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# 金融代写|金融衍生品代写Financial Derivatives代考|FINM7041 ESTIMATING VOLATILITIES

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## 金融代写|金融衍生品代写Financial Derivatives代考|ESTIMATING VOLATILITIES

In this unit we explain how historical data can be used to produce estimates of the current andfuture levels of volatilities and correlations. The unit is relevant both to the calculation of valueat risk using the model-building approach and to the valuation of derivatives. When calculatingvalue at risk, we are most interested in the current levels of volatilities and correlations because weare assessing possible changes in the value of a portfolio over a very short period of time. Whenvaluing derivatives, forecasts of volatilities and correlations over the whole life of the derivative areusually required.The unit considers models with imposing names such as exponentially weighted movingaverage (EWMA), autoregressive conditional heteroscedasticity $(\mathrm{ARCH})$, and generalized autoregressive conditional heteroscedasticity (GARCH). The distinctive feature of the models is thatthey recognize that volatilities and correlations are not constant. During some periods a particularvolatility or correlation may be relatively low, whereas during other periods it may be relativelyhigh. The models attempt to keep track of the variations in the volatility or correlation throughtime. $\sigma^2$
Consider a time series of returns $r t+i, i=1, \cdots, \tau$ and $T=t+\tau$, the sample variance,
$$\hat{\sigma}^2=\frac{1}{\tau-1} \sum_{i=1}^\tau\left(\gamma_{t+i}-\mu\right)^2$$
where $r t$ is the return at time $t$, and $\mu$ is the average return over the $\tau$-period, and $\tau=$ $\sqrt{\sigma^2}$ is the unconditional volatility for the period $t$ to $T$. If $T-t$ is e.g. a ten-year period and $t$ is measured in daily interval, then $b \sigma^2$ in (1) is the daily variance, $b \sigma^2 d$, over the ten-year period. If $t$ is measured in weekly interval, then $b \sigma^2$ in (1) is the weekly variance, $b \sigma^2 w$, over the ten-year period. Since variance is linear in time and can be aggregated but not standard deviation,

$$\hat{\sigma}w^2=5 \times \hat{\sigma}_d^2$$ with a multiplier of 5 since there are 5 trading days in a week. To derive volatility, which is often linked to the standard deviation, we have the weekly volatility \begin{aligned} \hat{\sigma}_w & =\sqrt{5 \times \hat{\sigma}_d^2} \ & =\sqrt{5} \times \hat{\sigma}_d \end{aligned} and daily volatility is simply $b \sigma d$. It is a well known fact that volatility does not remain constant through time, the conditional volatility, $\sigma t$, is a more relevant information for asset pricing and risk management at time $t$. So it is a common practice to break $T-t$ up into smaller superiors such that $$\begin{gathered} T-t=\left(T_n-T{n-1}\right)+\left(T_{n-1}-T_{n-2}\right)+\left(T_{n-3}-T_{n-3}\right)+\ldots+\left(T_1-t\right) \ =T_n+T_{n-1}+T_{n-2}+\ldots+T_1 \end{gathered}$$
and (1) becomes
$$\hat{\sigma}t^2=\frac{1}{T_j-1} \sum{i=1}^{T_j}\left(T_{t+i}-\mu\right)^2, \quad j=1, \ldots, n$$

## 金融代写|金融衍生品代写Financial Derivatives代考|USING SQUARED RETURN AS A PROXY FOR DAILY VOLATILITY

Volatility is a latent variable. Before high frequency data became widely available, many researchers resorted to using daily squared return, calculated from market closing prices, to proxy daily volatility. Lopez (2001) shows that $\varepsilon^{2 t}$ is an unbiased but extremely imprecise estimator of $\sigma^{2 t}$ due to its asymmetric distribution. Let
$$r_t=\mu+\varepsilon_t, \quad \varepsilon_t=\sigma_t z_t$$
If $r_t \sim N\left(0, \sigma_t^2\right)$, then $E\left(\left|r_t\right|\right)=\sigma_t \sqrt{2 / \pi}$. Hence $\hat{\sigma}_t=\frac{\left|r_t\right|}{\sqrt{2 / \pi}}$ if $r_t$ has conditional and $z_t \sim N(0,1)$. Then

$$E\left[\varepsilon_t^2 \mid \Phi_{t-1}\right]=\sigma_t^2 E\left[z_t^2 \mid \Phi_{t-1}\right]=\sigma_t^2$$
since $z_t^2 \sim \chi_{(1)}^2$. However, since the median of a $\chi_{(1)}^2$ distribution is $0.455$, is $\varepsilon_t^2$ less than $\frac{1}{2} \sigma_t^2$ more than $50 \%$ of the time. In fact
$$P_r\left(\varepsilon_t^2 \in\left[\frac{1}{2} \sigma_t^2, \frac{3}{2} \sigma_t^2\right]\right)=P_r\left(z_t^2 \in\left[\frac{1}{2}, \frac{3}{2}\right]\right)=0.2588,$$
which means that $\varepsilon^{2 t}$ is $50 \%$ greater or smaller than $\sigma^{2 t}$ nearly $75 \%$ of the time! Under the null hypothesis that rt in (4) is generated by a GARCH(1,1) process, Andersen and Bollerslev (1998) show that the population $R^2$ for the regression
$$\varepsilon_t^2=\alpha+\beta \hat{\sigma}_t^2+v_t$$
is equal to $k^{-1}$ where $k$ is the kurtosis of the standardized residuals, $z t$, and $k$ is finite. For conditional Gaussian error, the $R^2$ from a correctly specified GARCH $(1,1)$ model is bounded from above by 1/3. Christodoulakis and Satchell (1998) extend the results to include compound normals and the Gram-Charlier class of distributions and show that the mis-estimation offorecast performance is likely to be worsened by non-normality.

## 金融代写|金融衍生品代写Financial Derivatives代考|ESTIMATING VOLATILITIES

$$\hat{\sigma}^2=\frac{1}{\tau-1} \sum_{i=1}^\tau\left(\gamma_{t+i}-\mu\right)^2$$
$$\hat{\sigma} w^2=5 \times \hat{\sigma}d^2$$ $$\hat{\sigma}_w=\sqrt{5 \times \hat{\sigma}_d^2} \quad=\sqrt{5} \times \hat{\sigma}_d$$ 的信息.. 所以打破是一种常见的俼法 $T-t$ 上䍧到更小的上级，这样 $$T-t=\left(T_n-T n-1\right)+\left(T{n-1}-T_{n-2}\right)+\left(T_{n-3}-T_{n-3}\right)+\ldots+\left(T_1-t\right)=T_n+T_{n-1}+T_{n-2}+\ldots+T_1$$
(1) 变成
$$\hat{\sigma} t^2=\frac{1}{T_j-1} \sum i=1^{T_j\left(T_{t+i}-\mu\right)^2, \quad j=1, \ldots, n}$$

## 金融代写|金融衍生品代写Financial Derivatives代考|USING SQUARED RETURN AS A PROXY FOR DAILY VOLATILITY

$$r_t=\mu+\varepsilon_t, \quad \varepsilon_t=\sigma_t z_t$$

$$P_r\left(\varepsilon_t^2 \in\left[\frac{1}{2} \sigma_t^2, \frac{3}{2} \sigma_t^2\right]\right)=P_r\left(z_t^2 \in\left[\frac{1}{2}, \frac{3}{2}\right]\right)=0.2588,$$
Bollerslev (1998) 表朋总体 $R^2$ 对扣回忉
$$\varepsilon_t^2=\alpha+\beta \hat{\sigma}_t^2+v_t$$

## 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。