modeling代考|One-factor model

The covariance VaR model assumes that the portfolio value is linear to the risk factor and normally distributed. Then, the key parameters are the mean and variance of the $\mathrm{P} \& \mathrm{~L}$ distribution, which are measured from historical data.
Let us consider a portfolio with a single risk factor $x$, the value of which is denoted by $p(x)$. From the linearity assumption, we can model $p(x)$ as $p(x)=a+b x$ for some positive constants $a$ and $b$. Let $\mu$ and $\sigma^{2}$ be, respectively, the mean and variance of $x$. For convenience, we assume that $p(\mu)$ is equal to the current value of the portfolio. It is known from the normality of the distribution that the $95 \%$ confidence level of $x$ is $\mu-1.645 \sigma$, and the $99 \%$ confidence level is $\mu-2.326 \sigma$. From this, we can see that the $95 \% \mathrm{VaR}$ is given by
$$
\begin{aligned}
95 \% V a R &={a+\mu b}-{a+b(\mu-1.645 \sigma)} \
&=1.645 \sigma b .
\end{aligned}
$$
Fig. $1.7$ may help in understanding this scheme. Similarly, we have
$$
99 \% V a R=2.326 \sigma b .
$$
When $p(x)$ is not linear but almost linear, that is, when $p(x) \approx a+b x$, we can approximate the VaR from (1.42) and (1.43).

金融代写|利率建模代写Interest Rate Modeling代考|Historical Simulation Models

In historical simulation models, the empirical distribution of the P\&L is directly constructed from historical data.

For simplicity, we consider a portfolio consisting of two assets, $A$ and $B$. For a holding period $\Delta T$, let $t_{1}, \cdots, t_{J+1}$ be a sequence of past days with $t_{i}-t_{i+1}=\Delta T$ and $t_{1}=0$. We let $p_{A}(i)$ and $p_{B}(i)$ denote the prices of $A$ and $B$, respectively, at $t_{i}$. Let $r_{A}(i)$ and $r_{B}(i)$ be the rates of return for $A$ and $B$, respectively. These are defined by
$$
\begin{aligned}
&r_{A}(i)=\frac{p_{A}(i)-p_{A}(i+1)}{p_{A}(i+1)} \
&r_{B}(i)=\frac{p_{B}(i)-p_{B}(i+1)}{p_{B}(i+1)}
\end{aligned}
$$
for $i=1, \cdots, J$.
For the present values $p_{A}(0)$ and $p_{B}(0)$, the $\mathrm{P} \& \mathrm{~L}$ distribution of the portfolio is built from the set
$$
\left{p_{A}(0) r_{A}(i)+p_{B}(0) r_{B}(i)\right}_{i=1, \cdots, J}
$$
as shown in Fig. 1.6. From this distribution, we obtain the VaR, with the process working according to the idea shown in Fig. 1.6. Even for a very large portfolio, we can measure the VaR in the same way.

It is a point in favor of this model that it does not require assumptions about neither the distribution of prices nor the linearity of assets. However, it is difficult to collect the sufficient historical data for this method. Even when there are sufficiently many data points from a long period of observation, some data might be too old to reasonably reflect future risk, which is needed for measuring VaR.

金融代写|利率建模代写Interest Rate Modeling代考|Monte Carlo Simulation Models

A Monte Carlo method is a simulation technique that uses randomly generated numbers for simulation. Here, distribution functions of risk factors are created by using a sequence of random numbers.

Let $T$ be a holding period. For simplicity, we assume a single risk factor and let $p(x)$ and $x$ denote, respectively, the value of the portfolio and the value of the risk factor at $T$. We assume that $x$ is normally distributed with mean $\mu$ and variance $\sigma^{2}$, and that the current value of the portfolio is equal to $p(\mu)$.
Let $z_{i}, i=1, \cdots, J$ be a sequence of numbers generated randomly according to a standard normal distribution, where $J$ is the number of simulation runs. Then, we can assign $x_{i}=\mu+\sigma z_{i}$. By regarding generated sequences $x_{i}, i=$ $1, \cdots, J$ as scenarios for the risk factor $x$, we can obtain a distribution $p\left(x_{i}\right)$, where the probability of each scenario is $1 / J$. Next we rearrange $\left{p\left(x_{i}\right)\right}_{i=1, \cdots, J}$ to $\left{q_{k}\right}_{k=1, \cdots, J}$ such that
$$
\left{p\left(x_{i}\right) ; i=1, \cdots, J\right}=\left{q_{k} ; k=1, \cdots, J\right}
$$
with $q_{k} \leq q_{k+1}$ for all $k$.
To obtain a confidence level $1-\alpha$, we set $k_{\alpha}=\alpha J$. Then, $(1-\alpha) 100 \%$ VaR is given by
$$
V a R_{\alpha}=\frac{-1}{2}\left(q_{k_{\alpha}}+q_{k_{\alpha}+1}\right) .
$$
It is advantageous that Monte Carlo simulation models are applicable to nonlinear assets and path-dependent assets. This makes it possible to consider a great many scenarios. Moreover, we can employ this technique to the arbitrage-free model, which we will introduce in Chapter 3 . The disadvantages of Monte Carlo simulation are high computational time and increased model risk.

For a more advanced treatment of Monte Carlo simulation in financial engineering, interested readers are recommended to consult Glasserman (2004).

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|One-factor model

协方差 VaR 模型假设投资组合价值与风险因子成线性关系且呈正态分布。然后，关键参数是均值和方差磷& 大号分布，这是从历史数据中测得的。
让我们考虑一个具有单一风险因素的投资组合X，其值表示为p(X). 根据线性假设，我们可以建模p(X)作为p(X)=一个+bX对于一些正常数一个和b. 让μ和σ2分别是的均值和方差X. 为方便起见，我们假设p(μ)等于投资组合的当前价值。由分布的正态性可知，95%置信水平X是μ−1.645σ, 和99%置信水平是μ−2.326σ. 由此我们可以看出，95%在一个R是（谁）给的

95%在一个R=一个+μb−一个+b(μ−1.645σ) =1.645σb.
如图。1.7可能有助于理解这个方案。同样，我们有

99%在一个R=2.326σb.
什么时候p(X)不是线性的而是几乎是线性的，也就是说，当p(X)≈一个+bX，我们可以从（1.42）和（1.43）近似VaR。

金融代写|利率建模代写Interest Rate Modeling代考|Historical Simulation Models

在历史模拟模型中，损益的经验分布是直接从历史数据构建的。

为简单起见，我们考虑由两种资产组成的投资组合，一个和乙. 持有期Δ吨，让吨1,⋯,吨Ĵ+1是过去几天的序列吨一世−吨一世+1=Δ吨和吨1=0. 我们让p一个(一世)和p乙(一世)表示价格一个和乙，分别在吨一世. 让r一个(一世)和r乙(一世)是回报率一个和乙，分别。这些定义为

r一个(一世)=p一个(一世)−p一个(一世+1)p一个(一世+1) r乙(一世)=p乙(一世)−p乙(一世+1)p乙(一世+1)
为了一世=1,⋯,Ĵ.
对于现值p一个(0)和p乙(0)，这磷& 大号投资组合的分布是从集合中构建的

\left{p_{A}(0) r_{A}(i)+p_{B}(0) r_{B}(i)\right}_{i=1, \cdots, J}\left{p_{A}(0) r_{A}(i)+p_{B}(0) r_{B}(i)\right}_{i=1, \cdots, J}
如图 1.6 所示。从这个分布中，我们得到 VaR，过程按照图 1.6 所示的思路工作。即使对于非常大的投资组合，我们也可以用同样的方式衡量 VaR。

支持该模型的一点是，它不需要对价格分布和资产线性度进行假设。但是，这种方法很难收集到足够的历史数据。即使有足够多的长期观察数据点，一些数据可能太旧而无法合理反映未来风险，这是衡量 VaR 所必需的。

金融代写|利率建模代写Interest Rate Modeling代考|Monte Carlo Simulation Models

蒙特卡罗方法是一种使用随机生成的数字进行模拟的模拟技术。在这里，风险因素的分布函数是通过使用一系列随机数创建的。

让吨是一个持有期。为简单起见，我们假设一个风险因素并让p(X)和X分别表示投资组合的价值和风险因子的价值吨. 我们假设X正态分布，均值μ和方差σ2，并且投资组合的当前价值等于p(μ).
让和一世,一世=1,⋯,Ĵ是根据标准正态分布随机生成的数字序列，其中Ĵ是模拟运行的次数。然后，我们可以赋值X一世=μ+σ和一世. 通过考虑生成的序列X一世,一世= 1,⋯,Ĵ作为风险因素的情景X，我们可以得到一个分布p(X一世)，其中每个场景的概率是1/Ĵ. 接下来我们重新排列\left{p\left(x_{i}\right)\right}_{i=1, \cdots, J}\left{p\left(x_{i}\right)\right}_{i=1, \cdots, J}至\left{q_{k}\right}_{k=1, \cdots, J}\left{q_{k}\right}_{k=1, \cdots, J}这样

\left{p\left(x_{i}\right) ; i=1, \cdots, J\right}=\left{q_{k} ; k=1, \cdots, J\right}\left{p\left(x_{i}\right) ; i=1, \cdots, J\right}=\left{q_{k} ; k=1, \cdots, J\right}
和qķ≤qķ+1对所有人ķ.
获得置信水平1−一个，我们设置ķ一个=一个Ĵ. 然后，(1−一个)100%VaR 由下式给出

在一个R一个=−12(qķ一个+qķ一个+1).
蒙特卡罗模拟模型适用于非线性资产和路径相关资产是有利的。这使得可以考虑很多场景。此外，我们可以将这种技术应用于无套利模型，我们将在第 3 章中介绍。蒙特卡罗模拟的缺点是计算时间长和模型风险增加。

有关金融工程中蒙特卡罗模拟的更高级处理，建议感兴趣的读者参考 Glasserman (2004)。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

金融代写|利率建模代写Interest Rate Modeling代考|INTEREST RATE RISK

Posted on 2022年6月21日2022年6月21日 by statistics-lab

如果你也在怎样代写利率建模Interest Rate Modeling这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|INTEREST RATE RISK

金融代写|利率建模代写Interest Rate Modeling代考|Interest Rate and Discount Factor

This section introduces some fundamentals of interest rate theory, including discount factors, bond pricing, yield, forward rates, swap rates, and so on.
Discount factor
Let $r$ denote a simple interest rate for a period $[0, T]$. If we invest $1 \mathrm{unit}$ of currency today, this investment will be worth $1+r T$ at time $T$. Thus, a present value of 1 is economically equivalent to a value of $1+r T$ at time $T$. In other words, today’s $1 /(1+r T)$ has a future value of 1 at time $T$. Accordingly a present value of
$$
\frac{C}{1+r T}
$$
is equivalent to a future value of $C$ at time $T$.
The factor $1 /(1+r T)$ is called the discount factor, and $r$ is called the discount rate. The discount factor to time $T$ is denoted by $D(T)$, hence
$$
D(T)=\frac{1}{1+r T}
$$
Typically, the present value of a cash flow $C$ at time $T$ is defined as the result of discounting back the cash flow as $D(T) C$.

Next, let $r$ denote an annually compounded interest rate. If we repeatedly invest 1 at this rate $r$ for $n$ years, then the value of the investment would be $(1+r)^{n}$ at time $n$. The discount factor for $n$ years is thus formulated as
$$
D(n)=\frac{1}{(1+r)^{n}}
$$
Similarly, when $r$ denotes the semi-annually compounded interest rate, the discount factor for $n$ years is
$$
D(n)=\frac{1}{(1+r / 2)^{2 n}}
$$

金融代写|利率建模代写Interest Rate Modeling代考|Swap Rate and Forward LIBOR

This section introduces the fundamentals of the London Interbank Offered Rate (LIBOR) and swap rates, which are the most important interest rates in derivatives trading.
Interest rate swap
The most common example of a swap contract is a fixed-for-floating rate swap in a single currency. The floating rate is usually determined from the LIBOR, and the fixed rate is referred to as the swap rate. For example in a semi-annual swap, the six-month LIBOR rate is converted to a fixed rate.

Let $T_{i}, i=1, \cdots, n$ be a sequence of future days, and let $T_{n}$ be the maturity date of the swap. For simplicity, we assume that the notional principal is equal to 1 . In the swap contract, the payer of the fixed rate pays the swap rate and receives the forward LIBOR at every $T_{i}$. The receiver of the fixed rate pays the forward LIBOR and receives the swap rate at every $T_{i}$.

We denote by $S(n)$ a swap rate with maturity $T_{n}$, and by $L(i)$ a forward LIBOR for the period $\left[T_{i}, T_{i+1}\right] . D(i)$ denotes the discount factor to time $T_{i}$. For simplicity, we assume that each interval of the period $\left[T_{i}, T_{i+1}\right]$ is authorthe same length, $\delta>0$. That is, $T_{i+1}-T_{i}=\delta$ for all $i<n$. Note that the floating payment $\delta L(i)$ takes place at time $T_{i+1}$, and so its present value is given by $L(i) D(i+1)$. Since the present value of the fixed cash flows should be equal to that of the floating cash flows, we obtain
$$
S(n) \sum_{i=0}^{n-1} \delta D(i+1)=\sum_{i=0}^{n-1} \delta D(i+1) L(i)
$$
From (1.12), it holds that
$$
\delta D(i+1) L(i)=D(i)-D(i+1)
$$
Taking the sum of $(1.16)$ over $i=0, \cdots, n-1$, the right side of $(1.15)$ becomes
$$
\begin{aligned}
\delta \sum_{i=0}^{n-1} D(i+1) L(i) &=\sum_{i=0}^{n-1} D(i)-D(i+1) \
&=1-D(n)
\end{aligned}
$$

金融代写|利率建模代写Interest Rate Modeling代考|Term Structure of Interest Rates

The yield to maturity of bonds can be observed in the bond market. The yield curve is a graph that plots time to maturity on the horizontal axis and the yield to maturity on the vertical axis, which visually exhibits the term structure of interest rates. The swap rate curve and the forward LIBOR curve are similarly exhibited.

For example, from (1.15) the swap rate $S(n)$ is represented by a series of forward LIBORs $L(i), i=1, \cdots, n-1$ as
$$
S(n)=\frac{\sum_{i=0}^{n-1} \delta D(i+1) L(i)}{\sum_{i=0}^{n-1} \delta D(i+1)} .
$$

This means that the swap rate consists of forward LIBORs $L(i)$ for $i<n$. This property is the same when considering the yield to maturity.

In contrast, the forward rate is the obtained interest rate of the corresponding future period. Therefore, the forward rate curve shows us the term structure of interest rates more clearly than the yield curve and the swap rate curve can. For this reason, the forward rate curve is often used to analyze the interest rate market. In particular, the forward rates derived from a term structure of interest rates are called the implied forward rate. For example,the “implied” forward rate can be practically observed in a government bond market or a swap market.

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|Interest Rate and Discount Factor

本节介绍利率理论的一些基础知识，包括贴现因子、债券定价、收益率、远期利率、掉期利率等。
折扣系数
Letr表示一个时期的简单利率[0,吨]. 如果我们投资1在n一世吨今天的货币，这项投资将是值得的1+r吨有时吨. 因此，现值 1 在经济上等价于1+r吨有时吨. 换句话说，今天1/(1+r吨)未来值为 1吨. 因此，现值

C1+r吨
等于未来值C有时吨.
因素1/(1+r吨)称为折扣因子，并且r称为贴现率。时间折扣系数吨表示为D(吨)，因此

D(吨)=11+r吨
通常，现金流的现值C有时吨被定义为将现金流折现为的结果D(吨)C.

接下来，让r表示年复利利率。如果我们以这个速度重复投资 1r为了n年，那么投资的价值将是(1+r)n有时n. 折扣系数为n年因此被制定为

D(n)=1(1+r)n
同样，当r表示半年复利利率，折现因子为n年是

D(n)=1(1+r/2)2n

金融代写|利率建模代写Interest Rate Modeling代考|Swap Rate and Forward LIBOR

本节介绍伦敦银行同业拆借利率 (LIBOR) 和掉期利率的基本面，它们是衍生品交易中最重要的利率。
利率掉期
掉期合约最常见的例子是单一货币的固定浮动利率掉期。浮动利率通常由 LIBOR 确定，固定利率称为掉期利率。例如，在半年期掉期中，六个月的 LIBOR 利率转换为固定利率。

让吨一世,一世=1,⋯,n是未来几天的序列，让吨n是互换的到期日。为简单起见，我们假设名义本金等于 1。在掉期合约中，固定利率的支付方支付掉期利率并在每吨一世. 固定利率的接收方支付远期 LIBOR 并在每吨一世.

我们表示小号(n)有期限的掉期利率吨n，并由大号(一世)该期间的远期 LIBOR[吨一世,吨一世+1].D(一世)表示时间的折扣因子吨一世. 为简单起见，我们假设周期的每个区间[吨一世,吨一世+1]是作者相同的长度，d>0. 那是，吨一世+1−吨一世=d对所有人一世<n. 注意浮动支付d大号(一世)发生在时间吨一世+1，因此它的现值由下式给出大号(一世)D(一世+1). 由于固定现金流的现值应等于浮动现金流的现值，我们得到

小号(n)∑一世=0n−1dD(一世+1)=∑一世=0n−1dD(一世+1)大号(一世)
从（1.12），它认为

dD(一世+1)大号(一世)=D(一世)−D(一世+1)
取总和(1.16)超过一世=0,⋯,n−1, 的右侧(1.15)变成

d∑一世=0n−1D(一世+1)大号(一世)=∑一世=0n−1D(一世)−D(一世+1) =1−D(n)

金融代写|利率建模代写Interest Rate Modeling代考|Term Structure of Interest Rates

债券的到期收益率可以在债券市场上观察到。收益率曲线是以到期时间为横轴，以到期收益率为纵轴的图表，直观地展示了利率的期限结构。互换利率曲线和远期 LIBOR 曲线也有类似的表现。

例如，从 (1.15) 掉期利率小号(n)由一系列远期 LIBOR 表示大号(一世),一世=1,⋯,n−1作为

小号(n)=∑一世=0n−1dD(一世+1)大号(一世)∑一世=0n−1dD(一世+1).

这意味着掉期利率由远期 LIBOR 组成大号(一世)为了一世<n. 在考虑到期收益率时，此属性是相同的。

相反，远期利率是获得的相应未来期间的利率。因此，远期利率曲线比收益率曲线和掉期利率曲线更清楚地向我们展示了利率的期限结构。出于这个原因，远期利率曲线经常被用来分析利率市场。特别是，由利率期限结构得出的远期利率称为隐含远期利率。例如，“隐含”远期利率可以在政府债券市场或掉期市场中实际观察到。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

金融代写|利率建模代写Interest Rate Modeling代考|MTH5520

Posted on 2022年6月7日2022年6月7日 by statistics-lab

如果你也在怎样代写利率建模Interest Rate Modeling这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

Vasicek利率模型一词是指一种对利率的运动和演变进行建模的数学方法。它是一种基于市场风险的单因素短利率模型。瓦西克利率模型常用于经济学中，以确定利率在未来的移动方向。

我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|MTH5520

金融代写|利率建模代写Interest Rate Modeling代考|Bootstrapping the Zero-Coupon Yields

The determination of the zero-coupon yield curve (or discount curve) based on the yields of the on-the-run issues is an under-determined problem: we need to solve for infinitely many unknowns based on a few inputs. To define a meaningful solution, one must parameterize the zero-coupon yield curve. The simplest parameterization that is financially acceptable is to assume piecewise constant functional forms for the zero-coupon yield curve. Under such a parameterization, the zero-coupon yield curve can be derived sequentially. Such a procedure is often called bootstrapping in finance. Next, we describe the bootstrapping procedure with the construction of the zero-coupon yield curve for U.S. Treasuries.

Let $\left{B_{j}^{c}, T_{j}\right}_{j=1}^{7}$ be the prices and maturities of the seven on-the-run issues. Let $T_{0}=0$ and $\Delta T=0.5$. We assume that the zero-coupon yield for maturities between $\left[T_{0}, T_{7}\right]$ is a piecewise linear function. The determination of the YTMs is done sequentially. Because the first two on-the-run issues are zero-coupon bonds, we first back out $y(0.25)$ and $y(0.5)$, the zero yields for $\left(0, T_{1}\right]$ and $\left(T_{1}, T_{2}\right]$, using formula $3.18$. This will require a root-finding procedure. Once $y(0.5)$ is found, we proceed to determining $y(i \Delta T), i=2,3,4$ from the following equation:
$$
B_{3}^{c}=\frac{c_{3} \Delta T}{(1+y(\Delta T) \Delta T)^{i}}+\sum_{i=2}^{4} \frac{c_{3} \Delta T}{(1+y(i \Delta T) \Delta T)^{i}}+\frac{1}{(1+y(4 \Delta T) \Delta T)^{4}}
$$
where
$$
y(i \Delta T)=y(0.5)+\alpha \times(i \Delta T-0.5), \quad i=2,3,4
$$
So, our zero-coupon yield is a linear function over $T \in\left[T_{2}, T_{3}\right]$. Equation $9.20$ become the equation for $\alpha$, which can be determined through a root-finding procedure. This procedure can continue all the way to $j=7$. The entire A zero-coupon yield curve implies a discount curve. Suppose that the $y_{T}$ is the zero-coupon yield for maturity $T$. Then the corresponding zerocoupon bond price is calculated according to Equation 3.18. With discount bond prices, we can value any coupon bond using Equation 3.19.

金融代写|利率建模代写Interest Rate Modeling代考|Forward Rates and Forward-Rate Agreements

A forward-rate agreement (FRA) is a contract between two parties to lend and borrow a certain amount of money for some future period of time with a pre-specified interest rate. The agreement is so structured that neither party needs to make an upfront payment. This is equivalent to saying that, as a financial instrument, the value of the contract is zero when the agreement is entered. The key to such a contract lies in the lending rate that should be fair to both parties. Fortunately, this fair rate can be determined through arbitrage arguments.

Let the time now be $t$ and the fair lending rate for a future period, $[T, T+\Delta T]$, be $f_{\Delta T}(t, T)$. To finance the lending, the lender may short $P(t, T) / P(t, T+\Delta T)$ units of the $(T+\Delta T)$-maturity zero-coupon bond, and then long one unit of the $T$-maturity zero-coupon bond. At time $T$, the proceeds from the $T$-maturity zero are lent out for a period of $\Delta T$ with the interest rate $f_{\Delta T}(t, T)$. At time $T+\Delta T$, the loan is paid back from the borrower and the short position of $(T+\Delta T)$-maturity zero-coupon bond (which just matures) is covered, yielding a net cash flow of
$$
V=1+\Delta T f_{\Delta T}(t, T)-\frac{P(t, T)}{P(t, T+\Delta T)}
$$
Because this is a set of zero net transactions initially, in the absence of arbitrage, $V$ must be zero, which leads to the following expression of the fair lending rate:
$$
f_{\Delta T}(t, T)=\frac{1}{\Delta T}\left(\frac{P(t, T)}{P(t, T+\Delta T)}-1\right) .
$$
Hence, the arbitrage free or fair forward lending rate is totally determined by the prices of zero-coupon bonds. We call $f_{\Delta T}(t, T)$ the simple forward rate for the period $(T, T+\Delta T)$ seen at time $t$, or simply a forward rate.

Consider now the limiting case, $\Delta T \rightarrow 0$, for the forward rate. There is
$$
\begin{aligned}
f(t, T) & \triangleq \lim {\Delta T \rightarrow 0} f{\Delta T}(t, T) \
&=\lim {\Delta T \rightarrow 0} \frac{1}{\Delta T}\left(\frac{P(t, T)}{P(t, T+\Delta T)}-1\right) \ &=\frac{-1}{P(t, T)} \frac{\partial P(t, T)}{\partial T} \ &=-\frac{\partial \ln P(t, T)}{\partial T} \end{aligned} $$ We call $f(t, T)$ an instantaneous forward rate. According to Equation $3.23$, we can express the price of a $T$-maturity zero-coupon bond in terms of $f(t, s), t \leq$ $s \leq T$ : $$ P(t, T)=\mathrm{e}^{-\int{t}^{T} f(t, s) \mathrm{d} s}
$$

金融代写|利率建模代写Interest Rate Modeling代考|Duration and Convexity

In the bond market, bond prices change unpredictably on a daily basis. The changes in bond prices can be interpreted as the consequence of unpredictable changes in yields. The duration of a bond is a measure of risk exposure with respect to a possible change in the bond yield. It has been observed that the prices of long-maturity bonds are more sensitive to change in yields than are the prices of short-maturity bonds, and the impact of yield changes on bond prices seems proportional to the cash flow dates of the bonds. Intuitively, Macaulay (1938) introduced the weighted average of the cash flow dates as a measure of price sensitivity with respect to the bond yield:
$$
\begin{aligned}
D_{\text {mac }}=& \frac{\operatorname{Pr}}{B_{t}^{c}}\left[\sum_{i, T_{i}>t}^{n} \Delta T \cdot c(1+y \Delta T)^{-\left(T_{i}-t\right) / \Delta T}\left(T_{i}-t\right)\right.\
&\left.+(1+y \Delta T)^{-\left(T_{n}-t\right) / \Delta T}\left(T_{n}-t\right)\right] .
\end{aligned}
$$
This measure is called the Macaulay duration in the bond market. Note that, for a zero-coupon bond, the duration is simply its maturity. It was later understood that the Macaulay duration is closely related to the derivative of the bond price with respect to its yield. In fact, differentiating Equation $3.13$ with

respect to $y$ yields
$$
\begin{aligned}
\frac{\mathrm{d} B_{t}^{c}}{\mathrm{~d} y}=&-\frac{\operatorname{Pr}}{1+y \Delta T}\left[\sum_{i ; T_{i}>t}^{n} \Delta T \cdot c(1+y \Delta T)^{-\left(T_{i}-t\right) / \Delta T}\left(T_{i}-t\right)\right.\
&\left.+(1+y \Delta T)^{-\left(T_{n}-t\right) / \Delta T}\left(T_{n}-t\right)\right]
\end{aligned}
$$
In terms of $D_{\operatorname{mac}}$, the Macaulay duration just defined, we have
$$
\frac{\mathrm{d} B_{t}^{c}}{B_{t}^{c}}=-\frac{D_{\mathrm{mac}}}{1+y \Delta T} \mathrm{~d} y \quad \text { or } \quad \frac{1}{B_{t}^{c}} \frac{\mathrm{d} B_{t}^{c}}{\mathrm{~d} y}=-\frac{D_{\operatorname{mac}}}{1+y \Delta T}
$$
According to Equation 3.25, the Macaulay duration is essentially the rate of change with respect to the yield for each dollar of market value of the bond. After multiplying by the change in the yield, the Macaulay duration gives the percentage change in the value of the bond. For convenience, we define
$$
D_{\text {mod }}=\frac{D_{\text {mac }}}{1+y \Delta T},
$$
and call it the modified duration. Then the first equation of Equation $3.25$ can be written in the following simple form:
$$
\frac{\mathrm{d} B_{t}^{c}}{B_{t}^{c}}=-D_{\bmod } \mathrm{d} y
$$
Both $D_{\text {mac }}$ and $D_{\text {mod }}$ are called duration measures of bonds.
By using Equation 3.17, the succinct bond formula, we can obtain the following formula for the modified duration:
$$
D_{\text {mod }}=\frac{\operatorname{Pr}}{B^{c}}\left[\frac{c}{y^{2}}\left(1-\frac{1}{(1+y \Delta T)^{n}}\right)+\left(1-\frac{c}{y}\right) \frac{n \Delta T}{(1+y \Delta T)^{n+1}}\right]
$$
The above expression is simplified for par bonds. When $c=y$ and $B^{c}=\operatorname{Pr}$, we have
$$
D_{\mathrm{mod}}=\frac{1}{y}\left[1-(1+y \Delta T)^{-n}\right] .
$$
Note that Treasury bonds are quoted in yields and that recent issues are usually traded close to par, so Equation $3.28$ gives us an approximate value of the durations for bonds being traded close to par.

The next example shows how much the dollar value of a bond changes given its duration.

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|Bootstrapping the Zero-Coupon Yields

根据运行中问题的收益率确定零息票收益率曲线（或贴现曲线）是一个未确定问题：我们需要基于少量输入解决无限多的未知数。要定义一个有意义的解决方案，必须参数化零息票收益率曲线。财务上可接受的最简单的参数化是假设零息票收益率曲线的分段常数函数形式。在这样的参数化下，可以依次推导出零息票收益率曲线。这样的过程在金融中通常被称为引导。接下来，我们将描述构建美国国债零息票收益率曲线的引导程序。

让\left{B_{j}^{c}, T_{j}\right}_{j=1}^{7}\left{B_{j}^{c}, T_{j}\right}_{j=1}^{7}是七个在运行的问题的价格和到期日。让吨0=0和Δ吨=0.5. 我们假设到期日的零息票收益率介于[吨0,吨7]是分段线性函数。YTM 的确定是按顺序进行的。因为前两个在运行的问题是零息债券，我们首先退出是(0.25)和是(0.5)，零收益率(0,吨1]和(吨1,吨2], 使用公式3.18. 这将需要一个寻根程序。一次是(0.5)找到了，我们继续确定是(一世Δ吨),一世=2,3,4从以下等式：

乙3C=C3Δ吨(1+是(Δ吨)Δ吨)一世+∑一世=24C3Δ吨(1+是(一世Δ吨)Δ吨)一世+1(1+是(4Δ吨)Δ吨)4
在哪里

是(一世Δ吨)=是(0.5)+一个×(一世Δ吨−0.5),一世=2,3,4
所以，我们的零息票收益率是一个线性函数吨∈[吨2,吨3]. 方程9.20成为方程一个，这可以通过寻根程序来确定。这个程序可以一直持续到j=7. 整个 A 零息票收益率曲线意味着一条贴现曲线。假设是吨是到期的零息票收益率吨. 然后根据公式 3.18 计算相应的零息债券价格。通过折扣债券价格，我们可以使用公式 3.19 对任何息票债券进行估值。

金融代写|利率建模代写Interest Rate Modeling代考|Forward Rates and Forward-Rate Agreements

一个

远期利率协议 (FRA) 是两方之间以预先指定的利率在未来一段时间内借入和借入一定数量资金的合同。该协议的结构如此之好，任何一方都无需支付预付款。这相当于说，作为一种金融工具，在签订协议时，合同的价值为零。这种合同的关键在于对双方都应该公平的贷款利率。幸运的是，这个公平的利率可以通过套利论据来确定。

让现在的时间吨以及未来一段时间的公平贷款利率，[吨,吨+Δ吨]，是FΔ吨(吨,吨). 为了融资贷款，贷方可以卖空磷(吨,吨)/磷(吨,吨+Δ吨)的单位(吨+Δ吨)-到期零息债券，然后做多一个单位吨- 到期零息债券。当时吨，收益来自吨-到期时间为零被借出一段时间Δ吨与利率FΔ吨(吨,吨). 当时吨+Δ吨, 贷款由借款人偿还，空头头寸(吨+Δ吨)-到期零息债券（刚刚到期）被覆盖，产生的净现金流为

在=1+Δ吨FΔ吨(吨,吨)−磷(吨,吨)磷(吨,吨+Δ吨)
因为这是一组最初的零净交易，在没有套利的情况下，在必须为零，这导致公平贷款利率的表达式如下：

FΔ吨(吨,吨)=1Δ吨(磷(吨,吨)磷(吨,吨+Δ吨)−1).
因此，无套利或公平的远期贷款利率完全由零息债券的价格决定。我们称之为FΔ吨(吨,吨)该期间的简单远期利率(吨,吨+Δ吨)当时看到吨，或者只是一个远期利率。

现在考虑极限情况，Δ吨→0, 为远期利率。有

F(吨,吨)≜林Δ吨→0FΔ吨(吨,吨) =林Δ吨→01Δ吨(磷(吨,吨)磷(吨,吨+Δ吨)−1) =−1磷(吨,吨)∂磷(吨,吨)∂吨 =−∂ln⁡磷(吨,吨)∂吨我们称之为F(吨,吨)瞬时远期汇率。根据方程3.23，我们可以表示a的价格吨-到期零息债券F(吨,s),吨≤ s≤吨 :

磷(吨,吨)=和−∫吨吨F(吨,s)ds

金融代写|利率建模代写Interest Rate Modeling代考|Duration and Convexity

在债券市场上，债券价格每天都会发生不可预测的变化。债券价格的变化可以解释为收益率变化不可预测的结果。债券的久期是衡量债券收益率可能变化的风险敞口。据观察，长期债券的价格对收益率的变化比短期债券的价格更敏感，收益率变化对债券价格的影响似乎与债券的现金流量日期成正比。直观地说，Macaulay (1938) 引入了现金流量日期的加权平均值作为对债券收益率的价格敏感度的衡量标准：

D苹果电脑 =公关乙吨C[∑一世,吨一世>吨nΔ吨⋅C(1+是Δ吨)−(吨一世−吨)/Δ吨(吨一世−吨) +(1+是Δ吨)−(吨n−吨)/Δ吨(吨n−吨)].
这个度量在债券市场被称为麦考利久期。请注意，对于零息债券，久期只是其到期日。后来了解到，麦考利久期与债券价格对收益率的导数密切相关。实际上，微分方程3.13和

尊重是产量

d乙吨C d是=−公关1+是Δ吨[∑一世;吨一世>吨nΔ吨⋅C(1+是Δ吨)−(吨一世−吨)/Δ吨(吨一世−吨) +(1+是Δ吨)−(吨n−吨)/Δ吨(吨n−吨)]
按照D苹果电脑，刚刚定义的麦考利持续时间，我们有

d乙吨C乙吨C=−D米一个C1+是Δ吨 d是或者 1乙吨Cd乙吨C d是=−D苹果电脑1+是Δ吨
根据公式 3.25，麦考利久期本质上是债券每 1 美元市场价值的收益率相对于收益率的变化率。在乘以收益率的变化后，麦考利久期给出了债券价值的百分比变化。为方便起见，我们定义

D反对 =D苹果电脑 1+是Δ吨,
并将其称为修改后的持续时间。那么方程的第一个方程3.25可以写成以下简单的形式：

d乙吨C乙吨C=−D反对d是
两个都D苹果电脑和D反对称为债券的久期度量。
通过使用公式 3.17，简洁的债券公式，我们可以得到修正久期的以下公式：

D反对 =公关乙C[C是2(1−1(1+是Δ吨)n)+(1−C是)nΔ吨(1+是Δ吨)n+1]
上述表达式对于面值债券进行了简化。什么时候C=是和乙C=公关，我们有

D米○d=1是[1−(1+是Δ吨)−n].
请注意，国债以收益率报价，并且最近发行的债券通常以接近面值的价格交易，因此等式3.28为我们提供了接近面值交易的债券久期的近似值。

下一个例子显示了债券的美元价值在其久期的情况下发生了多少变化。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

金融代写|利率建模代写Interest Rate Modeling代考|MATH4513

Posted on 2022年6月7日2022年6月7日 by statistics-lab

如果你也在怎样代写利率建模Interest Rate Modeling这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|MATH4513

金融代写|利率建模代写Interest Rate Modeling代考|Yield to Maturity

In a free market, the price of a bond is determined by supply and demand. Due to discounting, the full price is normally smaller than the total notional value of coupons plus the principal. Denote the full price of a bullet bond as $B^{c}$. Suppose that all cash flows are discounted by a uniform rate, $y$, of compounding frequency $\omega$. Then $y$ should satisfy the following equation:
$$
B^{c}=\operatorname{Pr} \cdot\left(\sum_{i=1}^{n} \frac{c \Delta T}{(1+y \Delta t)^{i \Delta T / \Delta t}}+\frac{1}{(1+y \Delta t)^{n \Delta T / \Delta t}}\right)
$$
where $n$ is the number of coupons and $\Delta t=1 / \omega$. In bond mathematics, the compounding frequency is taken to be $\omega=1 / \Delta T$ by default, when there is $\Delta t=\Delta T$. This discount rate, which can be easily solved by a trial-and-error procedure using Equation 3.12, is defined to be the yield to maturity (YTM), as well as the internal rate of return (IRR) of the bond, and it is often simply called the bond yield.

As the function of the yield (for $\omega=1 / \Delta T$ ), the formula for a general time, $t \leq T$, is
$$
B_{t}^{c}=\operatorname{Pr} \cdot\left(\sum_{i ; i \Delta T>t}^{n} \frac{c \Delta T}{(1+y \Delta T)^{(i \Delta T-t) / \Delta T}}+\frac{1}{(1+y \Delta T)^{(n \Delta T-t) / \Delta T}}\right)_{(3.13)}
$$

Assuming that $t \in\left(T_{j}, T_{j+1}\right]$, and introducing
$$
q=\frac{t-T_{j}}{T_{j+1}-T_{j}}=\frac{t-T_{j}}{\Delta T}
$$
we then can write
$$
t=T_{j}+\Delta T q=(j+q) \Delta T \text { and } i \Delta T-t=(i-j-q) \Delta T, \quad \forall i .
$$
It follows that
$$
\begin{aligned}
B_{t}^{c} &=\operatorname{Pr} \cdot\left(\sum_{i=j+1}^{n} \frac{c \Delta T}{(1+y \Delta T)^{i-j-q}}+\frac{1}{(1+y \Delta T)^{n-j-q}}\right) \
&=\operatorname{Pr} \cdot(1+y \Delta T)^{q}\left(\sum_{i=1}^{n-j} \frac{c \Delta T}{(1+y \Delta T)^{i}}+\frac{1}{(1+y \Delta T)^{n-j}}\right)
\end{aligned}
$$
Given the bond price at any time, $t$, the bond yield is implied by Equation 3.14. A rough way to compare the relative cheapness/richness of two bonds with the same coupon frequency is to compare their yields. Intuitively, a bond with a higher yield is cheaper and thus may be more attractive.

There is a one-to-one price-yield relationship, as shown in Figure 3.2. Because of this relationship, a bond price is also quoted using its yield in the industry. As we can see in Figure $3.2$, a bond price is a convex function of the yield. Such a feature will be used later for convexity adjustment related to futures trading.
The price-yield relationship of a zero-coupon bond simplifies to
$$
P=\operatorname{Pr} \cdot(1+y \Delta T)^{-(T-t) / \Delta T} .
$$

金融代写|利率建模代写Interest Rate Modeling代考|Par Bonds, Par Yields, and the Par Yield Curve

The summation in Equation $3.12$ can be worked out so that
$$
\begin{aligned}
B^{c} &=\Delta T \cdot c \cdot \operatorname{Pr} \sum_{i=1}^{n}(1+y \Delta T)^{-i}+\operatorname{Pr}(1+y \Delta T)^{-n} \
&=\operatorname{Pr}\left[1-\left(1-\frac{c}{y}\right)\left(1-\frac{1}{(1+y \Delta T)^{n}}\right)\right]
\end{aligned}
$$
From the above expression, we can tell when the price is smaller, equal to, or larger than the principal value.

When $c<y, B^{c}<$ Pr. In such a case, we say that the bond is sold at discount (of the par value).
When the coupon rate is $c=y$, then $B^{c}=\operatorname{Pr}$, that is, the bond price equals the par value of the bond. In such a case, we call the bond a par bond, and the corresponding coupon rate a par yield.
When $c>y, B^{c}>\operatorname{Pr}$. In such a case, we call the bond a premium bond (it is traded at a premium to par).

Par yields play an important role in today’s interest-rate derivatives market. As we shall see later, there are many derivatives based on the par yields.

金融代写|利率建模代写Interest Rate Modeling代考|Yield Curves for U.S. Treasuries

A bond issuer may routinely issue bonds of various maturities, and, in a market, there can be many bonds of the same issuer being traded. For various reasons, some bonds are more liquid than others. The most liquid ones are often called benchmark bonds for the issuer. Their yields reflect the level of borrowing costs the market demands from the issuer. Moreover, the prices of the benchmark bonds imply a discount curve for cash flows from the issuer, and the discount curve can be used to gauge the relative cheapness/expensiveness of the issuer’s other bonds. If a relatively cheaper or more expensive bond is found, one may trade against this bond using the benchmark bonds and thus take an arbitrage profit. Hence, the prices or yields of the benchmark bonds carry essential information for the arbitrage pricing of the issuer’s other bonds, and they are treated as a summary of the status quo of all bonds offered by the same issuer.

In the U.S. Treasury market, newly issued bills and notes/bonds are called on-the-run Treasury securities. Traditionally, the on-the-run issues enjoy higher liquidity and are thus treated as benchmarks. Table $3.1$ provides the closing price quotes of the on-the-run issues for July 3 , 2008. As can be seen in the table, the on-the-run issues have maturities of 3 months, 6 months, 2 years, 3 years, 5 years, 10 years, and 30 years. When we connect the yields of the benchmark bonds through interpolation, we obtain a so-called yield curve. Since bond yields vary from day-to-day so does the yield curve. Figure $3.3$ shows the yield curves for the U.S. Treasuries constructed by linear interpolation for April 28 and May 1, 2006 , two consecutive trading days.
A yield curve is constructed based on yields of on-the-run issues using the interpolation technique. It provides a rough idea of the level of yields for various maturities. Further, the Treasury yield curve implies a discount curve, namely, the collection of prices of all zero-coupon bonds. The discount curve is used for pricing off-the-run Treasury securities, or marking to market Treasury portfolios. Moreover, the discount curve is also essential for pricing future cash flows of any security, either deterministic or stochastic. To price a portfolio of interest-rate derivatives, we may model the dynamics of the entire yield curve, in contrast to modeling the dynamics of a stock price for stock options. In the next section, we describe the technique for “backing” out the discount curve from the yield curve.

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|Yield to Maturity

在自由市场中，债券的价格由供求关系决定。由于贴现，全价通常小于票面总价值加上本金。将子弹债券的全价表示为乙C. 假设所有现金流都以统一的利率贴现，是, 的复利频率ω. 然后是应满足以下等式：

乙C=公关⋅(∑一世=1nCΔ吨(1+是Δ吨)一世Δ吨/Δ吨+1(1+是Δ吨)nΔ吨/Δ吨)
在哪里n是优惠券的数量和Δ吨=1/ω. 在债券数学中，复合频率被认为是ω=1/Δ吨默认情况下，当有Δ吨=Δ吨. 这个贴现率可以很容易地通过使用公式 3.12 的试错法求解，定义为到期收益率 (YTM) 以及债券的内部收益率 (IRR)，并且它通常简称为债券收益率。

作为产量的函数（对于ω=1/Δ吨)，一般时间的公式，吨≤吨，是

乙吨C=公关⋅(∑一世;一世Δ吨>吨nCΔ吨(1+是Δ吨)(一世Δ吨−吨)/Δ吨+1(1+是Δ吨)(nΔ吨−吨)/Δ吨)(3.13)

假如说吨∈(吨j,吨j+1], 并引入

q=吨−吨j吨j+1−吨j=吨−吨jΔ吨
然后我们可以写

吨=吨j+Δ吨q=(j+q)Δ吨和一世Δ吨−吨=(一世−j−q)Δ吨,∀一世.
它遵循

乙吨C=公关⋅(∑一世=j+1nCΔ吨(1+是Δ吨)一世−j−q+1(1+是Δ吨)n−j−q) =公关⋅(1+是Δ吨)q(∑一世=1n−jCΔ吨(1+是Δ吨)一世+1(1+是Δ吨)n−j)
给定任何时候的债券价格，吨，债券收益率由公式 3.14 暗示。比较具有相同票息频率的两种债券的相对便宜/丰富程度的粗略方法是比较它们的收益率。直观地说，收益率较高的债券更便宜，因此可能更具吸引力。

价格与收益之间存在一对一的关系，如图 3.2 所示。由于这种关系，债券价格也使用其在行业中的收益率进行报价。如图所示3.2，债券价格是收益率的凸函数。稍后将使用此功能进行与期货交易相关的凸性调整。
零息债券的价格-收益关系简化为

磷=公关⋅(1+是Δ吨)−(吨−吨)/Δ吨.

金融代写|利率建模代写Interest Rate Modeling代考|Par Bonds, Par Yields, and the Par Yield Curve

方程式中的求和3.12可以这样计算

乙C=Δ吨⋅C⋅公关⁡∑一世=1n(1+是Δ吨)−一世+公关⁡(1+是Δ吨)−n =公关⁡[1−(1−C是)(1−1(1+是Δ吨)n)]
从上面的表达式中，我们可以判断价格何时小于、等于或大于本金价值。

什么时候C<是,乙C<公关。在这种情况下，我们说债券以（面值的）折扣价出售。
当票面利率为C=是，然后乙C=公关，即债券价格等于债券面值。在这种情况下，我们称该债券为面值债券，相应的票面利率为面值收益率。
什么时候C>是,乙C>公关. 在这种情况下，我们称该债券为溢价债券（以面值溢价交易）。

面值收益率在当今的利率衍生品市场中发挥着重要作用。正如我们稍后将看到的，有许多基于面值收益率的衍生品。

金融代写|利率建模代写Interest Rate Modeling代考|Yield Curves for U.S. Treasuries

债券发行人可能会定期发行各种期限的债券，并且在一个市场上，可以有许多同一发行人的债券在交易。由于各种原因，一些债券比其他债券更具流动性。流动性最强的债券通常被称为发行人的基准债券。它们的收益率反映了市场要求发行人的借贷成本水平。此外，基准债券的价格暗示了发行人现金流的贴现曲线，而贴现曲线可用于衡量发行人其他债券的相对便宜/昂贵。如果发现相对便宜或更昂贵的债券，则可以使用基准债券与该债券进行交易，从而获得套利利润。因此，基准债券的价格或收益率为发行人其他债券的套利定价提供了重要信息，

在美国国债市场上，新发行的票据和票据/债券被称为流通国库券。传统上，在运行的问题享有更高的流动性，因此被视为基准。桌子3.1提供 2008 年 7 月 3 日在售股的收盘价。从表中可以看出，在售股的到期日为 3 个月、6 个月、2 年、3 年、5年、10 年和 30 年。当我们通过插值连接基准债券的收益率时，我们得到了一条所谓的收益率曲线。由于债券收益率每天都在变化，因此收益率曲线也是如此。数字3.3显示了 2006 年 4 月 28 日和 5 月 1 日连续两个交易日通过线性插值构建的美国国债收益率曲线。
收益率曲线是使用插值技术根据运行中问题的收益率构建的。它提供了不同期限的收益率水平的粗略概念。此外，国债收益率曲线意味着一条贴现曲线，即所有零息债券价格的集合。贴现曲线用于对非流通国库券进行定价，或将国库券投资组合标记为市场。此外，贴现曲线对于任何证券的未来现金流定价也是必不可少的，无论是确定性的还是随机的。为了给利率衍生品投资组合定价，我们可以对整个收益率曲线的动态进行建模，这与对股票期权的股价动态进行建模形成对比。在下一节中，我们将描述从收益率曲线中“退出”贴现曲线的技术。

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广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

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STATA代写	机器学习/统计学习代写
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金融代写|利率建模代写Interest Rate Modeling代考|MATH3075

Posted on 2022年6月7日2022年6月7日 by statistics-lab

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我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|MATH3075

金融代写|利率建模代写Interest Rate Modeling代考|The Black-Scholes Formula

Consider the pricing of a European call option on an asset, $S_{t}$, which has the payoff
$$
C_{T}=\left(S_{T}-K\right)^{+}
$$
at time $T$. Assume that the short rate is a constant, $r_{t}=r$. According to the Black-Scholes-Merton Equation $2.57$ and the terminal condition $2.58$, the value of the call option satisfies
$$
\left{\begin{array}{l}
\frac{\partial C_{t}}{\partial t}+\frac{1}{2} \sigma_{t}^{2} S^{2} \frac{\partial^{2} C_{t}}{\partial S^{2}}+r S \frac{\partial C_{t}}{\partial S}-r C_{t}=0 \
C_{T}=(S-K)^{+}
\end{array} .\right.
$$
By solving this terminal-value problem of the partial differential equation (PDE), we can obtain the price of the option.

Alternatively, we can derive the formula for the call options by working out the expectation in Equation $2.71$ directly. We write
$$
\begin{aligned}
C_{t} &=\mathrm{e}^{-r(T-t)} E_{t}^{\mathrm{Q}}\left[\left(S_{T}-K\right)^{+}\right] \
&=\mathrm{e}^{-r(T-t)}\left(E_{t}^{\mathrm{Q}}\left[S_{T} 1_{S_{T}>K}\right]-K E_{t}^{\mathrm{Q}}\left[1_{S_{T}>K}\right]\right)
\end{aligned}
$$
Since
$$
S_{T}=S_{t} \exp \left[\left(r-\frac{1}{2} \bar{\sigma}^{2}\right) \tau+\bar{\sigma} \sqrt{\tau} \cdot \varepsilon\right], \quad \varepsilon \sim N(0,1),
$$
where $\tau=T-t$ and $\bar{\sigma}$ is the mean volatility,
$$
\bar{\sigma}=\sqrt{\frac{1}{\tau} \int_{0}^{\tau} \sigma_{s}^{2} \mathrm{~d} s,}
$$
we have
$$
E_{t}^{\mathrm{Q}}\left[1_{S_{T}>K}\right]=\operatorname{Prob}\left(\varepsilon>-\frac{\ln \left(S_{t} / K\right)+\left(r-(1 / 2) \bar{\sigma}^{2}\right) \tau}{\bar{\sigma} \sqrt{\tau}}\right)=\Phi\left(d_{2}\right),
$$
with
$$
d_{2}=\frac{\ln \left(S_{t} / K\right)+\left(r-(1 / 2) \bar{\sigma}^{2}\right) \tau}{\bar{\sigma} \sqrt{\tau}}
$$
Meanwhile,
$$
\begin{aligned}
E_{t}^{\mathrm{Q}}\left[S_{T} 1_{S_{T}>K}\right] &=\frac{1}{\sqrt{2 \pi}} \int_{-d_{2}}^{\infty} S_{t} \exp \left[\left(r-\frac{1}{2} \bar{\sigma}^{2}\right) \tau+\bar{\sigma} \sqrt{\tau} x-\frac{1}{2} x^{2}\right] \mathrm{d} x \
&=\frac{S_{t} \mathrm{e}^{r \tau}}{\sqrt{2 \pi}} \int_{-d_{2}-\bar{\sigma} \sqrt{\tau}}^{\infty} \exp \left(-\frac{1}{2} y^{2}\right) \mathrm{d} y \
&=S_{t} \mathrm{e}^{r \tau} \Phi\left(d_{1}\right),
\end{aligned}
$$
where
$$
d_{1}=d_{2}+\bar{\sigma} \sqrt{\tau} .
$$
By substituting Equations $2.77$ and $2.79$ into $2.74$, we arrive at the celebrated Black-Scholes formula:
$$
C_{t}=S_{t} \Phi\left(d_{1}\right)-\mathrm{e}^{-r(T-t)} K \Phi\left(d_{2}\right) .
$$
By direct verification, we can show that the hedge ratio, $\varphi_{t}$, is
$$
\frac{\partial C_{t}}{\partial S_{t}}=\Phi\left(d_{1}\right) .
$$
Next, we proceed to derive the formula for a put option, which has the payoff function
$$
P_{T}=\left(K-S_{T}\right)^{+} .
$$

金融代写|利率建模代写Interest Rate Modeling代考|Short Rate and Money Market Accounts

The short rate is associated with a savings account in a bank. The short rate at time $t$ is conventionally denoted as $r_{t}$. Interest on a savings account is accrued daily, using the actual $/ 365$ convention. Let $B_{t}$ denote the account balance at time (or date) $t$, and let $\Delta t=1$ day $=1 / 365$ year. Then the new balance the next day at $t+\Delta t$ is
$$
B_{t+\Delta t}=B_{t}\left(1+r_{t} \Delta t\right)
$$

Because $\Delta t \leqslant 1$, daily compounding is very well approximated by continuous compounding: in the limit of $\Delta t \rightarrow 0$, Equation $3.1$ becomes
$$
\mathrm{d} B_{t}=r_{t} B_{t} \mathrm{~d} t
$$
Because $r_{t}$ is applied to $(t, t+d t)$, an infinitesimal interval of time, it is also called the instantaneous interest rate. As a mathematical approximation and idealization, continuous compounding is necessary to continuous-time finance. Suppose that a sum of money is deposited at $t=0$ into a savings account and that there has not been a deposit or withdrawal since. Then the balance at a later time, $t$, is
$$
B_{t}=B_{0} \mathrm{e}^{\int_{0}^{t} r_{s} \mathrm{~d} s}
$$
In the real world, the balance, $B_{t}$, is not known in advance due to the stochastic nature of the short rate. Nonetheless, the deposit in the savings account is considered a risk-free security, and its return is used as a benchmark to measure the profits and losses of other investments.

In reality, savings accounts for institutions and for individuals offer different interest rates, which reflect different overhead management costs for institutional and individual clients. To distinguish from an individual’s account, we call the savings account for an institution a money market account. Note that this is somewhat an abuse of terminology. In the United States, a money market account is also a type of savings account for retail customers, which offers higher interest rates under some restrictions, including minimum balances and limited numbers of monthly withdrawals. Its compounding rule is also different from continuous compounding. Hence, we need to emphasize here that, in fixed-income modeling, a money market account means a savings account for institutions that compounds continuously. Such a money market account plays an important role in continuous-time modeling of finance.

金融代写|利率建模代写Interest Rate Modeling代考|Term Rates and Certificates of Deposit

Term rates are associated to certificates of deposit (CD). A CD is a deposit that is committed for a fixed period of time, and the interest rate applied to the CD is called a term rate. For retail customers, the available terms are typically one month, three months, six months, and one year. Usually, the longer the term, the higher the term rate, as investors are awarded a higher premium for committing their money for a longer period of time. The interest payments of CDs use simple compounding. Let $r_{t, \Delta t}$ be the interest rate for the term $\Delta t$ and $I_{t}$ be the value of the deposit at time $t$. Then the balance at the maturity of the CD is
$$
I_{t+\Delta t}=I_{t}\left(1+r_{t, \Delta t} \Delta t\right) .
$$
Investors of CDs often roll over their CDs, meaning that after a CD matures, the entire amount (principal plus interest) is deposited into another CD with

the same terms but with the prevailing term rate at the time when the rolling over takes place. Suppose that a CD is rolled over $n$ times. Then the terminal balance at time $t+n \Delta t$ is
$$
I_{t+n \Delta t}=I_{t} \cdot \prod_{i=1}^{n}\left(1+r_{t+(i-1)} \Delta t, \Delta t \Delta t\right) .
$$
If the $\Delta t$ term rate remains unchanged over the investment horizon, that is, $r_{t+(i-1) \Delta t, \Delta t}=r_{t, \Delta t}, i=1, \ldots, n$, then there is
$$
I_{t+n \Delta T}=I_{t}\left(1+r_{t, \Delta t} \Delta t\right)^{n},
$$
and we say that the deposit is compounded $n$ times with interest rate $r_{t, \Delta t}$. We call $\omega=1 / \Delta t$ the compounding frequency, which is the number of compoundings per year. For example, when $\Delta t=3$ months or $0.25$ year, we have $\omega=1 / \Delta t=4$, corresponding to the so-called quarterly compounding. By the way, a savings account is compounded daily, corresponding to $\omega=365$.

Different term rates often mean different rates of return. One way to compare CDs of different terms is to check their effective annual yields (EAY), defined as the dollar-value return over a year for a $\$ 1$ initial investment:
$$
\mathrm{EAY}=\left(1+r_{t, \Delta t} \Delta t\right)^{1 / \Delta t}-1
$$
Should interest rates stay constant over the investment horizon, then a higher EAY gives a higher return in value. In reality, term rates change in a correlated yet random way. Hence, for any fixed investment horizon when rolling over is needed, it is difficult to judge in advance which term is optimal to an investor. In fact, investors often choose terms based on cash flow considerations.

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|The Black-Scholes Formula

考虑资产的欧式看涨期权的定价，小号吨, 有回报

C吨=(小号吨−ķ)+
有时吨. 假设短期利率是一个常数，r吨=r. 根据 Black-Scholes-Merton 方程2.57和终端条件2.58, 看涨期权的价值满足
$$
\left{

∂C吨∂吨+12σ吨2小号2∂2C吨∂小号2+r小号∂C吨∂小号−rC吨=0 C吨=(小号−ķ)+。\正确的。
$$
通过求解这个偏微分方程（PDE）的终值问题，我们可以得到期权的价格。

或者，我们可以通过计算公式中的期望来推导出看涨期权的公式2.71直接地。我们写

C吨=和−r(吨−吨)和吨问[(小号吨−ķ)+] =和−r(吨−吨)(和吨问[小号吨1小号吨>ķ]−ķ和吨问[1小号吨>ķ])
自从

小号吨=小号吨经验⁡[(r−12σ¯2)τ+σ¯τ⋅e],e∼ñ(0,1),
在哪里τ=吨−吨和σ¯是平均波动率，

σ¯=1τ∫0τσs2 ds,
我们有

和吨问[1小号吨>ķ]=概率⁡(e>−ln⁡(小号吨/ķ)+(r−(1/2)σ¯2)τσ¯τ)=披(d2),
和

d2=ln⁡(小号吨/ķ)+(r−(1/2)σ¯2)τσ¯τ
同时，

和吨问[小号吨1小号吨>ķ]=12圆周率∫−d2∞小号吨经验⁡[(r−12σ¯2)τ+σ¯τX−12X2]dX =小号吨和rτ2圆周率∫−d2−σ¯τ∞经验⁡(−12是2)d是 =小号吨和rτ披(d1),
在哪里

d1=d2+σ¯τ.
通过代入方程2.77和2.79进入2.74，我们得出著名的 Black-Scholes 公式：

C吨=小号吨披(d1)−和−r(吨−吨)ķ披(d2).
通过直接验证，我们可以证明对冲比率，披吨，是

∂C吨∂小号吨=披(d1).
接下来，我们继续推导出具有收益函数的看跌期权的公式

磷吨=(ķ−小号吨)+.

金融代写|利率建模代写Interest Rate Modeling代考|Short Rate and Money Market Accounts

短期利率与银行的储蓄账户相关联。短期利率吨通常表示为r吨. 储蓄账户的利息是每天累积的，使用实际的/365惯例。让乙吨表示当时（或日期）的账户余额吨，然后让Δ吨=1天=1/365年。然后第二天在新余额吨+Δ吨是

乙吨+Δ吨=乙吨(1+r吨Δ吨)

因为Δ吨⩽1，每日复利非常近似于连续复利：在Δ吨→0, 方程3.1变成

d乙吨=r吨乙吨 d吨
因为r吨应用于(吨,吨+d吨)，一个无限小的时间间隔，也称为瞬时利率。作为一种数学近似和理想化，连续复利是连续时间金融所必需的。假设有一笔钱存入吨=0存入储蓄账户，此后没有存款或取款。然后在稍后的时间余额，吨，是

乙吨=乙0和∫0吨rs ds
在现实世界中，平衡，乙吨, 由于短期利率的随机性，事先不知道。尽管如此，储蓄账户中的存款被认为是无风险的证券，其回报被用作衡量其他投资损益的基准。

实际上，机构和个人的储蓄账户提供不同的利率，这反映了机构和个人客户的不同管理费用。为了与个人账户区分开来，我们将机构的储蓄账户称为货币市场账户。请注意，这有点滥用术语。在美国，货币市场账户也是零售客户的一种储蓄账户，在某些限制下提供更高的利率，包括最低余额和有限的每月取款次数。它的复利规则也不同于连续复利。因此，我们需要在此强调，在固定收益模型中，货币市场账户是指机构的储蓄账户，可以持续复利。

金融代写|利率建模代写Interest Rate Modeling代考|Term Rates and Certificates of Deposit

定期利率与存款证 (CD) 相关联。CD 是在固定期限内承诺的存款，适用于 CD 的利率称为定期利率。对于零售客户，可用期限通常为一个月、三个月、六个月和一年。通常，期限越长，期限利率越高，因为投资者在较长时间内投入资金会获得更高的溢价。CD 的利息支付使用简单的复利。让r吨,Δ吨是该期限的利率Δ吨和我吨是当时存款的价值吨. 那么 CD 到期时的余额为

我吨+Δ吨=我吨(1+r吨,Δ吨Δ吨).
CD 的投资者经常将他们的 CD 展期，这意味着在 CD 到期后，全部金额（本金加利息）将存入另一张 CD

相同的条款，但采用展期时的现行条款利率。假设一张 CD 被翻转n次。然后是当时的终端余额吨+nΔ吨是

我吨+nΔ吨=我吨⋅∏一世=1n(1+r吨+(一世−1)Δ吨,Δ吨Δ吨).
如果Δ吨长期利率在投资期限内保持不变，即r吨+(一世−1)Δ吨,Δ吨=r吨,Δ吨,一世=1,…,n, 那么有

我吨+nΔ吨=我吨(1+r吨,Δ吨Δ吨)n,
我们说存款是复利的n与利率的倍数r吨,Δ吨. 我们称之为ω=1/Δ吨复利频率，即每年复利的次数。例如，当Δ吨=3月或0.25年，我们有ω=1/Δ吨=4，对应于所谓的季度复利。顺便说一句，储蓄账户每天复利，对应于ω=365.

不同的期限利率通常意味着不同的回报率。比较不同期限 CD 的一种方法是检查它们的有效年收益率 (EAY)，定义为一年内的美元价值回报$1初始投资：

和一个是=(1+r吨,Δ吨Δ吨)1/Δ吨−1
如果利率在投资期限内保持不变，那么较高的 EAY 会带来更高的价值回报。实际上，期限利率以一种相关但随机的方式变化。因此，对于任何需要展期的固定投资期限，很难提前判断哪个期限对投资者来说是最优的。事实上，投资者通常会根据现金流的考虑来选择条款。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

金融代写|利率建模代写Interest Rate Modeling代考|MTH5520

Posted on 2022年6月7日2022年6月7日 by statistics-lab

如果你也在怎样代写利率建模Interest Rate Modeling这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|Multi-Factor Extensions

In derivatives pricing, we often need to model simultaneously the dynamics of multiple risky securities, using multiple risk factors. Because of that, we must extend several major results established so far to the setting of multiple risk sources or assets. These results include the CMG theorem, the martingale representation theorem, and the option pricing formula, as in Equation 2.52. The proofs are parallel to those for the one-dimensional case and thus are omitted for brevity. Hereafter, we use a superscript “T” to denote the transposition of a matrix.

Theorem 2.6.1 (The CMG Theorem). Let $\mathbf{W}{t}=\left(W{1}(t), W_{2}(t), \ldots\right.$,

$\left.W_{n}(t)\right)^{\mathrm{T}}$ be an $n$-dimensional $\mathbb{P}$-Brownian motion, and let $\gamma_{t}=\left(\gamma_{1}(t)\right.$, $\left.\gamma_{2}(t), \ldots, \gamma_{n}(t)\right)^{\mathrm{T}}$ be an $n$-dimensional $\mathcal{F}{t}$-adaptive process, such that $$ E^{P}\left[\exp \left(\frac{1}{2} \int{0}^{T}\left|\gamma_{t}\right|_{2}^{2} d t\right)\right]<\infty
$$
Define a new measure, Q1, with a Radon-Nikodym derivative
$$
\left.\frac{d \mathbb{Q}}{d \mathbb{P}}\right|{\mathcal{F}{t}}=\exp \left(\int_{0}^{t}-\gamma_{s}^{\mathrm{T}} d \mathbf{W}{s}-\frac{1}{2} \int{0}^{t}\left|\gamma_{s}\right|_{2}^{2} d s\right) \text {. }
$$
Then $\mathbb{Q}$ is equivalent to $\mathbb{P}$, and
$$
\tilde{\mathbf{W}}{t}=\mathbf{W}{t}+\int_{0}^{t} \gamma_{s} d s
$$
is an n-dimensional (1-Brownian motion.
Theorem 2.6.2 (The Martingale Representation Theorem). Let $\mathbf{W}{t}$ be an $n$-dimensional Brownian motion and suppose that $\mathbf{M}{t}$ is an $n$-dimensional Q-martingale process, $\mathbf{M}{t}=\left(M{1}(t), M_{2}(t), \ldots, M_{n}(t)\right)^{\mathrm{T}}$, such that
$$
d M_{i}(t)=\sum_{j=1}^{n} a_{i j}(t) d W_{j}(t) .
$$
Let $\mathbf{A}=\left(a_{i j}\right)$ be a non-singular matrix. If $N_{t}$ is any one-dimensional $\mathbb{Q}$ martingale with $E^{Q}\left[N_{t}^{2}\right]<\infty$, there exists an $n$-dimensional $\mathcal{F}{t}$-adaptive process, $\Phi{t}=\left(\varphi_{1}(t), \varphi_{2}(t), \ldots, \varphi_{n}(t)\right)^{\mathrm{T}}$, such that
$$
E^{Q}\left[\int_{0}^{t}\left(\sum_{j} a_{i j}^{2}(s) \varphi_{j}^{2}(s) d s\right)\right]<\infty, \quad \forall i,
$$
and
$$
\begin{aligned}
N_{t} &=N_{0}+\sum_{j=1}^{n} \int_{0}^{t} \varphi_{j}(s) d M_{j}(s) \
& \triangleq N_{0}+\int_{0}^{t} \Phi^{\mathrm{T}}(s) d \mathbf{M}(s) .
\end{aligned}
$$

金融代写|利率建模代写Interest Rate Modeling代考|Existence of a Martingale Measure

We consider a standard model of a complete financial market with a money market account and $n$ risky securities. Let the time $t$ prices be $B_{t}$ and $S_{t}^{i}$, $1 \leq i \leq n$, respectively. We assume lognormal price processes for all assets:
$$
\begin{aligned}
\mathrm{d} B_{t} &=r_{t} B_{t} \mathrm{~d} t \
\mathrm{~d} S_{t}^{i} &=S_{t}^{i}\left(\mu_{t}^{i} \mathrm{~d} t+\sum_{j=1}^{n} \sigma_{i j} \mathrm{~d} W_{j}(t)\right) \
&=S_{t}^{i}\left(\mu_{t}^{i} \mathrm{~d} t+\sigma_{i}^{\mathrm{T}}(t) \mathrm{d} \mathbf{W}{t}\right), \quad i=1,2, \ldots, n . \end{aligned} $$ Here, $$ \sigma{i}^{\mathrm{T}}(t)=\left(\sigma_{i, 1}, \sigma_{i, 2}, \ldots, \sigma_{i, n}\right) .
$$
Let $Z_{t}^{i}=B_{t}^{-1} S_{t}^{i}$ denote the discounted asset price of the $i$ th asset. It then follows that
$$
\mathrm{d} Z_{t}^{i}=Z_{t}^{i}\left[\boldsymbol{\sigma}{i}^{\mathrm{T}}(t) \mathrm{d} \mathbf{W}{t}+\left(\mu_{t}^{i}-r_{t}\right) \mathrm{d} t\right], \quad i=1,2, \ldots, n
$$
To construct a martingale measure for $Z_{t}^{i}, \forall i$, we must “absorb” the drift terms in Equation $2.62$ into the Brownian motion. For that reason, we define an $\mathcal{F}{t}$-adaptive function, $\gamma{t}$, via the following equations:
$$
\boldsymbol{\sigma}{i}^{\mathrm{T}}(t) \gamma{t}=\mu_{t}^{i}-r_{t}, \quad i=1,2, \ldots, n .
$$
Suppose that $\gamma_{t}$, the solution to Equation 2.63, exists and satisfies
$$
E^{2}\left[\exp \left(\int_{0}^{T}\left|\gamma_{t}\right|^{2} \mathrm{~d} t\right)\right]<\infty $$ for some $T>0$. We then can define a new measure, $\mathbb{Q}$, according to Equation 2.59. Under this newly defined $\mathbb{Q}$,
$$
\overline{\mathbf{W}}{t}=\mathbf{W}{t}+\int_{0}^{t} \gamma_{s} \mathrm{~d} s
$$
is a multi-dimensional Brownian motion, with which we can rewrite the price processes for the discounted assets into
$$
\mathrm{d} Z_{t}^{i}=Z_{t}^{i} \sigma_{i}^{\mathrm{T}}(t) \mathrm{d} \hat{\mathbf{W}}{t}, \quad i=1,2, \ldots, n, $$ and $Z{t}^{i}, i=1,2, \ldots, n$ are lognormal $\mathbb{Q}$-martingales.
We now study the existence of $\gamma_{t}$ and condition 2.64. In matrix form, Equation $2.63$ can be recast into
$$
\boldsymbol{\Sigma} \gamma_{t}=\boldsymbol{\mu}{t}-r{t} \mathbf{I}
$$

金融代写|利率建模代写Interest Rate Modeling代考|Pricing Contingent Claims

Now we are ready to address the pricing of a contingent claim depending on the prices of multiple underlying securities. Having found the martingale measure, $\mathbb{Q}$, for the underlying securities, we define a $\mathbb{Q}$-martingale as
$$
N_{t}=E^{Q}\left(B_{T}^{-1} X_{T} \mid \mathcal{F}{t}\right), $$ using the discounted value of $X{T}$, the payoff function of the claim at time $T$. Without loss of generality, we assume that the volatility matrix of the underlying risky securities, $\Sigma$, is non-singular. ${ }^{2}$ According to the

martingale representation theorem, there exists an $\mathcal{F}{t}$-adaptive function, $\Phi{t}=\left(\varphi_{1}(t), \ldots, \varphi_{n}(t)\right)^{\mathrm{T}}$, such that
$$
\mathrm{d} N_{t}=\Phi_{t}^{\mathrm{T}} \mathrm{d} \mathbf{Z}{t} $$ where $\mathbf{Z}{t}$ is the vector of the discounted prices. We now define another process,
$$
\psi_{t}=N_{t}-\boldsymbol{\Phi}{t}^{\mathrm{T}} \mathbf{Z}{t}
$$
and form a portfolio with $\psi_{t}$ units of the money market account and $\phi_{i}(t)$ units of the $i$ th risky security, $i=1, \ldots, n$. The discounted value of the portfolio is
$$
\tilde{V}{t}=\Phi{t}^{\mathrm{T}} \mathbf{Z}{t}+\psi{t}=N_{t}
$$
The last equation implies replication of the payoff of the contingent portfolio. Furthermore, from Equation 2.70, we can derive
$$
\mathrm{d} V_{t}=\Phi_{t}^{\mathrm{T}} \mathrm{d} \mathbf{S}{t}+\psi{t} \mathrm{~d} B_{t}
$$
which implies that the replication strategy is a self-financing one. So, we conclude that the value of the contingent claim equals that of the portfolio and thus is given by
$$
V_{t}=B_{t} E^{\mathbb{Q}}\left[B_{T}^{-1} X_{T} \mid \mathcal{F}{t}\right]=E^{Q}\left[\mathrm{e}^{-\int{t}^{T} r_{x} \mathrm{~d}{s}} X{T} \mid \mathcal{F}_{t}\right]
$$
Formally, Equation $2.71$ is identical to Equation $2.52$, the formula for options on a single underlying security.

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|Multi-Factor Extensions

在衍生品定价中，我们经常需要使用多个风险因素同时对多个风险证券的动态进行建模。正因为如此，我们必须将迄今为止建立的几个主要结果扩展到多个风险源或资产的设置。这些结果包括 CMG 定理、鞅表示定理和期权定价公式，如公式 2.52 所示。证明与一维情况的证明是平行的，因此为简洁起见省略。此后，我们使用上标“T”来表示矩阵的转置。

定理 2.6.1（CMG 定理）。让在吨=(在1(吨),在2(吨),…,

在n(吨))吨豆n维磷-布朗运动，让C吨=(C1(吨),C2(吨),…,Cn(吨))吨豆n维F吨-自适应过程，使得

和磷[经验⁡(12∫0吨|C吨|22d吨)]<∞
使用 Radon-Nikodym 导数定义新的度量 Q1

d问d磷|F吨=经验⁡(∫0吨−Cs吨d在s−12∫0吨|Cs|22ds).
然后问相当于磷，和

在~吨=在吨+∫0吨Csds
是一个 n 维（1-布朗运动。
定理 2.6.2（鞅表示定理）。让在吨豆n维布朗运动并假设米吨是一个n-维Q-鞅过程，米吨=(米1(吨),米2(吨),…,米n(吨))吨, 这样

d米一世(吨)=∑j=1n一个一世j(吨)d在j(吨).
让一个=(一个一世j)是一个非奇异矩阵。如果ñ吨是任何一维的问鞅和问[ñ吨2]<∞, 存在一个n维F吨- 自适应过程，披吨=(披1(吨),披2(吨),…,披n(吨))吨, 这样

和问[∫0吨(∑j一个一世j2(s)披j2(s)ds)]<∞,∀一世,
和

ñ吨=ñ0+∑j=1n∫0吨披j(s)d米j(s) ≜ñ0+∫0吨披吨(s)d米(s).

金融代写|利率建模代写Interest Rate Modeling代考|Existence of a Martingale Measure

我们考虑一个具有货币市场账户的完整金融市场的标准模型，n风险证券。让时间吨价格是乙吨和小号吨一世, 1≤一世≤n，分别。我们假设所有资产的对数正态价格过程：

d乙吨=r吨乙吨 d吨 d小号吨一世=小号吨一世(μ吨一世 d吨+∑j=1nσ一世j d在j(吨)) =小号吨一世(μ吨一世 d吨+σ一世吨(吨)d在吨),一世=1,2,…,n.这里，

σ一世吨(吨)=(σ一世,1,σ一世,2,…,σ一世,n).
让从吨一世=乙吨−1小号吨一世表示资产折现价格一世资产。然后它遵循

d从吨一世=从吨一世[σ一世吨(吨)d在吨+(μ吨一世−r吨)d吨],一世=1,2,…,n
构造一个鞅测度从吨一世,∀一世，我们必须“吸收”方程中的漂移项2.62进入布朗运动。为此，我们定义一个F吨-自适应功能，C吨，通过以下等式：

σ一世吨(吨)C吨=μ吨一世−r吨,一世=1,2,…,n.
假设C吨, 方程 2.63 的解存在并且满足

和2[经验⁡(∫0吨|C吨|2 d吨)]<∞对于一些吨>0. 然后我们可以定义一个新的度量，问，根据公式 2.59。在这个新定义的问,

在¯吨=在吨+∫0吨Cs ds
是一个多维布朗运动，我们可以用它把贴现资产的价格过程改写为

d从吨一世=从吨一世σ一世吨(吨)d在^吨,一世=1,2,…,n,和从吨一世,一世=1,2,…,n是对数正态的问- 鞅。
我们现在研究存在C吨和条件 2.64。以矩阵形式，方程2.63可以重铸成

ΣC吨=μ吨−r吨我

金融代写|利率建模代写Interest Rate Modeling代考|Pricing Contingent Claims

现在我们已经准备好根据多个标的证券的价格来解决或有债权的定价问题。找到鞅测度后，问，对于标的证券，我们定义一个问- 鞅为

ñ吨=和问(乙吨−1X吨∣F吨),使用贴现值X吨, 索赔在时间的支付函数吨. 不失一般性，我们假设基础风险证券的波动率矩阵，Σ, 是非奇异的。2根据

鞅表示定理，存在一个F吨-自适应功能，披吨=(披1(吨),…,披n(吨))吨, 这样

dñ吨=披吨吨d从吨在哪里从吨是折扣价格的向量。我们现在定义另一个过程，

ψ吨=ñ吨−披吨吨从吨
并与ψ吨货币市场账户单位和φ一世(吨)的单位一世危险的安全性，一世=1,…,n. 投资组合的折现值为

在~吨=披吨吨从吨+ψ吨=ñ吨
最后一个等式意味着复制或有投资组合的收益。此外，从方程 2.70，我们可以推导出

d在吨=披吨吨d小号吨+ψ吨 d乙吨
这意味着复制策略是一种自筹资金的策略。因此，我们得出结论，或有债权的价值等于投资组合的价值，因此由下式给出

在吨=乙吨和问[乙吨−1X吨∣F吨]=和问[和−∫吨吨rX dsX吨∣F吨]
正式地，方程2.71与方程相同2.52, 单一基础证券期权的公式。

统计代写请认准statistics-lab™. statistics-lab™为您的留学生涯保驾护航。

金融工程代写

非参数统计代写

非参数统计指的是一种统计方法，其中不假设数据来自于由少数参数决定的规定模型；这种模型的例子包括正态分布模型和线性回归模型。

广义线性模型代考

广义线性模型（GLM）归属统计学领域，是一种应用灵活的线性回归模型。该模型允许因变量的偏差分布有除了正态分布之外的其它分布。

有限元方法代写

随机分析代写

时间序列分析代写

回归分析代写

MATLAB代写

R语言代写	问卷设计与分析代写
PYTHON代写	回归分析与线性模型代写
MATLAB代写	方差分析与试验设计代写
STATA代写	机器学习/统计学习代写
SPSS代写	计量经济学代写
EVIEWS代写	时间序列分析代写
EXCEL代写	深度学习代写
SQL代写	各种数据建模与可视化代写

金融代写|利率建模代写Interest Rate Modeling代考|MATH5985

Posted on 2022年6月7日2022年6月7日 by statistics-lab

如果你也在怎样代写利率建模Interest Rate Modeling这个学科遇到相关的难题，请随时右上角联系我们的24/7代写客服。

我们提供的利率建模Interest Rate Modeling及其相关学科的代写，服务范围广, 其中包括但不限于:

Statistical Inference 统计推断
Statistical Computing 统计计算
Advanced Probability Theory 高等概率论
Advanced Mathematical Statistics 高等数理统计学
(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|MATH5985

金融代写|利率建模代写Interest Rate Modeling代考|The Martingale Representation Theorem

The martingale representation theorem plays a critical role in the so-called martingale approach to derivatives pricing. This theorem has two important

consequences. First, it leads to a general principle for derivatives pricing. Second, it implies a replication or hedging strategy of a derivative using its underlying security. We first present a simple version of the theorem based on a single Brownian filtration, $\mathcal{F}{t}=\sigma\left(W{s}, 0 \leq s \leq t\right)$. We begin with a martingale process, $M_{t}$, such that
$$
\mathrm{d} M_{t}=\sigma_{t} \mathrm{~d} W_{t},
$$
and we call $\sigma_{t}$ the volatility of $M_{t}$.
Theorem 2.3.1 (The Martingale Representation Theorem). Suppose that $N_{t}$ is a $\mathbb{Q}$-martingale process that is adaptive to $\mathcal{F}{t}$ and satisfies $E^{\mathbb{Q}}\left[N{T}^{2}\right]<$ $\infty$ for some T. If the volatility of $M_{t}$ is non-zero almost surely, then there exists a unique $\mathcal{F}{t}-$ adaptive process, $\varphi{t}$, such that $E^{Q}\left[\int_{0}^{T} \varphi_{t}^{2} \sigma_{t}^{2} d t\right]<\infty$ almost surely, and
$$
N_{t}=N_{0}+\int_{0}^{t} \varphi_{s} d M_{s}, \quad t \leq T
$$
or, in differential form,
$$
d N_{t}=\varphi_{t} d M_{t}
$$
The proof combining the techniques of Steele $(2000)$ and $Ø k s e n d a l ~(2003)$ is provided in the appendix of this chapter. A different proof can be found in Korn and Korn (2000).

金融代写|利率建模代写Interest Rate Modeling代考|A Complete Market with Two Securities

We consider the first “complete market” in continuous time, which consists of a money market account and a risky security. The price processes for the two securities, $B_{t}$ and $S_{t}$, are assumed to be
$$
\begin{aligned}
\mathrm{d} B_{t} &=r_{t} B_{t} \mathrm{~d} t, & & B_{0}=1 \
\mathrm{~d} S_{t} &=S_{t}\left(\mu_{t} \mathrm{~d} t+\sigma_{t} \mathrm{~d} W_{t}\right), & S_{0} &=S_{0}
\end{aligned}
$$
Here, the volatility of the risky asset is $\sigma_{t} \neq 0$ almost surely, and the short rate, $r_{t}$, can be stochastic. Denote the discounted price of the risky asset as $Z_{t}=B_{t}^{-1} S_{t}$, which can be shown to follow the process
$$
\begin{aligned}
\mathrm{d} Z_{t} &=Z_{t}\left(\left(\mu_{t}-r_{t}\right) \mathrm{d} t+\sigma_{t} \mathrm{~d} W_{t}\right) \
&=Z_{t} \sigma_{t} \mathrm{~d}\left(W_{t}+\int_{0}^{t} \frac{\left(\mu_{s}-r_{s}\right)}{\sigma_{s}} \mathrm{~d} s\right)
\end{aligned}
$$
By introducing
$$
\gamma_{t}=\frac{\mu_{t}-r_{t}}{\sigma_{t}}
$$

which is $\mathcal{F}{t}$-adaptive, and by defining a new measure, Q, according to Equation $2.36$, we have $$ \tilde{W}{t}=W_{t}+\int_{0}^{t} \gamma_{s} \mathrm{~d} s,
$$
which is a Q-Brownian motion. In terms of $\tilde{W}{t}, Z{t}$ satisfies
$$
\mathrm{d} Z_{t}=\sigma_{t} Z_{t} \mathrm{~d} \tilde{W}_{t},
$$
which is a lognormal Q-martingale. Recall that in the binomial model for option pricing, we also derived the martingale measure for the underlying security.

金融代写|利率建模代写Interest Rate Modeling代考|Replicating and Pricing of Contingent Claims

Let $X_{T}$ be a contingent claim (or option) with payoff day or maturity $T$. The claim is an $\mathcal{F}{T}$-adaptive function whose value depends on $\left{S{t}, 0 \leq t \leq T\right}$. Define first a $\mathbb{Q}$-martingale with the discounted payoff:
$$
N_{t}=E^{Q}\left(B_{T}^{-1} X_{T} \mid \mathcal{F}{t}\right) . $$ Without loss of generality, we assume that $E^{\mathrm{Q}}\left[N{t}^{2}\right]<\infty$. According to the martingale representation theorem, there exists an $\mathcal{F}{t}$-adaptive function, $\varphi{t}$, such that
$$
\mathrm{d} N_{t}=\varphi_{t} \mathrm{~d} Z_{t},
$$
where $Z_{t}$, defined in the last section, is the discounted price of $S_{t}$. Next, we define
$$
\psi_{t}=N_{t}-\varphi_{t} Z_{t} .
$$
Consider now the portfolio with $\varphi_{t}$ units of the stock and $\psi_{t}$ units of the money market account, denoted as $\left(\varphi_{t}, \psi_{t}\right)$. According to the definition of $\psi_{t}$, the discount value of the replication portfolio is
$$
\tilde{V}{t}=\varphi{t} Z_{t}+\psi_{t}=N_{t}
$$
This portfolio has two important properties. First, at time $T$, when the option matures,
$$
\tilde{V}{T}=N{T}=B_{T}^{-1} X_{T}
$$
which suggests that the (discounted) value of the portfolio equals that of the option. In other words, the portfolio replicates the payoff of the contingent claim. Second, the replicating portfolio is a self-financing one, meaning that it can track the asset allocation, $\left(\varphi_{t}, \psi_{t}\right)$, without the need for either capital

infusion or capital withdrawal. In fact, based on Equations $2.45$ and $2.47$, we have
$$
\mathrm{d} \tilde{V}{t}=\mathrm{d} N{t}=\varphi_{t} \mathrm{~d} Z_{t}
$$
In terms of the spot value, $B_{t}$ and $S_{t}$, Equation $2.49$ becomes
$$
\begin{aligned}
\mathrm{d} V_{t} &=\mathrm{d}\left(\tilde{V}{t} B{t}\right) \
&=B_{t} \mathrm{~d} \tilde{V}{t}+\tilde{V}{t} \mathrm{~d} B_{t} \
&=B_{t} \varphi_{t} \mathrm{~d} Z_{t}+\left(\varphi_{t} Z_{t}+\psi_{t}\right) \mathrm{d} B_{t} \
&=\varphi_{t}\left(B_{t} \mathrm{~d} Z_{t}+Z_{t} \mathrm{~d} B_{t}\right)+\psi_{t} \mathrm{~d} B_{t} \
&=\varphi_{t} \mathrm{~d}\left(B_{t} Z_{t}\right)+\psi_{t} \mathrm{~d} B_{t} \
&=\varphi_{t} \mathrm{~d} S_{t}+\psi_{t} \mathrm{~d} B_{t}
\end{aligned}
$$
A direct consequence of the above equation is the equality
$$
\begin{aligned}
\varphi_{t+d t} S_{t+d t}+\psi_{t+d t} B_{t+d t} &=\varphi_{t} S_{t}+\psi_{t} B_{t}+\varphi_{t} \mathrm{~d} S_{t}+\psi_{t} \mathrm{~d} B_{t} \
&=\varphi_{t} S_{t+d t}+\psi_{t} B_{t+d t},
\end{aligned}
$$

利率建模代考

金融代写|利率建模代写Interest Rate Modeling代考|The Martingale Representation Theorem

鞅表示定理在衍生品定价的所谓鞅方法中起着关键作用。这个定理有两个重要的

结果。首先，它引出了衍生品定价的一般原则。其次，它意味着使用其基础证券的衍生品的复制或对冲策略。我们首先提出一个基于单个布朗过滤的简单版本的定理，F吨=σ(在s,0≤s≤吨). 我们从鞅过程开始，米吨, 这样

d米吨=σ吨 d在吨,
我们打电话给σ吨的波动性米吨.
定理 2.3.1（鞅表示定理）。假设ñ吨是一个问- 适应于鞅过程F吨并满足和问[ñ吨2]< ∞对于一些 T。如果米吨几乎肯定是非零的，那么存在唯一的F吨−适应过程，披吨, 这样和问[∫0吨披吨2σ吨2d吨]<∞几乎可以肯定，并且

ñ吨=ñ0+∫0吨披sd米s,吨≤吨
或者，以微分形式，

dñ吨=披吨d米吨
结合斯蒂尔技术的证明(2000)和ØØķs和nd一个l (2003)在本章的附录中提供。在 Korn and Korn (2000) 中可以找到不同的证明。

金融代写|利率建模代写Interest Rate Modeling代考|A Complete Market with Two Securities

我们考虑连续时间的第一个“完整市场”，它由货币市场账户和风险证券组成。两种证券的价格过程，乙吨和小号吨, 假设为

d乙吨=r吨乙吨 d吨,乙0=1 d小号吨=小号吨(μ吨 d吨+σ吨 d在吨),小号0=小号0
在这里，风险资产的波动率为σ吨≠0几乎可以肯定，短期利率，r吨, 可以是随机的。将风险资产的折现价格表示为从吨=乙吨−1小号吨, 可以显示遵循该过程

d从吨=从吨((μ吨−r吨)d吨+σ吨 d在吨) =从吨σ吨 d(在吨+∫0吨(μs−rs)σs ds)
通过介绍

C吨=μ吨−r吨σ吨

这是F吨-自适应，并通过定义一个新的度量，Q，根据方程2.36，我们有

在~吨=在吨+∫0吨Cs ds,
这是一个Q-布朗运动。按照在~吨,从吨满足

d从吨=σ吨从吨 d在~吨,
这是一个对数正态 Q-鞅。回想一下，在期权定价的二项式模型中，我们还推导出了基础证券的鞅测度。

金融代写|利率建模代写Interest Rate Modeling代考|Replicating and Pricing of Contingent Claims

让X吨是具有支付日或到期日的或有债权（或期权）吨. 索赔是一个F吨- 自适应函数，其值取决于\left{S{t}, 0 \leq t \leq T\right}\left{S{t}, 0 \leq t \leq T\right}. 首先定义一个问- 贴现收益的鞅：

ñ吨=和问(乙吨−1X吨∣F吨).不失一般性，我们假设和问[ñ吨2]<∞. 根据鞅表示定理，存在一个F吨-自适应功能，披吨, 这样

dñ吨=披吨 d从吨,
在哪里从吨，在最后一节中定义，是折扣价小号吨. 接下来，我们定义

ψ吨=ñ吨−披吨从吨.
现在考虑投资组合披吨股票的单位和ψ吨货币市场账户的单位，表示为(披吨,ψ吨). 根据定义ψ吨，复制组合的折现值为

在~吨=披吨从吨+ψ吨=ñ吨
这个投资组合有两个重要的属性。首先，在时间吨，当期权到期时，
$$
\tilde{V} {T}=N {T}=B_{T}^{-1} X_{T}
$$
这表明投资组合的（贴现）价值等于的选项。换言之，投资组合复制了或有债权的回报。其次，复制投资组合是一种自筹资金的投资组合，这意味着它可以跟踪资产配置，(披吨,ψ吨), 不需要任何资本

注入或撤资。事实上，基于方程2.45和2.47，我们有

d在~吨=dñ吨=披吨 d从吨
从现货价值来看，乙吨和小号吨, 方程2.49变成

d在吨=d(在~吨乙吨) =乙吨 d在~吨+在~吨 d乙吨 =乙吨披吨 d从吨+(披吨从吨+ψ吨)d乙吨 =披吨(乙吨 d从吨+从吨 d乙吨)+ψ吨 d乙吨 =披吨 d(乙吨从吨)+ψ吨 d乙吨 =披吨 d小号吨+ψ吨 d乙吨
上述等式的直接结果是等式

披吨+d吨小号吨+d吨+ψ吨+d吨乙吨+d吨=披吨小号吨+ψ吨乙吨+披吨 d小号吨+ψ吨 d乙吨 =披吨小号吨+d吨+ψ吨乙吨+d吨,

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