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• 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
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## 金融代写|量化风险管理代写Quantitative Risk Management代考|CORNISH-FISHER VAR

The delta-normal VaR model assumes that underlying returns are normally distributed and that option returns can be approximated using their delta-adjusted exposure. The Cornish-Fisher VaR model maintains the first assumption, while trying to improve the approximation for options. The method relies on what is known as the Cornish-Fisher expansion. The Cornish-Fisher expansion is a general method that allows us to approximate the confidence intervals for a random variable based on the central moments of that variable. As with the delta-normal approach, the Cornish-Fisher approach can easily be extended to portfolios containing multiple securities.

To start with, we introduce some notation. Define the value of an option as $V$, and the value of the option’s underlying security as $U$. Next, define the option’s exposure-adjusted Black-Scholes-Merton Greeks as
\begin{aligned} &\tilde{\Delta}=\frac{d V}{d U} U=\Delta U \ &\tilde{\Gamma}=\frac{d^{2} V}{d U^{2}} U^{2}=\Gamma U^{2} \ &\theta=\frac{d V}{d t} \end{aligned}
Given a return on the underlying security, $R$, we can approximate the change in value of the option using the exposure-adjusted Greeks as
$$d V \approx \tilde{\Delta} R+\frac{1}{2} \tilde{\Gamma} R^{2}+\theta d t$$
If the returns of the underlying asset are normally distributed with a mean of zero and a standard deviation of $\sigma$, then we can calculate the moments of $d V$ based on Equation 3.4. The first three central moments and skewness of $d V$ are
\begin{aligned} \mu_{d V} &=\mathrm{E}[d V]=\frac{1}{2} \tilde{\Gamma} \sigma^{2}+\theta d t \ \sigma_{d V}^{2} &=\mathrm{E}\left[(d V-\mathrm{E}[d V])^{2}\right]=\tilde{\Delta}^{2} \sigma^{2}+\frac{1}{2} \tilde{\Gamma}^{2} \sigma^{4} \ \mu_{3, d V} &=3 \tilde{\Delta}^{2} \tilde{\Gamma} \sigma^{4}+\tilde{\Gamma}^{3} \sigma^{6} \ s_{d V} &=\frac{\mu_{3, d V}}{\sigma_{d V}^{3}} \end{aligned}

## 金融代写|量化风险管理代写Quantitative Risk Management代考|BACKTESTING

An obvious concern when using $\mathrm{VaR}$ is choosing the appropriate confidence interval. As mentioned, $95 \%$ has become a very popular choice in risk management. In some settings there may be a natural choice, but, most of the time, the specific value chosen for the confidence level is arbitrary.

A common mistake for newcomers is to choose a confidence level that is too high. Naturally, a higher confidence level sounds more conservative. A risk manager who measures one-day VaR at the $95 \%$ confidence level will, on average, experience an exceedance event every 20 days. A risk manager who measures VaR at the $99.9 \%$ confidence level expects to see an exceedance only once every 1,000 days. Is an event that happens once every 20 days really something that we need to worry about? It is tempting to believe that the risk manager using the $99.9 \%$ confidence level is concerned with more serious, riskier outcomes, and is therefore doing a better job.

The problem is that, as we go further and further out into the tail of the distribution, we become less and less certain of the shape of the distribution. In most cases, the assumed distribution of returns for our portfolio will be based on historical data. If we have 1,000 data points, then there are 50 data points to back up our $95 \%$ confidence level, but only one to back up our $99.9 \%$ confidence level. As with any parameter, the variance of our estimate of the parameter decreases with the sample size. One data point is hardly a good sample size on which to base a parameter estimate.

A related problem has to do with backtesting. Good risk managers should regularly backtest their models. Backtesting entails checking the predicted outcome of a model against actual data. Any model parameter can be backtested.

## 金融代写|量化风险管理代写Quantitative Risk Management代考|Translation Invariance

A portfolio composed solely of risk-free assets-cash or short-term Treasuries, for example-has, by definition, zero risk. Adding or subtracting risk-free assets to a portfolio should not alter the risk of that portfolio. A risk measure that is unaltered by the addition or subtraction of a risk-free asset is said to obey translation invariance. Both standard deviation and $\mathrm{VaR}$ are translation invariant.

Translation invariance is sometimes defined in a slightly different fashion, which can lead to some confusion. By far the most common way of defining risk is in terms of uncertainty about the change in the value of an asset. This is the approach we have used up until now, and the approach we will use throughout the rest of this book.

## 有限元方法代写

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## 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。