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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考| A Binomial Tree for the Ho–Lee Model

The IIo-Lee model was first presented with a binomial tree. For a Gaussian short-rate model with mean and variance of change over $(t, t+\Delta t)$ given by
\begin{aligned} E^{\mathbb{Q}}\left[\Delta r_t\right] &=\theta_t \Delta t \ \operatorname{VaR}\left(\Delta r_t\right) &=\sigma^2 \Delta t \end{aligned}
we consider a rather natural binomial tree approximation as illustrated in Figure 5.1, where, without loss of generality, the branching probabilities are uniformly one half.
For notational efficiency, we let
$$r_{i, n}=r_{0,0}+\Delta t \sum_{k=1}^{n-1} \theta_k+(2 i-n) \sigma \sqrt{\Delta t}, \quad i=0,1, \ldots, n$$

Then we have a multi-period tree as shown in Figure 5.2.
Before being applied to derivatives pricing, such a tree must first be calibrated to the current term structure of the interest rate. For the Ho-Lee model, we need to determine the drift, $\theta_t$, by reproducing the prices of zero-coupon bonds of all maturities. This task can be efficiently achieved with the help of the so-called Arrow-Debreu prices.

## 金融代写|利率建模代写Interest Rate Modeling代考|Arrow–Debreu Prices

An Arrow-Debreu (1954) security is a canonical asset that has a cash flow of $\$ 1$if a particular state (of interest rate) is realized, or nothing otherwise. The pattern of payment is shown in Figure$5.3$, where we let$Q_{i, j}$denote the price of the security at time 0 that would pay$\$1$ at time $j$ if the state $i$ is realized, or nothing if otherwise.

Note that a zero-coupon bond can be regarded as a portfolio of ArrowDebreu securities. By linearity, the price of the zero-coupon bond maturing in time $j$ is equal to
$$P(0, j)=\sum_{i=0}^j Q_{i, j} .$$
Given an interest-rate tree as in Figure 5.2, we can construct the ArrowDebreu tree through a forward induction process. We begin with
$$Q_{0,0}=1 .$$
The calculations of $Q_{1,1}$ and $Q_{0,1}$ are done by “expectation pricing” using the trees in Figure $5.4$, where $r_{0,0}$ is the discount rate at node $(0,0)$. Intuitively, the prices of the two Arrow-Debreu securities are given by
$$Q_{1,1}=Q_{0,1}=\frac{1}{2} \mathrm{e}^{-r_{0,0} \Delta t}$$

## 金融代写|利率建模代写利率建模代考| Ho-Lee模型的二叉树

IIo-Lee模型首先用二叉树表示。对于均值和方差大于$(t, t+\Delta t)$的高斯短期速率模型(
\begin{aligned} E^{\mathbb{Q}}\left[\Delta r_t\right] &=\theta_t \Delta t \ \operatorname{VaR}\left(\Delta r_t\right) &=\sigma^2 \Delta t \end{aligned}
)，我们考虑一种相当自然的二叉树近似，如图5.1所示，在不丧失一般性的情况下，分支概率一致为1 / 2。

$$r_{i, n}=r_{0,0}+\Delta t \sum_{k=1}^{n-1} \theta_k+(2 i-n) \sigma \sqrt{\Delta t}, \quad i=0,1, \ldots, n$$

## 金融代写|利率建模代写Interest Rate Modeling代考| 阿罗-德布鲁·普莱斯

a Arrow-Debreu(1954)证券是一种典型资产，如果实现了特定的(利率)状态，或者其他什么都没有，那么它的现金流为$\$ 1$。支付模式如图$5.3$所示，其中我们让$Q_{i, j}$表示时间0时的证券价格，如果状态$i$实现，则在时间$j$时支付$\$1$，否则则不支付

$$P(0, j)=\sum_{i=0}^j Q_{i, j} .$$

$$Q_{0,0}=1 .$$

$$Q_{1,1}=Q_{0,1}=\frac{1}{2} \mathrm{e}^{-r_{0,0} \Delta t}$$ 给出

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考| GENERAL MARKOVIAN MODELS

Existing short-rate models are Markovian models. A no-arbitrage shortrate model should also be derived from the HJM framework. However, this can be quite difficult. In this section, we address the opposite question: under what kind of forward-rate volatility specifications should the resulting short-rate model be a Markovian random variable? Answering this question will help us to calibrate and implement a short-rate model more efficiently.
According to Equation 4.21, the short rate can be expressed as
$$r_t=f(t, t)=f(0, t)+\int_0^t\left[-\boldsymbol{\sigma}^{\mathrm{T}}(s, t) \mathbf{\Sigma}(s, t) \mathrm{d} s+\boldsymbol{\sigma}^{\mathrm{T}}(s, t) \mathrm{d} \mathbf{W}_s\right]$$

where $\mathbf{W}t$ is the $n$-dimensional Brownian motion under the risk-neutral measure, $\sigma(t, T)$ the forward-rate volatility, and $\boldsymbol{\Sigma}(t, T)$ the volatility of the $T$-maturity zero-coupon bond, given by $\boldsymbol{\Sigma}(t, T)=-\int_t^T \boldsymbol{\sigma}(t, u) \mathrm{d} u$. The stochastic differentiation of the short rate is \begin{aligned} \mathrm{d} r_t=& {\left[f_t(0, t)+\int_0^t\left(-\frac{\partial}{\partial t}\left(\boldsymbol{\sigma}^{\mathrm{T}}(s, t) \boldsymbol{\Sigma}(s, t)\right) \mathrm{d} s+\frac{\partial \boldsymbol{\sigma}^{\mathrm{T}}(s, t)}{\partial t} \mathrm{~d} \mathbf{W}_s\right)\right] \mathrm{d} t } \ &+\boldsymbol{\sigma}^{\mathrm{T}}(t, t) \mathrm{d} \mathbf{W}_t \ =& {\left[f_t(t, T)\right]{T=t} \mathrm{~d} t+\boldsymbol{\sigma}^{\mathrm{T}}(t, t) \mathrm{d}t . } \end{aligned} Based on Equation $5.17$ we can make the following judgment: for the shortrate model to be a Markovian process, we need the drift term, $\left[f_t(t, T)\right]{T=t}$, to be a function of a finite set of state variables that are jointly Markovian in their evolution.

To write the short rate as a function of several state variables, we introduce auxiliary functions
$$b_i(t, T)=\sigma_i(t, T) \int_t^T \sigma_i(t, s) \mathrm{d} s, \quad i=1,2, \ldots, n .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Monte Carlo Simulations for Options Pricing

Owing to the Markovian property of short-rate models, path simulations by Monte Carlo methods can be carried out efficiently, which is important for pricing exotic and path-dependent options. Take the pricing of the option on a zero-coupon for example. The value can be expressed as
$$V_t=E_t^{\mathbb{Q}}\left[\mathrm{e}^{-\int_t^T r_s \mathrm{~d} s}(P(T, \tau)-K)^{+}\right], \quad t<T<\tau$$
where $\mathbb{Q}$ stands for the risk-neutral measure, $r_t$ is given by Equation $5.20$, and the bond price is given by Equation 5.39. Both variables are expressed in terms of $\chi_i(t)$ and $\varphi_i(t), i=1, \ldots, n$, which evolve according to Equation 5.23. The corresponding simulation scheme for $\chi_i(t)$ and $\varphi_i(t)$ is
\begin{aligned} &\varphi_i(t+\Delta t)-\varphi_i(t)+\left(\sigma_i^2(t, t)-2 \kappa_i(t) \varphi_i(t)\right) \Delta t \ &\chi_i(t+\Delta t)=\chi_i(t)+\left(\varphi_i(t)-\kappa_i(t) \chi_i(t)\right) \mathrm{d} t+\sigma_i(t, t) \Delta W_i(t) \end{aligned}
which is simply the so-called Euler scheme. The bond option is priced by simulating many payoffs before taking an average.
In Inui and Kijima (1998), the following example is considered:
$$\boldsymbol{\sigma}(t, T)=\left(\begin{array}{c} c_1 r_t^\alpha \ c_2 r_t^\beta \mathrm{e}^{-\kappa(T-t)} \end{array}\right)$$
where $c_i, i=1,2, \alpha, \beta$, and $\kappa$ are non-negative constants. It can be verified that the components of the volatility vector satisfy
$$\frac{\partial \sigma_1(t, T)}{\partial T}=0, \quad \frac{\partial \sigma_2(t, T)}{\partial T}=-\kappa \sigma_2(t, T) .$$

## 金融代写|利率建模代写Interest Rate Modeling代考| 一般马尔可夫模型

$$r_t=f(t, t)=f(0, t)+\int_0^t\left[-\boldsymbol{\sigma}^{\mathrm{T}}(s, t) \mathbf{\Sigma}(s, t) \mathrm{d} s+\boldsymbol{\sigma}^{\mathrm{T}}(s, t) \mathrm{d} \mathbf{W}_s\right]$$

where $\mathbf{W}t$ 是 $n$风险中性测度下的-维布朗运动， $\sigma(t, T)$ 远期利率波动 $\boldsymbol{\Sigma}(t, T)$ 波动率 $T$到期零息债券，由 $\boldsymbol{\Sigma}(t, T)=-\int_t^T \boldsymbol{\sigma}(t, u) \mathrm{d} u$。短期汇率的随机微分是 \begin{aligned} \mathrm{d} r_t=& {\left[f_t(0, t)+\int_0^t\left(-\frac{\partial}{\partial t}\left(\boldsymbol{\sigma}^{\mathrm{T}}(s, t) \boldsymbol{\Sigma}(s, t)\right) \mathrm{d} s+\frac{\partial \boldsymbol{\sigma}^{\mathrm{T}}(s, t)}{\partial t} \mathrm{~d} \mathbf{W}_s\right)\right] \mathrm{d} t } \ &+\boldsymbol{\sigma}^{\mathrm{T}}(t, t) \mathrm{d} \mathbf{W}_t \ =& {\left[f_t(t, T)\right]{T=t} \mathrm{~d} t+\boldsymbol{\sigma}^{\mathrm{T}}(t, t) \mathrm{d}t . } \end{aligned} 基于方程 $5.17$ 我们可以做出以下判断:要使短期模型为马尔可夫过程，我们需要漂移项， $\left[f_t(t, T)\right]{T=t}$，是一组有限状态变量的函数，这些状态变量在演化过程中共同具有马尔可夫性 为了将短期汇率写成几个状态变量的函数，我们引入了辅助函数
$$b_i(t, T)=\sigma_i(t, T) \int_t^T \sigma_i(t, s) \mathrm{d} s, \quad i=1,2, \ldots, n .$$

## 金融代写|利率建模代写利率建模代考|期权定价的蒙特卡洛模拟

$$V_t=E_t^{\mathbb{Q}}\left[\mathrm{e}^{-\int_t^T r_s \mathrm{~d} s}(P(T, \tau)-K)^{+}\right], \quad t<T<\tau$$
，其中$\mathbb{Q}$表示风险中性测度，$r_t$由式$5.20$给出，债券价格由式5.39给出。两个变量都用$\chi_i(t)$和$\varphi_i(t), i=1, \ldots, n$表示，根据公式5.23进行演化。$\chi_i(t)$和$\varphi_i(t)$对应的模拟方案是
\begin{aligned} &\varphi_i(t+\Delta t)-\varphi_i(t)+\left(\sigma_i^2(t, t)-2 \kappa_i(t) \varphi_i(t)\right) \Delta t \ &\chi_i(t+\Delta t)=\chi_i(t)+\left(\varphi_i(t)-\kappa_i(t) \chi_i(t)\right) \mathrm{d} t+\sigma_i(t, t) \Delta W_i(t) \end{aligned}
，这就是所谓的欧拉方案。债券期权的定价是通过在取平均值之前模拟许多支付进行的。在Inui和Kijima(1998)中，考虑以下例子:
$$\boldsymbol{\sigma}(t, T)=\left(\begin{array}{c} c_1 r_t^\alpha \ c_2 r_t^\beta \mathrm{e}^{-\kappa(T-t)} \end{array}\right)$$

$$\frac{\partial \sigma_1(t, T)}{\partial T}=0, \quad \frac{\partial \sigma_2(t, T)}{\partial T}=-\kappa \sigma_2(t, T) .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考| ON THE LOGNORMAL SPECIFICATION OF FORWARD RATES

We now explore the possibility of using the state-dependent volatility function in the HJM model. Without loss of generality, we consider the forward-rate volatility function of the form
$$\sigma(t, T)=\sigma_0(t, T) f^\alpha(t, T),$$

where $\sigma_0(t, T)$ is a deterministic function and $\alpha$ a positive exponent. In the special case, $\alpha=0$, we obtain a Gaussian model.

Similar to Avellaneda and Laurence (1999), we show that the “lognormal” model, corresponding to $\alpha=1$, blows up in finite time in the sense that a forward rate reaches infinity. This result was first obtained by Morton (1988). One can imagine that similar results may apply to the case of $\alpha>0$. Hence, volatility specification in the form of Equation $4.136$ is denied.

It suffices to show the result with a one-factor model. The no-arbitrage condition dictates that the drift must be
$$\mu(t, T)=f(t, T) \sigma_0(t, T) \int_t^T f(t, s) \sigma_0(t, s) \mathrm{d} s,$$
which depends on the entire curve of $f(t, s), t \leq s \leq T$. Consider the simplest specification of $\sigma_0(t, T)$ : $\sigma_0(t, T)=\sigma_0=$ constant. The HJM equation then becomes
$$\frac{\mathrm{d} f(t, T)}{f(t, T)}=\sigma_0 \mathrm{~d} \tilde{W}_t+\left(\sigma_0^2 \int_t^T f(t, s) \mathrm{d} s\right) \mathrm{d} t .$$
The formal solution to the above equation is
\begin{aligned} f(t, T) &=f(0, T) \exp \left(\sigma_0 \tilde{W}_t-\frac{\sigma_0^2}{2} t+\sigma_0^2 \int_0^t\left(\int_s^T f(s, u) \mathrm{d} u\right) \mathrm{d} s\right) \ &=f(0, T) M(t) \exp \left(\sigma_0^2 \int_0^t\left(\int_s^T f(s, u) \mathrm{d} u\right) \mathrm{d} s\right) \end{aligned} where $M(t)=\exp \left(\sigma_0 \tilde{W}_t-\left(\sigma_0^2 / 2\right) t\right)$. Assume for simplicity that the initial term structure is flat, that is, $f(0, T)=f_0=$ constant.

## 金融代写|利率建模代写Interest Rate Modeling代考|FROM SHORT-RATE MODELS TO FORWARD-RATE MODELS

Short-rate models dominated fixed-income modeling before the emergence of the no-arbitrage framework of Heath, Jarrow, and Morton (1992), which is based on forward rates. Short-rate models can be made arbitrage free by taking appropriate drift terms, such as the Ho-Lee model and the I Iull-White model. But this is not always easy. One way to derive the correct drift term is to identify the corresponding forward-rate volatility and then to solve for the expression of the forward rates, which include the short rate as an extreme case, from the HJM equation. The focus in this section is on how to derive the corresponding forward-rate volatility in order to identify the model as a special case of the HJM framework.

Consider in general an Ito’s process for the short rate under the riskneutral measure, $\mathbb{Q}$,
$$\mathrm{d} r_t=v\left(r_t, t\right) \mathrm{d} t+\rho\left(r_t, t\right) \mathrm{d} W_t,$$ where the drift, $v\left(r_t, t\right)$, and volatility, $\rho\left(r_t, t\right)$, are deterministic functions of their arguments. Note that, for notational simplicity, we hereafter drop ” $\sim$ ” over the $\mathbb{Q}$-Brownian motion, $W_t$. Define an auxiliary function
$$g(x, t, T)=-\ln E^{\mathbb{Q}}\left[\exp \left(-\int_t^T r_s \mathrm{~d} s\right) \mid r_t=x\right]$$
We have the following result (Baxter and Rennie, 1996).

## 金融代写|利率建模代写Interest Rate Modeling代考| 关于远期汇率的对数正规规范

$$\sigma(t, T)=\sigma_0(t, T) f^\alpha(t, T),$$

$$\mu(t, T)=f(t, T) \sigma_0(t, T) \int_t^T f(t, s) \sigma_0(t, s) \mathrm{d} s,$$
，这取决于$f(t, s), t \leq s \leq T$的整条曲线。考虑$\sigma_0(t, T)$: $\sigma_0(t, T)=\sigma_0=$常量的最简单规范。HJM方程于是变成
$$\frac{\mathrm{d} f(t, T)}{f(t, T)}=\sigma_0 \mathrm{~d} \tilde{W}_t+\left(\sigma_0^2 \int_t^T f(t, s) \mathrm{d} s\right) \mathrm{d} t .$$

\begin{aligned} f(t, T) &=f(0, T) \exp \left(\sigma_0 \tilde{W}_t-\frac{\sigma_0^2}{2} t+\sigma_0^2 \int_0^t\left(\int_s^T f(s, u) \mathrm{d} u\right) \mathrm{d} s\right) \ &=f(0, T) M(t) \exp \left(\sigma_0^2 \int_0^t\left(\int_s^T f(s, u) \mathrm{d} u\right) \mathrm{d} s\right) \end{aligned}，其中$M(t)=\exp \left(\sigma_0 \tilde{W}_t-\left(\sigma_0^2 / 2\right) t\right)$。为简单起见，假设初始期限结构是平坦的，即$f(0, T)=f_0=$常数。

## 金融代写|利率建模代写Interest Rate Modeling代考|从短期利率模型到远期利率模型

$$\mathrm{d} r_t=v\left(r_t, t\right) \mathrm{d} t+\rho\left(r_t, t\right) \mathrm{d} W_t,$$，其中漂移，$v\left(r_t, t\right)$和波动率，$\rho\left(r_t, t\right)$是其参数的确定性函数。注意，为了表示法的简单性，我们以后在$\mathbb{Q}$ -布朗运动$W_t$上省略“$\sim$”。定义一个辅助函数
$$g(x, t, T)=-\ln E^{\mathbb{Q}}\left[\exp \left(-\int_t^T r_s \mathrm{~d} s\right) \mid r_t=x\right]$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考|CHANGE OF MEASURES UNDER BROWNIAN FILTRATION

2.2.1 The Radon-Nikodym Derivative of a Brownian Path
Consider a path of $\mathbb{P}$-Brownian motion over $(0, t)$ with discrete time stepping,
$${W(0)=0, W(\Delta t), W(2 \Delta t), \ldots, W(n \Delta t)}$$
where $\Delta t=t / n$. With the probability ratio in mind, our immediate question is what the path probability is. The answer, unfortunately, is zero.

The implication that we cannot define the notion of the probability ratio given that the same path is realized under two different probability measures. To circumvent this problem, we first seek to calculate the probability for the Brownian motion to travel in a corridor (the so-called corridor probability), as is shown in Figure 2.5, and then we define the ratio of the corridor probabilities. The ratio of the path probabilities is finally defined through a limiting procedure. The corridor can be represented by the intervals $A_{i}=\left(x_{i}-(\Delta x / 2), x_{i}+(\Delta x / 2)\right), i=1,2, \ldots, n$, where $x_{i}=W(i \Delta t)$ and $\Delta x>0$ is a small number.

For a Brownian motion, the marginal distribution at $t_{i}=i \Delta t$ is known to be
$$f_{\mathrm{P}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}\right)^{2} / \Delta t\right]} \sim N\left(x_{i}, \Delta t\right) .$$
Hence, the probability for the next step to fall in $A_{i+1}$ is
\begin{aligned} \operatorname{Prob}{\mathbb{P}}\left(A{i+1}\right) &=\int_{x_{i+1}-\Delta x / 2}^{x_{i+1}+\Delta x / 2} f_{\mathrm{P}}(x) \mathrm{d} x \ & \approx f_{\mathrm{P}}\left(x_{i+1}\right) \Delta x=\frac{\Delta x}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x_{i+1}-x_{i}\right)^{2} / \Delta t\right]} . \end{aligned}
Approximately, we can define the corridor probability to be
$$\prod_{i=1}^{n} \operatorname{Prob}{\mathbb{P}}\left(A{i}\right)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}\right)^{2}}$$
Next, suppose that the same path is realized under a different marginal probability,
$$f_{\mathbb{Q}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}+\gamma \Delta t\right)^{2} / \Delta t\right]} \sim N\left(x_{i}-\gamma \Delta t, \Delta t\right), \quad \forall i$$

where $\gamma$ is taken to be constant for simplicity. Then the corresponding corridor probability can be similarly obtained to be
$$\prod_{i=1}^{n} \operatorname{Prob}{\mathrm{Q}}\left(A{i}\right)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}+\gamma \Delta t\right)^{2}}$$

## 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 socalled 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 2.2 (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 $\int_{0}^{T} \varphi_{t}^{2} \sigma_{t}^{2} \mathrm{~d} t<\infty$ almost surely, and
$$N_{t}=N_{0}+\int_{0}^{t} \varphi_{s} \mathrm{~d} M_{s}, \quad t \leq T$$
or, in differential form,
$$\mathrm{d} N_{t}=\varphi_{t} \mathrm{~d} M_{t} .$$
A sketchy proof along the lines of Steele (2000) is provided at the end 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, $\mathbb{Q}$, according to Equation 2.36, we have $$\tilde{W}{t}=W_{t}+\int_{0}^{t} \gamma_{s} \mathrm{~d} s$$
which is a $\mathbb{Q}$-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 $\mathbb{Q}$-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|CHANGE OF MEASURES UNDER BROWNIAN FILTRATION

$2.2 .1$ 布朗路径的 Radon-Nikodym 导数

$$W(0)=0, W(\Delta t), W(2 \Delta t), \ldots, W(n \Delta t)$$

$A_{i}=\left(x_{i}-(\Delta x / 2), x_{i}+(\Delta x / 2)\right), i=1,2, \ldots, n$ ， 在哪里 $x_{i}=W(i \Delta t)$ 和 $\Delta x>0$ 是一个数字。

$$f_{\mathrm{P}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}\right)^{2} / \Delta t\right]} \sim N\left(x_{i}, \Delta t\right)$$

$$\operatorname{Prob} \mathbb{P}(A i+1)=\int_{x_{i+1}-\Delta x / 2}^{x_{i+1}+\Delta x / 2} f_{\mathrm{P}}(x) \mathrm{d} x \quad \approx f_{\mathrm{P}}\left(x_{i+1}\right) \Delta x=\frac{\Delta x}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x_{i+1}-x_{i}\right)^{2} / \Delta t\right]} .$$

$$\prod_{i=1}^{n} \operatorname{Prob} \mathbb{P}(A i)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}\right)^{2}}$$

$$f_{\mathbb{Q}}(x)=\frac{1}{\sqrt{2 \pi \Delta t}} \mathrm{e}^{-(1 / 2)\left[\left(x-x_{i}+\gamma \Delta t\right)^{2} / \Delta t\right]} \sim N\left(x_{i}-\gamma \Delta t, \Delta t\right), \quad \forall i$$

$$\prod_{i=1}^{n} \operatorname{Prob} \mathrm{Q}(A i)=\left(\frac{\Delta x}{\sqrt{2 \pi \Delta t}}\right)^{n} \mathrm{e}^{-(1 / 2 \Delta t) \sum_{i=0}^{n-1}\left(x_{i+1}-x_{i}+\gamma \Delta t\right)^{2}}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|THE MARTINGALE REPRESENTATION THEOREM

$$\mathrm{d} M_{t}=\sigma_{t} \mathrm{~d} W_{t},$$

$$N_{t}=N_{0}+\int_{0}^{t} \varphi_{s} \mathrm{~d} M_{s}, \quad t \leq T$$

$$\mathrm{d} N_{t}=\varphi_{t} \mathrm{~d} M_{t}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|A COMPLETE MARKET WITH TWO SECURITIES

$$\mathrm{d} B_{t}=r_{t} B_{t} \mathrm{~d} t, \quad B_{0}=1, \mathrm{~d} S_{t}=S_{t}\left(\mu_{t} \mathrm{~d} t+\sigma_{t} \mathrm{~d} W_{t}\right), S_{0}=S_{0}$$

$$\mathrm{d} Z_{t}=Z_{t}\left(\left(\mu_{t}-r_{t}\right) \mathrm{d} t+\sigma_{t} \mathrm{~d} W_{t}\right) \quad=Z_{t} \sigma_{t} \mathrm{~d}\left(W_{t}+\int_{0}^{t} \frac{\left(\mu_{s}-r_{s}\right)}{\sigma_{s}} \mathrm{~d} s\right)$$

$$\gamma_{t}=\frac{\mu_{t}-r_{t}}{\sigma_{t}}$$

$$\tilde{W} t=W_{t}+\int_{0}^{t} \gamma_{s} \mathrm{~d} s$$

$$\mathrm{d} Z_{t}=\sigma_{t} Z_{t} \mathrm{~d} \tilde{W}_{t}$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

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

We finish this chapter with the introduction of martingales, which is a key concept in derivatives modeling. The definition is given below.

Definition $1.5$ A stochastic process, $M_{t}$, is called a $\mathbb{P}$-martingale if and only if it has the following properties:

1. $E^{\mathbb{P}}\left[\left|M_{t}\right|\right]<\infty, \quad \forall t$.
2. $E^{\mathbb{P}}\left[M_{t} \mid \mathcal{F}{s}\right]=M{s}, \quad \forall s \leq t$.
The martingale properties are associated with fair games in investments or speculations. Let us think of $M_{t}-M_{s}$ as the profit or loss (P\&L) of a gamble between two parties over the time period $(s, t)$. Then the game is considered fair if the expected P\&L is zero. Daily life examples of fair games include the coin tossing game and futures investments in financial markets. In mathematics, there are plenty of examples as well. In fact, we have already seen several of them so far, of which we remind readers below.
Example $1.4$
3. The simple random walk, $X_{n}$, is a martingale because $E\left[\left|X_{n}\right|\right]<$ $n \sqrt{\Delta t}$ and $E\left[X_{n} \mid \mathcal{F}{m}\right]=X{m}, m \leq n$
1. A $\mathbb{P}$-Brownian motion, $W_{t}$, is a martingale by definition.
2. The stochastic integral $X_{t}=\int_{0}^{t} f(u) \mathrm{d} W_{u}$ is a martingale, since
\begin{aligned} E^{\mathbb{P}}\left[X_{t} \mid \mathcal{F}{s}\right] &=E^{\mathbb{P}}\left[\int{0}^{s}+\int_{s}^{t} f(u) \mathrm{d} W_{u} \mid \mathcal{F}{s}\right] \ &=\int{0}^{s} f(u) \mathrm{d} W_{u}=X_{s}, \quad \forall s \leq t \end{aligned}
Here, we have applied the first property of stochastic integrals (see page 11).
3. The process $M_{t}=\exp \left(\int_{0}^{t} \sigma_{s} \mathrm{~d} W_{s}-\frac{1}{2} \sigma_{s}^{2} \mathrm{~d} s\right)$ is an exponential martingale. In fact, using the Ito’s lemma, we can show that
$$\mathrm{d} M_{t}=\sigma_{t} M_{t} \mathrm{~d} W_{t}$$
which is an Ito’s process without drift. It follows that
$$M_{t}=M_{s}+\int_{s}^{t} M_{u} \sigma_{u} \mathrm{~d} W_{u}$$
Based on the conclusion of the last example, we know that $M_{t}$ is a martingale.

We emphasize here that an Ito’s process is a martingale process if and only if its drift term is zero. Finally, we present two additional examples.

## 金融代写|利率建模代写Interest Rate Modeling代考|A Motivating Example

Consider the simplest option-pricing model with an underlying asset following a one-period binomial process, as depicted in Figure 2.1. In Figure $2.1,0 \leq p \leq 1$ and $\bar{p}=1-p$. The option’s payoffs at time $1, f\left(S_{u}\right)$ and $f\left(S_{d}\right)$, are given explicitly, and we want to determine $f(S)$, the value of the option at time 0 . Without loss of generality, we assume that there is a zero interest rate in the model. To avoid arbitrage, we must impose the order $S_{d} \leq S \leq S_{u}$. We call $\mathbb{P}={p, \bar{p}}$ the objective measure of the underlying process.
It may be tempting to price the option by expectation under $\mathbb{P}$ :
\begin{aligned} f(S) &=E^{\mathbb{P}}\left[f\left(S_{1}\right)\right] \ &=p f\left(S_{u}\right)+\bar{p} f\left(S_{d}\right) \end{aligned}
However, except for a special $p$, the above price generates arbitrage and thus is wrong. To see that, we replicate the payoff of the option at time $l$ using a portfolio of the underlying asset and a cash bond, with respective numbers of units, $\alpha$ and $\beta$, such that, at time 1 ,
\begin{aligned} &\alpha S_{u}+\beta=f\left(S_{u}\right) \ &\alpha S_{d}+\beta=f\left(S_{d}\right) \end{aligned}

Solving for $\alpha$ and $\beta$, we obtain
\begin{aligned} \alpha &=\frac{f\left(S_{u}\right)-f\left(S_{d}\right)}{S_{u}-S_{d}}, \ \beta &=\frac{S_{u} f\left(S_{d}\right)-S_{d} f\left(S_{u}\right)}{S_{u}-S_{d}} . \end{aligned}
Equation $2.2$ implies that the time-1 values of the portfolio and option are identical. To avoid arbitrage, their values at time 0 must be identical as well, ${ }^{*}$ which yields the arbitrage price of the option at time 0 :
\begin{aligned} f(S) &=\alpha S+\beta \ &=q f\left(S_{u}\right)+\bar{q} f\left(S_{d}\right) \ &=E^{\mathbb{Q}}\left[f\left(S_{1}\right)\right], \end{aligned}
where $\mathbb{Q}={q, \bar{q}}$, and
$$q=\frac{S-S_{d}}{S_{u}-S_{d}}, \quad \bar{q}=1-q$$
is a different set of probabilities. Note that Equation $2.4$ gives the noarbitrage price of the option. Any other price will induce arbitrage to the market. Hence, the expectation price, in Equation 2.1, is correct only if $p=q$. In fact, ${q, \bar{q}}$ is the only set of probabilities that satisfies
$$S=q S_{u}+\bar{q} S_{d}=E^{\mathbb{Q}}\left(S_{1}\right) .$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Binomial Trees and Path Probabilities

Let us move one step further and consider the binomial tree model up to two time steps, as shown in Figure 2.2, where each pair of numbers represents a state (which can be associated with the price of an asset if necessary). Out of each state at time $j$, two possible states are generated at time $j+1$. Hence, we have $2^{j}$ states at time $j$, starting with a single state at time 0 . The branching probabilities for reaching the next two states from one state, $(i, j)$, are $p_{i, j} \in[0,1]$ and $\bar{p}{i, j}=1-p{i, j}$, respectively. The collection of branching probabilities, $\mathbb{P}=\left{p_{i, j}, \bar{p}{i, j}\right}$, is again called a measure. As is shown in Figure 2.2, there are two paths over the time horizon from 0 to 1 , whereas there are four paths over the time horizon from 0 to 2 . The corresponding path probabilities for the horizon from 0 to 1 are $$\pi{0,1}=\bar{p}{0,0} \quad \text { and } \quad \pi{1,1}=p_{0,0},$$

whereas for the horizon from 0 to 2 , they are
$$\pi_{0,2}=\bar{p}{0,0} \bar{p}{0,1}, \pi_{1,2}=\bar{p}{0,0} p{0,1}, \pi_{2,2}=p_{0,0} \bar{p}{1,1} \text {, and } \pi{3,2}=p_{0,0} p_{1,1} \text {. }$$
The path probabilities can also be marked in a binomial tree as is shown in Figure 2.3.

Consider now another set of branching probabilities, $\mathbb{Q}=\left{q_{i, j}, \bar{q}{i, j}=\right.$ $\left.1-q{i, j}\right}$, for the same tree. The corresponding path probabilities are
$$\pi_{0,1}^{\prime}=\bar{q}{0,0} \quad \text { and } \quad \pi{1,1}^{\prime}=q_{0,0}$$
up to time 1 , and
$$\pi_{0,2}^{\prime}=\bar{q}{0,0} \bar{q}{0,1}, \pi_{1,2}^{\prime}=\bar{q}{0,0} q{0,1}, \pi_{2,2}^{\prime}=q_{0,0} \bar{q}{1,1} \text {, and } \pi{3,2}^{\prime}=q_{0,0} q_{1,1}$$
up to time 2. Suppose that the $\mathbb{P}$-probability of paths $\pi_{i, j} \neq 0$ for all $i, j$. We then can define the ratio of path probabilities as follows:
$$\zeta_{i, j}=\frac{\pi_{i, j}^{\prime}}{\pi_{i, j}} .$$

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

1. $E^{\mathbb{P}}\left[\left|M_{t}\right|\right]<\infty, \quad \forall t$
2. $E^{\mathbb{P}}\left[M_{t} \mid \mathcal{F}{s}\right]=M s, \quad \forall s \leq t$ 鞅属性与投资或投机中的公平游戏有关。让我们想想 $M{t}-M_{s}$ 作为两方在一段时间内赌博的损益 $(\mathrm{P} \backslash \& \mathrm{Q})(s, t)$. 如 果预期盈亏为零，则认为该游戏是公平的。日常生活中公平游戏的例子包括抛硬币游戏和金融市场的期货投傝。在 数学中，也有很茤例子。事实上，到目前为止，我们已经看到了其中的几个，我们在下面提酲读者。
例子 $1.4$
3. 简单的随机游走， $X_{n}$, 是鞅，因为 $E\left[\left|X_{n}\right|\right]<n \sqrt{\Delta t}$ 和 $E\left[X_{n} \mid \mathcal{F} m\right]=X m, m \leq n$
$\mathrm{~ 2 . ~ 一 个 巴}$
4. 随机积分 $X_{t}=\int_{0}^{t} f(u) \mathrm{d} W_{u}$ 是鞅，因为
$$E^{P}\left[X_{t} \mid \mathcal{F} s\right]=E^{P}\left[\int 0^{s}+\int_{s}^{t} f(u) \mathrm{d} W_{u} \mid \mathcal{F}{s}\right] \quad=\int 0^{s} f(u) \mathrm{d} W{u}=X_{s}, \quad \forall s \leq t$$
在这里，我们应用了随机积分的第一个性质（参见第 11 页）。
5. 过程 $M_{t}=\exp \left(\int_{0}^{t} \sigma_{s} \mathrm{~d} W_{s}-\frac{1}{2} \sigma_{s}^{2} \mathrm{~d} s\right)$ 是指数鞅。事实上，使用伊藤引理，我们可以证明
$$\mathrm{d} M_{t}=\sigma_{t} M_{t} \mathrm{~d} W_{t}$$
这是一个没有漂移的伊藤工艺。它曎循
$$M_{t}=M_{s}+\int_{s}^{t} M_{u} \sigma_{u} \mathrm{~d} W_{u}$$
根据上一个例子的结论，我们知道 $M_{t}$ 是鞅。
我们在此强调，Ito 过程是鞅过程当且仅当其漂移项为零。最后，我们提出两个额外的例子。

## 金融代写|利率建模代写Interest Rate Modeling代考|A Motivating Example

$$f(S)=E^{\mathbb{P}}\left[f\left(S_{1}\right)\right] \quad=p f\left(S_{u}\right)+\bar{p} f\left(S_{d}\right)$$

$$\alpha S_{u}+\beta=f\left(S_{u}\right) \quad \alpha S_{d}+\beta=f\left(S_{d}\right)$$

$$\alpha=\frac{f\left(S_{u}\right)-f\left(S_{d}\right)}{S_{u}-S_{d}}, \beta=\frac{S_{u} f\left(S_{d}\right)-S_{d} f\left(S_{u}\right)}{S_{u}-S_{d}}$$

$$f(S)=\alpha S+\beta \quad=q f\left(S_{u}\right)+\bar{q} f\left(S_{d}\right)=E^{\mathbb{Q}}\left[f\left(S_{1}\right)\right]$$

$$q=\frac{S-S_{d}}{S_{u}-S_{d}}, \quad \bar{q}=1-q$$

$$S=q S_{u}+\bar{q} S_{d}=E^{\mathbb{Q}}\left(S_{1}\right)$$

## 金融代写|利率建模代写Interest Rate Modeling代考|Binomial Trees and Path Probabilities

Imathbb{P}=\left{p_{i, j}, Vbar{p}{,j}}rright} , 又称为测度。如图 $2.2$ 所示，在从 0 到 1 的时间范围内有两条路径，而在从 0 到 2 的时间范围内有四条路径。地平线从 0 到 1 的相应路径概率为
$$\pi 0,1=\bar{p} 0,0 \quad \text { and } \quad \pi 1,1=p_{0,0},$$

$$\pi_{0,2}=\bar{p} 0,0 \bar{p} 0,1, \pi_{1,2}=\bar{p} 0,0 p 0,1, \pi_{2,2}=p_{0,0} \bar{p} 1,1, \text { and } \pi 3,2=p_{0,0} p_{1,1} .$$

$\mathrm{~ 现 在 考 虑 另 一 组 分 支 概 率 ， ~ I m a t h b b { Q } = \ l e f t { q _ { i , j } , \ b a r { q } { i , j } =}$ 概率是
$$\pi_{0,1}^{\prime}=\bar{q} 0,0 \quad \text { and } \quad \pi 1,1^{\prime}=q_{0,0}$$

$$\pi_{0,2}^{\prime}=\bar{q} 0,0 \bar{q} 0,1, \pi_{1,2}^{\prime}=\bar{q} 0,0 q 0,1, \pi_{2,2}^{\prime}=q_{0,0} \bar{q} 1,1, \text { and } \pi 3,2^{\prime}=q_{0,0} q_{1,1}$$

$$\zeta_{i, j}=\frac{\pi_{i, j}^{\prime}}{\pi_{i, j}} .$$

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考|Simple Random Walks

Simple random walks are discrete time series, $\left{X_{i}\right}$, defined as
\begin{aligned} X_{0} &=0, \ X_{n+1} &= \begin{cases}X_{n}-\sqrt{\Delta t}, & p=\frac{1}{2} \ X_{n}+\sqrt{\Delta t}, & 1-p=\frac{1}{2}\end{cases} \end{aligned}
where $\Delta t>0$ stands for the interval of time for stepping forward. One can verify that $\left{X_{i}\right}$ have the following properties:

1. The increment of $X_{n+1}-X_{n}$ is independent of $\left{X_{i}\right}, \forall i \leq n$.
2. $E\left[X_{n} \mid X_{m}\right]=X_{m}, m \leq n$.
3. $\operatorname{Var}\left[X_{n} \mid X_{m}\right]=(n-m) \Delta t, m \leq n$.
An interesting feature of the simple random walk is the linearity of $X_{i}$ ‘s variance in time: given $X_{0}$, the variance of $X_{i}$ is equal to $i \Delta t$, the time it takes the time series to evolve from $X_{0}$ to $X_{i}$.

Out of the simple Brownian random walk, we can construct a continuous-time process through linear interpolation:
$$\bar{X}(t)=X_{i}+\frac{t-i \Delta t}{\Delta t}\left(X_{i+1}-X_{i}\right), \quad t \in[i \Delta t,(i+1) \Delta t]$$
We are interested in the limiting process of $\bar{X}(t)$ as $\Delta t \rightarrow 0$, in the hope that the limit remains a meaningful stochastic process. The next theorem confirms just that.

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

A continuous stochastic process is a collection of real-valued random variables, ${X(t, \omega), 0 \leq t \leq T}$ or $\left{X_{t}(\omega), 0 \leq t \leq T\right}$, that are defined on a probability space $(\Omega, \mathcal{F}, \mathbb{P})$. Here $\Omega$ is the collection of all $\omega$ s, which are so-called sample points, $\mathcal{F}$ the smallest $\sigma$-algebra that contains $\Omega$, and $\mathbb{P}$ a probability measure on $\Omega$. Each random outcome, $\omega \in \Omega$, corresponds to an entire time series
$$t \rightarrow X_{t}(\omega), \quad t \in T$$
which is called a path of $X_{t}$. In view of Equation 1.7, we can regard $X_{t}(\omega)$ as a function of two variables, $\omega$ and $t$. For notational simplicity, however, we often suppress the $\omega$ variable when its explicit appearance is not necessary.

In the context of financial modeling, we are particularly interested in the Brownian motion introduced earlier. Its formal definition is given below.

Definition 1.1 A Brownian motion or a Wiener process is a realvalued stochastic process, $W_{t}$ or $W(t), 0 \leq t \leq \infty$, that has the following properties:

1. $W(0)=0$.
2. $W(t+s)-W(t)$ is independent of ${W(u), 0 \leq u \leq t}$.
3. For $t \geq 0$ and $s>0$, the increment $W(t+s)-W(t) \sim N(0, s)$.
4. $W(t)$ is continuous almost surely (a.s.).
Here $N(0, s)$ stands for a normal distribution with mean zero and variance s. Note that in some literature, property 4 is not part of the definition, as it can be proved to be implied by the first three properties (Varadhan, $1980 \mathrm{a}$ or Ikeda and Watanabe, 1989). A sample path of $W(t)$ is shown in Figure $1.1$, which is generated with a step size of $\Delta t=2^{-10}$.

Brownian motion plays a major role in continuous time stochastic modeling in physics, engineering and finance. In finance, it has been used to model the random behavior of asset returns. Several major properties of Brownian motion are listed below.

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

Stochastic calculus considers the integration and differentiation of general $\mathcal{F}{t}$-adaptive functions. The purpose of developing such a stochastic calculus is to model financial time series (with random dynamics) with either integral or differential equations. According to Lemma 1.1, a Brownian motion, $W(t)$, is nowhere differentiable in the usual sense of differentiation for deterministic functions. To define differentials of stochastic processes in a proper sense, we must first study the notion of stochastic integrals. Stochastic integrals can be defined for functions in the square-integrable space, $H^{2}[0, T]=L^{2}(\Omega \times[0, T], \mathrm{d} \mathbb{P} \times \mathrm{d} t)$, which is defined to be the collection of functions satisfying $$E\left[\int{0}^{T}|f(t, \omega)|^{2} \mathrm{~d} t\right]<\infty$$
Note that, without indicated otherwise, $E[\cdot]$ means $E^{\mathbb{P}}[\cdot]$, the unconditional expectation under $\mathbb{P}$. The definition consists of a three-step procedure. First, we make the definition for elementary or piecewise constant functions in an intuitive way. Second, we define the integrals of a bounded continuous function as a limit of integrals of elementary functions. Finally, we define the integral of a general square-integrable function as a limit of integrals of bounded continuous functions. The key in this three-step procedure is of course to ensure the convergence of the limits in $L^{2}(\Omega, \mathcal{F}, \mathbb{P})$, the Hilbert space of random variables satisfying
$$E\left[X^{2}(\omega)\right]<\infty$$
This definition approach is taken by Oksendal (1992). Alternative treatments of course also exist; see, for example, Mikosch (1998).

## 金融代写|利率建模代写Interest Rate Modeling代考|Simple Random Walks

$$X_{0}=0, X_{n+1} \quad=\left{X_{n}-\sqrt{\Delta t}, \quad p=\frac{1}{2} X_{n}+\sqrt{\Delta t}, \quad 1-p=\frac{1}{2}\right.$$

1. 的增量 $X_{n+1}-X_{n}$ 独立于⿴left $\left{X_{-}{i} \backslash\right.$ Iight}, Iforall i leq n.
2. $E\left[X_{n} \mid X_{m}\right]=X_{m}, m \leq n$.
3. $\operatorname{Var}\left[X_{n} \mid X_{m}\right]=(n-m) \Delta t, m \leq n$.
简单随机游走的一个有趣特征是 $X_{i}$ 的时间变化: 给定 $X_{0}$, 的方差 $X_{i}$ 等于 $i \Delta t$, 时间序列从 $X_{0}$ 至 $X_{i}$.
从简单的布朗随机游走中，我们可以通过线性揷值构造一个连续时间的过程:
$$\bar{X}(t)=X_{i}+\frac{t-i \Delta t}{\Delta t}\left(X_{i+1}-X_{i}\right), \quad t \in[i \Delta t,(i+1) \Delta t]$$
我们对限制过程感兴趣 $\bar{X}(t)$ 作为 $\Delta t \rightarrow 0$ ，希望极限仍然是一个有意义的随机过程。下一个定理证实了这一点。

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

$$t \rightarrow X_{t}(\omega), \quad t \in T$$

1. $W(0)=0$.
2. $W(t+s)-W(t)$ 独立于 $W(u), 0 \leq u \leq t$.
3. 为了 $t \geq 0$ 和 $s>0$, 增量 $W(t+s)-W(t) \sim N(0, s)$.
4. $W(t)$ 几平肯定是连续的 (as)。
这里 $N(0, s)$ 代表均值为零且方差为 $\mathrm{s}$ 的正态分布。请注意，在某些文献中，属性 4 不是定义的一部分，因 为可以证明前三个属性暗示了它 (Varadhan，1980a或池田和渡边，1989) 。一个示例路径 $W(t)$ 如图1.1 ，它的生成步长为 $\Delta t=2^{-10}$.
布朗运动在物理学、工程和金融领域的连续时间随机建模中发挥着重要作用。在金融领域，它已被用于模拟资产回 报的随机行为。下面列出了布朗运动的几个主要性质。

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

$$E\left[\int 0^{T}|f(t, \omega)|^{2} \mathrm{~d} t\right]<\infty$$

$$E\left[X^{2}(\omega)\right]<\infty$$
Oksendal (1992) 采用了这种定义方法。当然也存在替代疗法；例如，参见 Mikosch (1998)。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考|Discretization of Forward Rate Process

We recall the forward rate process in the H.JM model as
\begin{aligned} f(t, T)=& f(0, T)+\int_{0}^{t}\left{-\sigma(s, T) v(s, T)+\sigma(s, T) \varphi_{s}\right} d s \ &+\int_{0}^{t} \sigma(s, T) d W_{s} \end{aligned}
where $W_{t}$ is a $d$-dimensional P-Brownian motion. The HJM model is called Gaussian if $\sigma(t, T)$ is a deterministic function of $t$ and $T$. In this book, we always assume that volatility $\sigma(t, T)$ is deterministic and continuous with respect to $t$ and $T$. Then, $v(t, T)$ is also deterministic and continuous. In the following, the market price of risk is assumed to be constant. The validity of the constancy assumption will be examined in the context of risk management in Chapter $7 .$
Data observation
Denoting the market price of risk as a constant $\varphi$, the above forward rate process is expressed by
\begin{aligned} f(t, T)=& f(0, T)+\int_{0}^{t}{-\sigma(s, T) v(s, T)+\sigma(s, T) \varphi} d s \ &+\int_{0}^{t} \sigma(s, T) d W_{s} . \end{aligned}
Next, we specify a historical dataset as follows. Let a time interval $\Delta t>0$ be fixed, and $\left{t_{k}\right}_{k=1, \cdots, J+1}$ be a sequence of observation dates such that $t_{1}=0$ and $t_{k+1}-t_{k}=\Delta t$, where $J+1$ is the number of observation times. We denote the time length to a maturity $T$ from $t$ by $x=T-t$. For an integer $n \geq d$, $x_{1}, \cdots, x_{n}$ denotes a sequence of time lengths to maturity.

Typically, we observe the instantaneous forward rate $F\left(t_{k}, x_{i}\right)$ with respect to fixed $x_{i}$. Fig. $6.1$ illustrates an example of forward rate curves observed at $t_{k}, t_{k+1}$, and $t_{k+2}$, showing $F\left(t_{k}, x_{i}\right), F\left(t_{k+1}, x_{i}-\Delta t\right)$, and so on. We assume that the dynamics of these observations follow equation (6.2).

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

We recall the volatility structure associated with PCA in Section 4.3. A sample covariance matrix $V$ is defined by
\begin{aligned} V_{i j}=& \frac{1}{\Delta t} \operatorname{Cov}\left(F\left(t_{k}+\Delta t, x_{i}-\Delta t\right)-F\left(t_{k}, x_{i}\right)\right.\ &\left.F\left(t_{k}+\Delta t, x_{j}-\Delta t\right)-F\left(t_{k}, x_{j}\right)\right) ; \quad i, j=1, \cdots, n . \end{aligned}
We assume that $V$ has rank $d \leq n$. By the argument in Appendix $\mathrm{B}$, the covariance matrix is decomposed into $V_{i j}=\sum_{l=1}^{d} e_{i}^{l} \rho_{l}^{2} e_{j}^{l}$ for $i, j \leq n$, where $\rho_{l}^{2}$ is the lth eigenvalue, and $e^{l}=\left(e_{1}^{l}, \cdots, e_{n}^{l}\right)^{T}$ is the $l$ th principal component of the covariance for $l=1, \cdots, d$.
We always assume that
$$e_{1}^{l}>0, \quad \rho_{l}>0 ; \quad l=1, \cdots, d$$
This assumption is significant in the interpretation of the meaning of the market price of risk in Sections $6.4$ and 6.5. Recall the equation (B.4), that is, that the principal components $e^{1}, \cdots, e^{d}$ form an orthonormal set. Thus,
$$\sum_{i=1}^{n} e_{i}^{l} e_{i}^{h}=\delta_{l h} \quad ; \quad l, h=1, \cdots, d$$

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

This section introduces another method to estimate the market price of risk: working in a state space.

Denoting the market price of risk by $\varphi^{\prime}=\left(\varphi_{1}^{\prime}, \cdots, \varphi_{d}^{\prime}\right)^{T}$, we return to the discretization as equation (6.13), which we reproduce below:
$$\Delta F_{i}\left(t_{k}\right)=-\sigma_{0 i} v_{0 i} \Delta t+\sigma_{0 i} \varphi^{\prime} \Delta t+\sqrt{\Delta t} \sigma_{0 i} W_{1} ; i=1, \cdots, n, k=1, \cdots, J$$
We remark that the volatility is assumed to be determined by a principal component. Our objective here is to directly obtain $\varphi^{\prime}$ from the above equations.
We denote by $\epsilon\left(\varphi^{\prime}\right)$ the sum of the squared difference between each side of equation $(6.26)$ in the time series and cross sections, neglecting the random part, such that
$$\epsilon\left(\varphi^{\prime}\right)=\frac{1}{J} \sum_{k=1}^{J} \sum_{i=1}^{n}\left{\Delta F_{i}\left(t_{k}\right)+\left(\sigma_{0 i} v_{0 i}-\sigma_{0 i} \varphi^{\prime}\right) \Delta t\right}^{2}$$
Let $\varphi^{\prime}$ be the solution that minimizes $\epsilon\left(\varphi^{\prime}\right)$. We call this setting a state space setup, and call that used in the previous section a $P C A$ setup to distinguish between the two approaches. We note the implications of both definitions below.

• $\varphi$ is the solution that minimizes $\theta_{l}\left(\varphi_{l}\right)$ in equation $(6.19)$ in the principal component space, and also is the maximum likelihood estimate.
• $\varphi^{\prime}$ is the solution that minimizes $\epsilon\left(\varphi^{\prime}\right)$ of equation $(6.27)$ in the state space.

## 金融代写|利率建模代写Interest Rate Modeling代考|Discretization of Forward Rate Process

\begin{对齐} f(t, T)=& f(0, T)+\int_{0}^{t}\left{-\sigma(s, T) v(s, T)+\sigma( s, T) \varphi_{s}\right} d s \ &+\int_{0}^{t} \sigma(s, T) d W_{s} \end{aligned}\begin{对齐} f(t, T)=& f(0, T)+\int_{0}^{t}\left{-\sigma(s, T) v(s, T)+\sigma( s, T) \varphi_{s}\right} d s \ &+\int_{0}^{t} \sigma(s, T) d W_{s} \end{aligned}

F(吨,吨)=F(0,吨)+∫0吨−σ(s,吨)在(s,吨)+σ(s,吨)披ds +∫0吨σ(s,吨)d在s.

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

∑一世=1n和一世l和一世H=dlH;l,H=1,⋯,d

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

ΔF一世(吨ķ)=−σ0一世在0一世Δ吨+σ0一世披′Δ吨+Δ吨σ0一世在1;一世=1,⋯,n,ķ=1,⋯,Ĵ

\epsilon\left(\varphi^{\prime}\right)=\frac{1}{J} \sum_{k=1}^{J} \sum_{i=1}^{n}\left{\ Delta F_{i}\left(t_{k}\right)+\left(\sigma_{0 i} v_{0 i}-\sigma_{0 i} \varphi^{\prime}\right) \Delta t \对}^{2}\epsilon\left(\varphi^{\prime}\right)=\frac{1}{J} \sum_{k=1}^{J} \sum_{i=1}^{n}\left{\ Delta F_{i}\left(t_{k}\right)+\left(\sigma_{0 i} v_{0 i}-\sigma_{0 i} \varphi^{\prime}\right) \Delta t \对}^{2}

• 披是最小化的解决方案θl(披l)在等式中(6.19)在主成分空间中，也是最大似然估计。
• 披′是最小化的解决方案ε(披′)方程的(6.27)在状态空间。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

## 金融代写|利率建模代写Interest Rate Modeling代考|MATH3075

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考|LIBOR Market Model

The LIBOR market model was introduced by Miltersen et al. (1997), Brace et al. (1997; hereinafter, BGM), Musiela and Rutkowski (1997), and Jamshidian $(1997)$. The notable points of this model are listed here:

• The model has positive LIBOR.
• The model admits an arbitrary deterministic volatility structure.
• The price formulae of a caplet and a floorlet are derived so as to be consistent with the corresponding Black’s price.
• An approximated price formula for a swaption is derived.
From these, the LIBOR market model has a usability advantage in calibration, and so it is widely applied as a standard model for derivatives pricing. As a particular example, the BGM model is the most well-known type of LIBOR market model, and is built in the HJM framework. The BGM approach requires a kind of differentiability for LIBOR volatility. It is impossible to satisfy this smoothness in practice because the volatility cannot be constructed except as a piecewise continuous, but not necessarily smooth, function. Because of this, the BGM model is not strictly supported in the HJM framework. For more advanced study of this problem, see Yasuoka (2001, 2013b).

At one end of the spectrum of models, the approaches by Musiela and Rutkowski (1997) and Jamshidian (1997) stand on a martingale pricing theory, with no theoretical imperfections. However, their models are constructed under a risk-neutral measure without referring to the real-world measure.

In this section and the next, we introduce the LIBOR market model as described by Jamshidian (1997). Because the topic of this book is risk management, pricing of derivatives is not addressed here at length. For a more advanced treatment of pricing, readers are recommended to consult Brigo and Mercurio (2007) or Gatarek et al. (2007).

Similarly to the argument for the HJM model, when the LIBOR and bond prices are represented under a risk-neutral measure, we call the resulting system a risk-neutral model. When, instead, they are represented under $\mathbf{P}$, the resulting system is referred to as a real-world model. Strict definitions of these terms will be given later.

## 金融代写|利率建模代写Interest Rate Modeling代考|Existence of LIBOR Market Model

The existence of the LIBOR model is shown in the following theorem.
Theorem 5.2.1 For arbitrary deterministic volatility $\lambda_{i}(t), i=1, \cdots, n-1$, the LIBOR market model exists.

The LIBOR model can be constructed under any of several risk-neutral measures. Applying this, we will show the existence of the LIBOR model under the real-world measure in the next section, and show how the models are implied under other measures in Sections $5.4$ and $5.5$ of this chapter. It is thought that this approach is the simplest method of constructing the LIBOR market model for practical use. Therefore, we here only sketch Jamshidian’s LIBOR market model under a forward measure, omitting the proof.

Let each of $\lambda_{i}(t)$ be an arbitrary deterministic function in $t$ for $i=1, \cdots, n-$

1. Consider the following equation:
$$\frac{d L_{i}(t)}{L_{i}(t)}=\sum_{j=i+1}^{n-1} \frac{\delta_{j} L_{j}(t) \lambda_{i}(t) \lambda_{j}(t)}{1+\delta_{j} L_{j}} d t+\lambda_{i}(t) d Z_{t}$$
Here, $Z_{t}$ is a $d$-dimensional Brownian motion with respect to a measure $\mathbf{Q}(\sim$ $\mathbf{P})$. With this setup, the following proposition is given in Jamshidian ( 1997 , Corollary 2.1).

Proposition 5.2.2 The equation (5.4) admits a unique positive solution for an arbitrary initial condition $L_{i}(0)>0$ for all i. Further, $Y_{i}(t)=(1+$ $\left.\delta_{i} L_{i}(t)\right) \cdots\left(1+\delta_{i} L_{n-1}(t)\right)$ is a $\mathbf{Q}$-martingale.
Let $B_{n}(t)$ be an arbitrary bond price process such that $B_{n}\left(T_{n}\right)=1$ and
$$B_{n}\left(T_{i}\right)=\frac{1}{\prod_{j=i}^{n-1}\left(1+\delta_{j} L_{j}\left(T_{j}\right)\right)}$$ at each $T_{i}$. Accordingly, we define $B_{i}(t)$ for $i<n$ by
$$\frac{B_{i}(t)}{B_{n}(t)}=\prod_{j=i}^{n-1}\left(1+\delta_{j} L_{j}(t)\right)$$
From these, we see that $B_{i}\left(T_{i}\right)=1$ and the relation (5.2) is satisfied for all $i$. By Proposition 5.2.2, $\prod_{j=i}^{n-1}\left(1+\delta_{j} L_{j}(t)\right)$ is a Q-martingale for every $i$. Hence $B_{i}(t) / B_{n}(t)$ is a Q-martingale for all $i$.

Along these lines, $\mathbf{Q}$ is a $B_{n}$ numéraire measure and is referred to as a forward measure. As a result, the bond market $\mathcal{B}$ is arbitrage-free from Theorem $3.2 .2$

## 金融代写|利率建模代写Interest Rate Modeling代考|LIBOR Market Model under a Real-world Measure

Within the same setting as in Sections $5.1$ and 5.2, we give a definition of the LMRW and show the existence of the model, following Yasuoka (2013a).

Definition 5.3 The bond market $\mathcal{B}$ is called the $L M R W$ when the following conditions are satisfied.

1. The LIBOR processes $L_{i}, i=1, \cdots, n$, with $L_{i}(t)>0$, are represented under the real-world measure $\mathbf{P}$ such that each volatility $\lambda_{i}(t)$ and the market price of risk $\varphi_{t}$ are deterministic in $t$.
1. The bond market $\mathcal{B}$ is arbitrage-free; here this means that $B_{i}(t) \in \mathcal{B}, i=$ $1, \cdots, n$ and the state price deflator $\xi_{t}$ are positive Ito processes represented under $\mathbf{P}$.

For this, we define a left-continuous function $m(t)$ by $m(t)=j$, while $t \in$ $\left(T_{j-1}, T_{j}\right]$. Succinctly, $m(t)$ represents the index of the next maturity date $.$ Examination of Fig. $5.1$ may help to see the features of $m(t)$.

To show the existence of the LMRW, it is sufficient to give the simplest example for arbitrarily given volatility $\lambda$ and market price of risk $\varphi$. For this, we define a process $\bar{\mu}(t)$ by $\bar{\mu}(t)=\bar{\mu}\left(T_{m(t)}\right)$ such that
$$\bar{\mu}\left(T_{i}\right)=\frac{1}{\delta_{i-1}} \log \left{1+\delta_{i-1} L_{i-1}\left(T_{i-1}\right)\right}$$
at each time $T_{i}$. Specifically, $\bar{\mu}(t)$ represents the yield for the shortest maturity bond, with the next maturity $T_{m(t)}$. As a consequence, $\bar{\mu}(t)$ is constant on each period $\left(T_{i-1}, T_{i}\right], i=1, \cdots, n$.

Let $\varphi_{t}$ be an arbitrarily given market price of risk such that $\varphi_{l}$ is an $\mathbf{R}^{d}-$ valued deterministic function with
$$\int_{0}^{T}\left|\varphi_{t}\right|^{2} d s<\infty$$ Let $\lambda_{i}(t), i=1, \cdots, n$ be deterministic volatilities. We set $\chi_{i}(t)$ as $$\chi_{i}(t)=\frac{\lambda_{i}(t) \delta_{i} L_{i}(t)}{1+\delta_{i} L_{i}(t)} ; i=1, \cdots, n .$$ Consider the following equation with the initial LIBOR $L_{i}(0)>0$,
$$\frac{d L_{i}(t)}{L_{i}(t)}=\left{\lambda_{i}(t) \sum_{j=m(t)}^{i} \chi_{j}(t)+\lambda_{i}(t) \varphi_{t}\right} d t+\lambda_{i}(t) d W_{t}$$
for $i=1, \cdots, n$. It is known that the solution $L_{i}(t)$ exists uniquely and $L_{i}(t)>$ 0 . We assume that bond price processes $B_{i}(t), i=1, \cdots, n$ are Ito processes with initial values $B_{0}(0)=1$ and
$$B_{i}(0)=\prod_{j=0}^{i-1}\left(1+\delta_{j} L_{j}(0)\right)^{-1}$$ such that
$$\frac{d B_{i}(t)}{B_{i}(t)}=\left{\bar{\mu}(t)-\sum_{j=m(t)}^{i-1} \chi_{j}(t) \varphi_{t}\right} d t-\sum_{j=m(t)}^{i-1} \chi_{j}(t) d W_{t} .$$
Under this setup, we give the following theorem, which shows the existence of the LMRW.

## 金融代写|利率建模代写Interest Rate Modeling代考|LIBOR Market Model

LIBOR 市场模型由 Miltersen 等人引入。（1997 年），布雷斯等人。（1997 年；以下称为 BGM）、Musiela 和 Rutkowski（1997 年）和 Jamshidian(1997). 此处列出了此模型的值得注意的点：

• 该模型具有正 LIBOR。
• 该模型允许任意确定的波动率结构。
• 导出caplet和floorlet的价格公式以与对应的布莱克价格一致。
• 导出了互换期权的近似价格公式。
由此可见，LIBOR 市场模型在校准方面具有可用性优势，因此被广泛用作衍生品定价的标准模型。作为一个具体的例子，BGM 模型是最知名的 LIBOR 市场模型类型，并且构建在 HJM 框架中。BGM 方法需要一种 LIBOR 波动性的可微性。在实践中不可能满足这种平滑性，因为波动率只能构建为分段连续但不一定平滑的函数。因此，HJM 框架并不严格支持 BGM 模型。有关此问题的更高级研究，请参阅 Yasuoka (2001, 2013b)。

## 金融代写|利率建模代写Interest Rate Modeling代考|Existence of LIBOR Market Model

LIBOR 模型的存在如下定理所示。

LIBOR 模型可以在多种风险中性措施中的任何一种下构建。应用这一点，我们将在下一节中展示 LIBOR 模型在真实世界度量下的存在，并展示模型在其他度量下的隐含部分5.4和5.5本章的。这种方法被认为是构建 LIBOR 市场模型以供实际使用的最简单方法。因此，我们这里只勾勒出Jamshidian 的LIBOR 市场模型在前向测度下，省略了证明。

1. 考虑以下等式：
d大号一世(吨)大号一世(吨)=∑j=一世+1n−1dj大号j(吨)λ一世(吨)λj(吨)1+dj大号jd吨+λ一世(吨)d从吨
这里，从吨是一个d关于测度的一维布朗运动问(∼ 磷). 通过这种设置，Jamshidian (1997, Corollary 2.1) 给出了以下命题。

## 金融代写|利率建模代写Interest Rate Modeling代考|LIBOR Market Model under a Real-world Measure

1. LIBOR 流程大号一世,一世=1,⋯,n， 和大号一世(吨)>0, 在真实世界的度量下表示磷这样每个波动率λ一世(吨)和风险的市场价格披吨是确定性的吨.
2. 债券市场乙无套利；这意味着乙一世(吨)∈乙,一世= 1,⋯,n和国家价格平减指数X吨是正 Ito 过程表示下磷.

\bar{\mu}\left(T_{i}\right)=\frac{1}{\delta_{i-1}} \log \left{1+\delta_{i-1} L_{i-1 }\left(T_{i-1}\right)\right}\bar{\mu}\left(T_{i}\right)=\frac{1}{\delta_{i-1}} \log \left{1+\delta_{i-1} L_{i-1 }\left(T_{i-1}\right)\right}

∫0吨|披吨|2ds<∞让λ一世(吨),一世=1,⋯,n是确定性的波动率。我们设置χ一世(吨)作为

χ一世(吨)=λ一世(吨)d一世大号一世(吨)1+d一世大号一世(吨);一世=1,⋯,n.考虑以下具有初始 LIBOR 的等式大号一世(0)>0,

\frac{d L_{i}(t)}{L_{i}(t)}=\left{\lambda_{i}(t) \sum_{j=m(t)}^{i} \chi_{ j}(t)+\lambda_{i}(t) \varphi_{t}\right} d t+\lambda_{i}(t) d W_{t}\frac{d L_{i}(t)}{L_{i}(t)}=\left{\lambda_{i}(t) \sum_{j=m(t)}^{i} \chi_{ j}(t)+\lambda_{i}(t) \varphi_{t}\right} d t+\lambda_{i}(t) d W_{t}

\frac{d B_{i}(t)}{B_{i}(t)}=\left{\bar{\mu}(t)-\sum_{j=m(t)}^{i-1 } \chi_{j}(t) \varphi_{t}\right} d t-\sum_{j=m(t)}^{i-1} \chi_{j}(t) d W_{t} 。\frac{d B_{i}(t)}{B_{i}(t)}=\left{\bar{\mu}(t)-\sum_{j=m(t)}^{i-1 } \chi_{j}(t) \varphi_{t}\right} d t-\sum_{j=m(t)}^{i-1} \chi_{j}(t) d W_{t} 。

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考|The Hull–White Model

In the early days, many stochastic models were introduced to describe the dynamics of the short rate. As examples, see Cox et al. (1985; hereinafter,CIR), Ho and Lee (1986), Hull and White (1990), and Vasicek (1977), among others. A strong point of these models is their parsimoniousness. Additionally, these models are described by affine term structures. For details of affine models, readers are recommended to consult Duffie and Kan (1996), Björk $(2004)$, or Munk (2011).

It is known that the Ho-Lee model and the Hull-White model are special cases of the Gaussian HJM model. The Hull-White model, in particular, is one of the most popular models in many financial institutions. Following along these lines, this section introduces the Hull-White model as a special case of the HJM model.
Short rate process
Let us consider a one-dimensional process of the short rate $r(t)$ represented by
$$d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}$$
where $W_{t}$ is a one-dimensional Brownian motion under the real-world measure $\mathbf{P} ; \kappa$ and $\sigma$ are positive constants; $\theta(t)$ is a positive process; and $\varphi_{t}$ denotes the market price of risk.

It is empirically observed that the volatility of long-term interest rates is less than that of short term rates, reflecting a general phenomenon referred to as mean reversion. To model this feature, the rate at which $r(t)$ reverts to $\theta(t)$ is the speed $\kappa$, called the mean reversion rate.

The savings account $B_{t}=\exp \left{\int_{0}^{t} r(s) d s\right}$ is taken as a numéraire. We set $Z_{t}=\int_{0}^{t} \varphi_{s} d s+W_{t}$. By the Girsanov theorem, there exists a risk-neutral measure $\mathbf{Q}$ equivalent to $\mathbf{P}$ such that $Z_{t}$ is a Brownian motion under $\mathbf{Q}$. From these, the short rate $r(t)$ is represented under $\mathbf{Q}$ as
$$d r(t)=\kappa(\theta(t)-r(t)) d t+\sigma d Z_{t}$$
It is known that the price of a zero-coupon bond with maturity $T$ is given by
$$B(t, T)=\exp {-a(t, T)-b(T-t) r(t)}$$

## 金融代写|利率建模代写Interest Rate Modeling代考|VaR Computed in the Real-world

This section studies the reason that the VaR should be computed using a real-world model. For this purpose, the valuation of the VaR depends on the choice of measure. We use the following simple example to illustrate this. For simplicity, we assume a null discount rate in the following argument (i.e. the forward price is equal to the present price).

Suppose a binary bond with expiry at time $T$ and with payoff $X$ at $T$ is given as follows.
$$\left{\begin{array}{l} \text { If } L>5 \% \text { at } T, \text { then } X=0 \ \text { If } L \leq 5 \% \text { at } T, \text { then } X=1.01, \end{array}\right.$$
where $L$ indicates the 6 -month LIBOR at $T$. Succinctly, the payoff is determined by the level of the 6 -month LIBOR at the expiry date.

The price of this security is computed by using some interest rate model under some risk-neutral measure $\mathbf{Q}$. For the model, we assume the probability distribution of $L$ as
$$\left{\begin{array}{r} \mathrm{Q}(L>5 \%)=0.09 \% \ \mathbf{Q}(L \leq 5 \%)=99.01 \% \end{array}\right.$$
With this distribution, the arbitrage price of this bond at $t=0$ is calculated by
$$(1.01 \times 0.9901+0 \times 0.0009) \times 1=1.00$$ because of the assumption of a null discount rate.
We buy this bond at price $1.00$. Let us valuate the $99 \% \mathrm{VaR}$ of this bond for holding period $T$. We can sell this for the price $1.01$ at time $T$ at a probability of more than $99 \%$. The $99 \% \mathrm{VaR}$ is valuated as the profit of $-0.01(=1.00-1.01)$ under Q.

Next, we assume that historical observation estimates for the 6-month LIBOR are
$$\left{\begin{array}{l} \mathbf{P}(L>5 \%)=2 \% \ \mathbf{P}(L \leq 5 \%)=98 \% \end{array}\right.$$

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

In empirical analysis concerning the term structure of interest rates, we are observing historical data under the real-world measure. To give an example, when we use the Hull-White model, the dynamics of the short rate is described from equation $(4.41)$ as
$$d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}$$
To calibrate this model such that this equation explains the historical dynamics of the short rate, we must estimate the parameters $\sigma$ and $\kappa$ and the market price of risk $\varphi_{t}$. In this way, we inevitably estimate $\varphi_{t}$ as part of fitting any interest rate model with the historical dynamics of the interest rates.

Along these lines, there are many studies on estimating the market price of risk in the field of economics. Some papers in this vein are Ahn and Gao (1999), Cheridito et al. (2007), Cochrane and Piazzesi (2010), De Jong (2000), and Duffee (2002). However, there are few papers that explicitly describe the method used in estimating the market price of risk in short rate models. It is even more difficult to find such papers that work with forward rate models.
In this section, we briefly describe three approaches to estimating the market price of risk in short rate models. For a more advanced treatment of this subject, we study theoretical methods for estimating the market price of risk in the forward rate model from Chapter $6 .$

## 金融代写|利率建模代写Interest Rate Modeling代考|The Hull–White Model

d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}

dr(吨)=ķ(θ(吨)−r(吨))d吨+σd从吨

## 金融代写|利率建模代写Interest Rate Modeling代考|VaR Computed in the Real-world

$$\左{ 如果 大号>5% 在 吨, 然后 X=0 如果 大号≤5% 在 吨, 然后 X=1.01,\正确的。$$

$$\left{ 问(大号>5%)=0.09% 问(大号≤5%)=99.01%\正确的。 在一世吨H吨H一世sd一世s吨r一世b在吨一世○n,吨H和一个rb一世吨r一个G和pr一世C和○F吨H一世sb○nd一个吨吨=0一世sC一个lC在l一个吨和db是 (1.01 \times 0.9901+0 \times 0.0009) \times 1=1.00$$ 因为假设零贴现率。

$$\left{ 磷(大号>5%)=2% 磷(大号≤5%)=98%\正确的。$$

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

d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}d r(t)=\kappa\left{\theta(t)-r(t)+\frac{\sigma}{\kappa} \varphi_{t}\right} d t+\sigma d W_{t}

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

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

statistics-lab™ 为您的留学生涯保驾护航 在代写利率建模Interest Rate Modeling方面已经树立了自己的口碑, 保证靠谱, 高质且原创的统计Statistics代写服务。我们的专家在代写利率建模Interest Rate Modeling代写方面经验极为丰富，各种代写利率建模Interest Rate Modeling相关的作业也就用不着说。

• Statistical Inference 统计推断
• Statistical Computing 统计计算
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 金融代写|利率建模代写Interest Rate Modeling代考|Heath-Jarrow-Morton Framework

We introduced a bond market $\mathcal{B}$ in Chapter 3 that did not admit a term structure of interest rates. If, instead, we assume a term structure in the bond market, it becomes possible to relate the bond price to the interest rate. We represent the dynamics of bond price by a stochastic process and use this to specify the corresponding interest rates. Such a system is referred to as a term structure model of interest rates (or put simply, an interest rate model).
A model specified by the dynamics of a short rate is referred to as a short rate model. A model specified by the dynamics of forward rates is referred to as a forward rate model. For management of interest rate risk, it is better to suppose various types of changes in the yield curve, and specifically to suppose changes in the forward rates. Because of this, the forward rate model is more useful in risk management than the short rate model is.

This section briefly introduces the HJM model, which is the most general forward rate model. For additional details, readers are recommended to consult Cairns (2004), Munk (2011), or Shreve (2004), among others. For details on calibration, readers are recommended to consult Wu (2009).
Forward rate process
Let $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ be a filtered probability space, where $\mathcal{F}{t \in[0, \tau]}$ is the augmented filtration and $\mathbf{P}$ denotes the real-world measure. The instantaneous forward rate with maturity $T$ observed at time $t$ is denoted by $f(t, T)$. When the usage is unambiguous, $f(t, T)$ will be called the forward rate. Typically $f(0, T)$ represents an initial forward rate.

We assume that the dynamics of $f(t, T)$ on $\left(\Omega, \mathcal{F}{t \in[0, \tau]}, \mathbf{P}\right)$ is represented by $$d f(t, T)=\alpha(t, T) d t+\sigma(t, T) d W{t},$$
where $W_{t}=\left(W_{t}^{1}, \cdots, W_{t}^{d}\right)^{T}$ is a $d$-dimensional P-Brownian motion, and $\alpha(t, T)$ and $\sigma(t, T)$ are predictable processes satisfying some technical conditions. Here, $\sigma(t, T)=\left(\sigma^{1}(t, T), \cdots, \sigma^{d}(t, T)\right)^{T}$ is a $d$-dimensional process. The second term, $\sigma(t, T) d W_{t}$ denotes the inner product of $\sigma(t, T)$ and $d W_{t}$ in $\mathbf{R}^{d}$, specifically
$$\sigma(t, T) d W_{t}=\sum_{l=1}^{d} \sigma^{l}(t, T) d W_{t}^{l} .$$

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

This section briefly studies some fundamental subjects in the HJM model, specifically, forward rate process, arbitrage pricing, the market price of risk, and state price deflator.
Forward rate process
Here let us represent the forward rate process under the risk-neutral measure Q. Differentiating equation (4.9) with respect to $T$, we have
$$-\alpha(s, T)-\sigma(s, T) \int_{s}^{T} \sigma(s, u) d u=\frac{\partial b(s, T)}{\partial T} .$$
From equations (4.10) and (4.12), it follows that
\begin{aligned} -\alpha(s, T)-\sigma(s, T) \int_{s}^{T} \sigma(s, u) d u &=\frac{\partial v(s, T)}{\partial T} \varphi_{t} \ &=-\sigma(s, T) \varphi_{t} \end{aligned}
Substituting the above into equation (4.1), we obtain
$$d f(t, T)={-\sigma(t, T) v(t, T)+\sigma(t, T) \varphi(t)} d t+\sigma(t, T) d W_{t}$$
Recall the Q-Brownian motion $Z_{t}=\int_{0}^{t} \varphi_{s} d s+W_{t}$. Substituting this into the above, we have
$$d f(t, T)=-\sigma(t, T) v(t, T) d t+\sigma(t, T) d Z_{t}$$
where the drift under $\mathbf{Q}$ is completely determined by the volatility $\sigma(t, T)$. This form is the well-known forward rate process in the HJM model. For pricing interest rate derivatives, the dynamics of the forward rates are typically simulated by equation (4.18), and the bond pricing is performed by using this form. This method is essentially the same as used with short rate models.

## 金融代写|利率建模代写Interest Rate Modeling代考|Volatility and Principal Components

This section introduces a method for constructing the volatility in the H.JM model. There are two major approaches to do so. One is a market approach; the other is a historical approach.

In the market approach, the volatility is estimated such that the model implies option prices consistent with their market prices. In the historical approach, the volatility is constructed to represent a historical dynamics of an interest rate, for example, the short rate or the forward rate. Experimentally, these two approaches result in quite different volatility structures.

When we calibrate the model for derivatives pricing, the market approach should be employed. In this, it is understood that historical volatility cannot explain market prices because the option prices are determined mostly by traders’ forecasts for the future market rather than by historical volatility. Therefore, adopting a historical approach will result in a model that misprices major derivatives. Such a model is not valid for derivatives trading.

However, when we intend to calibrate a model for interest-risk-management, the historical approach is recommended, rather than the market approach. In the historical approach, principal component analysis (PCA) is a standard technique for reducing the dimensionality of the model. PCA will be repeatedly used in this book, and we introduce the construction of volatility by applying PCA. The fundamentals of PCA and the relevant linear algebra are given in Appendix B.

## 金融代写|利率建模代写Interest Rate Modeling代考|Heath-Jarrow-Morton Framework

Let(Ω,F吨∈[0,τ],磷)是一个过滤的概率空间，其中F吨∈[0,τ]是增强过滤和磷表示真实世界的度量。到期的瞬时远期利率吨当时观察到吨表示为F(吨,吨). 用法明确时，F(吨,吨)将被称为远期利率。通常F(0,吨)表示初始远期利率。

dF(吨,吨)=一个(吨,吨)d吨+σ(吨,吨)d在吨,

σ(吨,吨)d在吨=∑l=1dσl(吨,吨)d在吨l.

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

−一个(s,吨)−σ(s,吨)∫s吨σ(s,在)d在=∂b(s,吨)∂吨.

−一个(s,吨)−σ(s,吨)∫s吨σ(s,在)d在=∂在(s,吨)∂吨披吨 =−σ(s,吨)披吨

dF(吨,吨)=−σ(吨,吨)在(吨,吨)+σ(吨,吨)披(吨)d吨+σ(吨,吨)d在吨

dF(吨,吨)=−σ(吨,吨)在(吨,吨)d吨+σ(吨,吨)d从吨

## 有限元方法代写

tatistics-lab作为专业的留学生服务机构，多年来已为美国、英国、加拿大、澳洲等留学热门地的学生提供专业的学术服务，包括但不限于Essay代写，Assignment代写，Dissertation代写，Report代写，小组作业代写，Proposal代写，Paper代写，Presentation代写，计算机作业代写，论文修改和润色，网课代做，exam代考等等。写作范围涵盖高中，本科，研究生等海外留学全阶段，辐射金融，经济学，会计学，审计学，管理学等全球99%专业科目。写作团队既有专业英语母语作者，也有海外名校硕博留学生，每位写作老师都拥有过硬的语言能力，专业的学科背景和学术写作经验。我们承诺100%原创，100%专业，100%准时，100%满意。

## MATLAB代写

MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。