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# 数学代写|金融数学代写Financial Mathematics代考|MATHS1009 Illustrative Examples

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## 数学代写|金融数学代写Financial Mathematics代考|Illustrative Examples

As mentioned in the introduction to this chapter, the application of multivariate regression and time series models is in the area of linking multiple markets or multiple assets, and the tools are also useful in understanding the efficiency of the portfolios. There is some commonality among certain equities due to common fundamentals or industry factors that have similar impact on their performance. Whenever there is a deviation from expected commonality or co-movement, it provides trading opportunities. This is the basis of ‘pairs trading’ that is discussed in detail in Chapter $5 .$ Also the methodologies discussed in this chapter can be useful for portfolio analysis (Chapter 6) as well. The example that we present in this section is mainly for illustrating the methodologies discussed in this chapter as other applications will be taken up elsewhere.

We consider the daily exchange rates of five currencies British Pound (GBP), Euro (EUR), Japanese Yen (JPY), Canadian Dollar (CAD), Singapore Dollar (SGD), and Australian Dollar (AUD) all stated against US Dollar. The data is from January 1,1999 to May 2, 2016. Thus $Y_{t}$ is a five dimensional vector of time series. The return series $r_{t}$ are calculated as $r_{t}=\ln \left(Y_{t}\right)-\ln \left(Y_{t-1}\right)$. The plot of $Y_{t}$ is given in Figure $3.1$ and to make the comparison easier, the first observation in all series are set to unity. Clearly the series are non-stationary and exhibit common upward and downward movements in certain durations. The plot of returns and volatilities are given in Figures $3.2$ and 3.3, respectively.

Broadly defining the regimes as dollar strengthening-strong versus weak, the following regimes are identified: 1999-2002, 2002-2008, 2008-2009, 2009-2013 and 2013-present. The typical autocorrelation functions (ACFs) for the rate, return and volatility are given in Figures $3.4,3.5$, and 3.6, respectively.
Some general observations can be confirmed as follows:

• Rates are non-stationary.
• Returns are generally white-noise.
• Volatilities exhibit time-dependence.

## 数学代写|金融数学代写Financial Mathematics代考|State-Space Modeling

The state-space models were initially developed by control systems engineers to measure a signal contaminated by noise. The signal at time ” $t$ ” is taken to be a linear combination of variables, called state variables that form the so-called state vector at time, $t$. The key property of the state vector is that it contains information from past and present data but the future behavior of the system is independent of the past values and depends only on the present values. Thus the latent state vector evolves according to the Markov property. The state equation is stated as,
$$Z_{t}=\Phi_{t} Z_{t-1}+a_{t}$$
$$Y_{t}=H_{t} Z_{t}+N_{t},$$
where it is assumed that $a_{t}$ and $N_{t}$ are independent white-noise processes; $a_{t}$ is a vector white noise with covariance matrix $\Sigma_{a}$ and $N_{t}$ has variance $\sigma_{N}^{2}$. The matrix $\Phi_{t}$ in (4.1) is an $r \times r$ transition matrix and $H_{t}$ in (4.2) is an $1 \times r$ vector; both are allowed to vary in time. In engineering applications, the structure of these matrices is governed by some physical phenomena.

## 数学代写|金融数学代写Financial Mathematics代考|State-Space Modeling

\begin{aligned} &Z_{t}=\Phi_{t} Z_{t-1}+a_{t} \ &Y_{t}=H_{t} Z_{t}+N_{t} \end{aligned}

## 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。