### 分类： 交易策略作业代写

• 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

Power generation costs consist of fixed costs (e.g., equipment depreciation costs, labor costs, and maintenance costs) and variable costs (e.g., fuel costs and exhaust processing costs). However, the variable cost is almost the fuel cost. Therefore, the unit cost of gas-fired generation expressed as Cost and the corresponding natural gas procurement cost expressed as Gas satisfy the following equation:
$$\text { Cost }=\alpha_0 \times \text { Gas }+\alpha_1,$$
where $\alpha_0$ and $\alpha_1$ are the coefficients. Although the Henry Hub futures price HenryHub $b_{f, t}$ and the PJM futures price $P J M_{f, t}$, both of which are unit root processes, are cointegrated, the long-term equilibrium equation, which is a stochastic process, can have large outliers. Then, when the own spot spread, that is, the difference between the power generation unit cost and the corresponding gas procurement unit price, is smaller than the future spread, that is, the future price difference between the PJM and Henry Hub, we swap the spot spread, Spread $_s$ and the future spread, Spread $_f$, which we express as
$$\begin{gathered} \text { Spread }s=\alpha_0 \times \text { HenryHub }{f, t}+\alpha_1-\text { HenryHub }{f, t} \ \text { Spread }_f=\text { PJM }{f, t}-\text { HenryHub } b_{f, t} . \end{gathered}$$
Therefore, the difference between these spreads is
$$\text { Spread }s-\text { Spread }_f=\alpha_0 \times \text { HenryHub } b{f, t}+\alpha_1-P J M_{f, t} .$$
In the following equation:
$$\alpha_0 \times \text { HenryHub } b_{f, t}+\alpha_1-P J M_{f, t}<0 .$$
If we take the Henry Hub long position and the PJM short position corresponding to the electric energy planned for generation, we can lock in profit.

By estimating the long-term equilibrium equation of $H e n r y H u b_{f, t}$ and $P J M_{f, t}$ in a cointegration relationship, we can determine whether the futures spread on a candidate trading date is wider or narrower than the expected spread. This determination enables statistical arbitrage trading between Henry Hub and PJM.

Since the prices in period $t$ are not available for trading in period $t$, we estimate the following long-term equilibrium equation using the price series up to period $t-1$ :
$$P J M_{f, t}=\beta_{f, 0} \times \text { HenryHub} b_{f, t}+\beta_{f, 1^{\circ}} .$$
If the futures spread is higher than the expected value, then we express it as
$$P J M_{f, t}>\beta_{f, 0} \times \text { HenryHub } b_{f, t}+\beta_{f, 1} .$$
We can consider that the PJM price is higher and the Henry Hub price is lower; therefore, we take the PJM short position and Henry Hub long position. Then, the condition for closing these arbitrage positions is
\begin{aligned} \text { PJM }{f, t} &-\text { avgShort } P J M_f+\operatorname{avg} \text { LongHenryHub } \ &-\text { HenryHub } b{f, t}>0 \end{aligned}
where avg Short $P J M_f$ is the average price of the PJM futures short positions taken, and avgLong Henry Hub $b_f$ is the average price of the Henry Hub futures long positions taken. The clearance of all these futures positions under this condition leads to profit.

Conversely, if the futures spread is below the expected value, then we express it as
$$P J M_{f, t}<\beta_{f, 0} \times \text { HenryHub} b_{f, t}+\beta_{f, 1} .$$
We determine that the PJM price is lower and the Henry Hub price is higher; therefore, we take the PJM long position and Henry Hub short position.

# 交易策略代考

$$\text { Cost }=\alpha_0 \times \text { Gas }+\alpha_1,$$

Spread $s=\alpha_0 \times$ HenryHub $f, t+\alpha_1-$ HenryHub $f, t$ Spread $_f=\operatorname{PJM} f, t-$ HenryHub $b_{f, t}$.

Spread $s-$ Spread $_f=\alpha_0 \times$ HenryHub $b f, t+\alpha_1-P J M_{f, t}$.

$$\alpha_0 \times \text { HenryHub } b_{f, t}+\alpha_1-P J M_{f, t}<0 .$$

$$P J M_{f, t}=\beta_{f, 0} \times \text { HenryHub } b_{f, t}+\beta_{f, 1^{\circ}} .$$

$$P J M_{f, t}>\beta_{f, 0} \times \text { HenryHub } b_{f, t}+\beta_{f, 1} .$$

$$\text { PJM } f, t-\operatorname{avgShort} P J M_f+\text { avg LongHenryHub } \quad-\text { HenryHub } b f, t>0$$

$$P J M_{f, t}<\beta_{f, 0} \times \text { HenryHubb } b_{f, t}+\beta_{f, 1} .$$

## 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。

• 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

We can estimate the cointegrating vectors by using dynamic OLS (DOLS). OLS estimates the following equation with lag terms for the explanatory variables to climinate autocorrelation:
$$x_{v, t}=\varphi_0+\sum_{i=1}^{v-1}\left(\beta_i \varphi_{i, t}+\sum_{j=-K}^K \phi_{i, j} \Delta x_{i, t-j}\right) .$$
Since Sect. 2.2.4 utilizes a two-variable model, the model for estimating the longterm equilibrium is
$$P J M_t=\varphi_0+\varphi_1 \text { HenryHub}t+\sum{j=-K}^K \phi_j \Delta \text { Henry Hub } b_{t-j} .$$
The lag order $K$ was determined using SBIC. The long-term equilibrium equation for future prices is
The long-term equilibrium equation for the spot prices is
$$P J M_{\text {spot }}=11.142 \times \text { HenryHub }_{\text {spot }}+5.732 .$$

The only way to profit by trading goods is to “buy at a lower price and sell at a higher price.” If we trade only one item, then price forecasting is the most important matter. Is this realistically possible? A market is efficient if the information that affects the market price is comprehensive, constant, and has a timely effect on the price. Markets for securities and commodities listed on exchanges are almost efficient and depend on liquidity. In other words, we cannot forecast the price because the price already reflects all the currently available information, and any information that affects the price will occur independently of the price. Unfortunately, it is impossible for market participants to earn returns above the market average. Certainly, a “fully efficient market” is theoretical or virtual. Therefore, some investors and speculators try to collect information before it is reflected in the price. However, these actions make the market more efficient. Because the stationary hypothesis for most energy prices is rejected by the unit root test using daily data, energy companies should consider energy markets as efficient, and energy prices as unpredictable.

In general, power companies procure various types of fuels from various markets, produce electricity using various power generation methods, and sell the power through various sales channels. Section $2.3$ assumes a simple model of purchasing natural gas at the Henry Hub price and selling electricity at the PJM price, as Fig. $2.8$ illustrates. We propose two trading strategies. Section $2.4$ will simulate these methods using actual historical data. Both focus not on these prices but on the price difference between Henry Hub prices and PJM prices. We cannot expect profit owing to market efficiency, even if we analyze each price in detail. On the other hand, we demonstrate the potential to make a profit by investigating price differences, which is a stationary process. When buying the gas required to produce one unit of electricity and selling it, the gross margin is often called the spark spread.

The trading strategy introduced in Sect. 2.3.1 is the arbitrage between the futures market spreads and a company’s spreads expected from its power generation efficiency. This takes advantage of the spread of futures as a stochastic process. All we have to do is take the Henry Hub long position and the PJM short position to secure profits when a favorable futures spread occurs stochastically. The strategy proposed in Sect. 2.3.2 is statistical arbitrage utilizing the cointegration relationship between Henry Hub prices and PJM prices in the futures market. Making use of the longterm equilibrium equation in the futures market that expresses the futures spread, the lower PJM long positions and the higher Henry Hub short positions are expected to yield profit in the narrower spreads than the market when the spread approaches the long-term equilibrium.

# 交易策略代考

$$x_{v, t}=\varphi_0+\sum_{i=1}^{v-1}\left(\beta_i \varphi_{i, t}+\sum_{j=-K}^K \phi_{i, j} \Delta x_{i, t-j}\right) .$$

$$P J M_t=\varphi_0+\varphi_1 \text { HenryHubt }+\sum j=-K^K \phi_j \Delta \text { Henry Hub } b_{t-j} .$$

$$P J M_{\text {spot }}=11.142 \times \text { HenryHub }_{\text {spot }}+5.732 .$$

Sect. 中介绍的交易策略。2.3.1 是期货市场价差与公司发电效率预期价差之间的套利。这利用了期货的价差作为一个随机过程。我们所要做的就是在随机出现有利的期货价差时，持有 Henry Hub 多头头寸和 PJM 空头头寸以确保获利。节中提出的策略。2.3.2是统计套利，利用期货市场上Henry Hub价格和PJM价格之间的协整关系。利用期货市场中表示期货价差的长期均衡方程，当价差接近长期均衡时，较低的 PJM 多头头寸和较高的 Henry Hub 空头头寸有望在比市场更窄的价差中产生利润.

## 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。

• 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

Before conducting various analyses and simulations, it is extremely important to interpret the representative statistics of the data. Table $2.1$ provides the summary statistics of the Henry Hub and the PJM.

Considering that each future has a maturity of one month, we set each spot price to January 29, 2021 and each future to December 30, 2020 to simulate the spot-future arbitrage described later in Sect. 2.3.1. Because we extract only the days when both the Henry Hub and PJM data are available, we have 1511 and 1477 observations for the futures and spot prices, respectively.

The mean and median are numerical values located in the center of the economic variables. The mean $\bar{x}$ of the series $\left(x_i \mid i=1,2, \ldots, N\right)$ is calculated as
$$\bar{x}=\frac{1}{N} \sum_{i=1}^N x_i .$$
On the other hand, the median is a value located in the center of each series arranged in descending order. The medians of these futures and spot series are at the 756th and 739th values, respectively. If the number of observations is even, then the median is the average of the two data points in the center. Thus, the median is a more stable index expressing the middle than the mean because outlier values have less effect. Figure $2.1$ shows three distribution examples with the same mean, but different medians. Table $2.1$ indicates that both the mean and median of each future are higher than those of each spot. In other words, both Henry Hub and PJM tend to be contango. We can infer that the supply and demand are not very tight during this period. We can express the relationship between the future price $p_f$ and its spot price $p_s$ as
$$p_f=p_s e^{c_c \Delta T},$$ where $C_c$ is the cost of carry expressed in terms of yield and $\Delta T$ is the period from the present to maturity. The cost of carry is the sum of the risk-free interest rate and holding cost, expressed as yield minus the convenience yield. Therefore, if their supply and demand remained tight during the period, then the utility of holding their spots would be increasing. Thus, their costs of carry should become negative, and their futures should become lower than their spots. In addition, the medians of both the Henry Hub future and spot prices are higher than their respective means. Therefore, we can expect to find many outliers in the left tail of each distribution. On the contrary, the medians of both the PJM future and spot prices are lower than their respective means. Therefore, we can expect to find many outliers in the right tail of each distribution.

Figures $2.1$ and $2.2$ bring to mind the long-term equilibrium relationship between Henry Hub and the PJM in both futures and spot markets. However, as all four variables accept the unit root hypothesis, we must suspect a spurious regression.
Engle and Granger [7] introduced the concept of “cointegration,” which connects multiple unpredictable stochastic variables with a unit root. If a linear combination of multiple unit root processes is stationary, then these variables have a cointegrated relationship. In other words: suppose that the following vector consists of $v$ variables in a unit root process:
$$\mathbf{X}t={ }^T\left(x{1 t}, x_{2 t}, \ldots, x_{v t}\right)$$
The following linear combination is derived from the inner product of the $v$ dimensional coefficient vector and $\mathbf{X}t$ : $$\boldsymbol{\beta} \mathbf{X}_t=\left(\beta_1, \beta_2, \ldots, \beta_v\right)^T\left(x{1 t}, x_{2 t}, \ldots, x_{v t}\right)$$
If $\boldsymbol{\beta} \boldsymbol{X}t$ is a stationary process, then $x{1 t}, x_{2 t}, \ldots, x_{v t}$ have a cointegrated relationship. Additionally,
$$\boldsymbol{\beta}=\left(\beta_1, \beta_2, \ldots, \beta_v\right)$$
is the cointegrating vector. If there is cointegration between some variables, then the deviation of the observed values from their long-term equilibrium is a stable stochastic process. Because many economic variables have unit roots, this concept is very often applied in a wide range of fields to examine the relationships between economic variables.

Therefore, we test whether the Henry Hub and PJM prices are cointegrated and expect to use this cointegrated relationship in the trading strategies.

Engle and Granger’s [7] proposed test for cointegration has limitations. First, it does not expect a system with three or more variables to have two or more cointegration relationships. Second, the test results may change when the variables are interchanged.

# 交易策略代考

$$\bar{x}=\frac{1}{N} \sum_{i=1}^N x_i$$

$$p_f=p_s e^{c_c \Delta T},$$

Engle 和 Granger [7] 引入了”协整”的概念，它将多个不可预测的随机变量与一个单位根联系起来。如果多个单 位根过程的线性组合是平稳的，则这些变量具有协整关系。换句话说：假设以下向量由 $v$ 单位根过程中的变量:
$$\mathbf{X} t={ }^T\left(x 1 t, x_{2 t}, \ldots, x_{v t}\right)$$

$$\boldsymbol{\beta} \mathbf{X}t=\left(\beta_1, \beta_2, \ldots, \beta_v\right)^T\left(x 1 t, x{2 t}, \ldots, x_{v t}\right)$$

$$\boldsymbol{\beta}=\left(\beta_1, \beta_2, \ldots, \beta_v\right)$$

## 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。