### 统计代写|风险建模代写Financial risk modeling代考|Covariance Estimation

statistics-lab™ 为您的留学生涯保驾护航 在代写风险建模Financial risk 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。我们的专家在代写风险建模Financial risk modeling代写方面经验极为丰富，各种代写风险建模Financial risk 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

## 统计代写|风险建模代写Financial risk modeling代考|Dynamic Asset-Allocation

The recent availability of large, high-frequency financial data sets potentially provides a rich source of information about asset-price dynamics. Specifically, nonparametric variance/covariance measures constructed by summing intra-daily return data (i.e. realized variances and covariances) have the potential to provide very accurate estimates of the underlying quadratic variation and covariation and, as a consequence, accurate estimation of betas for asset pricing, index autocorrelation and lead-lag patterns. These measures, however, have been shown to be sensitive to market microstructure noise inherent in the observed asset prices. Moreover, it is well known from Epps (1979) that the nonsynchronicity of observed data leads to a bias towards zero in correlations among stocks as the sampling frequency increases. Motivated by these difficulties, some modifications of realized covariance-type estimators have been proposed in the literature (Martens, 2004; Hayashi and Yoshida, 2005; Large, 2007; Voev and Lunde, 2007; Barndorff-Nielsen et al., 2008a; Kinnebrock and Podolskij, 2008).

A different methodology has been proposed in Malliavin and Mancino $(2002)$, which is explicitly conceived for multivariate analysis. This method is based on Fourier analysis and does not rely on any datasynchronization procedure but employs all the available data. Therefore, from the practitioner’s point of view the Fourier estimator is easy to implement as it does not require any choice of synchronization method or sampling scheme.

## 统计代写|风险建模代写Financial risk modeling代考|Some properties of the Fourier estimator

The Fourier method for estimating co-volatilities was proposed in Malliavin and Mancino $(2002)$ considering the difficulties arising in the multivariate setting when applying the quadratic covariation theorem to the true returns data, given the nonsynchronicity of observed prices for different assets. In fact, the quadratic covariation formula is

unfeasible when applied to estimate cross-volatilities because it requires synchronous observations which are not available in real situations. Being based on the integration of “all” data, the Fourier estimator does not need any adjustment to fit nonsynchronous data. We briefly recall the methodology below (see also Malliavin and Mancino, 2009).

Assume that $p(t)=\left(p^{1}(t), \ldots, p^{k}(t)\right)$ are Brownian semi-martingales satisfying the following Itô stochastic differential equations
$$d p^{j}(t)=\sum_{i=1}^{d} \sigma_{i}^{j}(t) d W^{i}(t)+b^{j}(t) d t \quad j=1, \ldots, k$$
where $W=\left(W^{1}, \ldots, W^{d}\right)$ are independent Brownian motions. The price process $p(t)$ is observed on a fixed time window, which can always be reduced to $[0,2 \pi]$ by a change of the origin and rescaling, and $\sigma_{}^{}$ and $b^{}$ are adapted random processes satisfying the hypothesis $$E\left[\int_{0}^{2 \pi}\left(b^{i}(t)\right)^{2} d t\right]<\infty, E\left[\int_{0}^{2 \pi}\left(\sigma_{i}^{j}(t)\right)^{4} d t\right]<\infty \quad i=1, \ldots, d, j=1, \ldots, k .$$ From the representation (1.1) we define the “volatility matrix,” which in our hypothesis depends upon time $$\Sigma^{i j}(t)=\sum_{r=1}^{d} \sigma_{r}^{i}(t) \sigma_{r}^{i}(t)$$ The Fourier method reconstructs $\Sigma^{ *}(t)$ on $[0,2 \pi]$ using the Fourier transform of $d p^{\star}(t)$.

The main result in Malliavin and Mancino (2009) relates the Fourier transform of $\Sigma^{* }$ to the Fourier transform of the log-returns $d p^{}$. More precisely, the following result is proved: compute the Fourier transform of $d p^{j}$ for $j=1, \ldots, k$, defined for any integer $z$ by
$$F\left(d p^{j}\right)(z)=\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{-i z t} d p^{j}(t)$$
and consider the Fourier transform of the cross-volatility function defined for any integer $z$ by
$$F\left(\Sigma^{i j}\right)(z)=\frac{1}{2 \pi} \int_{0}^{2 \pi} e^{-i z t} \Sigma^{i j}(t) d t$$

then the following convergence in probability holds
$$F\left(\Sigma^{i j}\right)(z)=\lim {N \rightarrow \infty} \frac{2 \pi}{2 N+1} \sum{|s| \leq N} F\left(d p^{i}\right)(s) F\left(d p^{j}\right)(z-s)$$

## 统计代写|风险建模代写Financial risk modeling代考|Forecasting and asset allocation

We use the methodology suggested by Fleming et al. (2001) and Bandi et al. (2008) to evaluate the economic benefit of the Fourier estimator of integrated covariance in the context of an asset-allocation strategy. Specifically, we compare the utility obtained by virtue of covariance forecasts based on the Fourier estimator to the utility obtained through covariance forecasts constructed using the more familiar realized covariance and other recently proposed estimators. In the following, we adopt a notation which is common in the literature about portfolio management. It will not be difficult for the reader to match it with the one in the previous section.

Let $R f$ and $R_{t+1}$ be the risk-free return and the return vector on $k$ risky assets over a day $[t, t+1]$, respectively. Define $\mu_{t}=E_{t}\left[R_{t+1}\right]$ and $\Phi_{t}=E_{t}\left[\left(R_{t+1}-\mu_{t}\right)\left(R_{t+1}-\mu_{t}\right)^{\prime}\right]$ as the conditional expected value and the conditional covariance matrix of $R_{t+1}$. We consider a mean-variance investor who solves the problem
$$\min {\omega{t}} \omega_{t}^{\prime} \Phi_{t} \omega_{t}$$
subject to
$$\omega_{t}^{\prime} \mu_{t}+\left(1-\omega_{t}^{\prime} \mathbf{1}{k}\right) R^{f}=\mu p,$$ where $\omega{t}$ is a $k$-vector of portfolio weights, $\mu_{p}$ is a target expected return on the portfolio and $1_{k}$ is a $k \times 1$ vector of ones. The solution to this program is
$$\omega_{t}=\frac{\left(\mu_{p}-R^{f}\right) \Phi_{t}^{-1}\left(\mu_{t}-R^{f} \mathbf{1}{k}\right)}{\left(\mu{t}-R^{f} \mathbf{1}{k}\right)^{\prime} \Phi{t}^{-1}\left(\mu_{t}-R \mathbf{R}^{\prime}\right)} .$$
We estimate $\Phi_{t}$ using one-day-ahead forecasts $\hat{C}_{t}$ given a time series of daily covariance estimates obtained using the Fourier estimator, the

realized covariance estimator, the realized covariance plus leads and lags estimator, the $\mathrm{AO}$ estimator, its subsampled version and the kernel estimator. The out-of-sample forecast is based on a univariate ARMA model.

Given sensible choices of $R^{f}, \mu_{p}$ and $\mu_{t}$, each one-day-ahead forecast leads to the determination of a daily portfolio weight $\omega_{t}$. The time series of daily portfolio weights then leads to daily portfolio returns. In order to concentrate on volatility approximation and to abstract from the issues that would be posed by expected stock-return predictability, for all times $t$ we set the components of vector $\mu_{t}=E_{t}\left[R_{t+1}\right]$ equal to the sample means of the returns on the risky assets over the forecasting horizon. Finally, we employ the investor’s long-run mean-variance utility as a metric to evaluate the economic benefit of alternative covariance forecasts $\hat{C}{t}$, that is, $$U^{*}=\bar{R}^{p}-\frac{\lambda}{2} \frac{1}{m} \sum{t=1}^{m}\left(R_{t+1}^{p}-\bar{R}^{p}\right)^{2},$$
where $R_{t+1}^{p}=R^{f}+\omega_{t}^{\prime}\left(R_{t+1}-R^{f} \mathbf{1}{k}\right)$ is the return on the portfolio with estimated weights $\omega{t}, \bar{R}^{p}=\frac{1}{m} \sum_{t=1}^{m} R_{t+1}^{p}$ is the sample mean of the portfolio returns across $m \leq n$ days and $\lambda$ is a coefficient of risk aversion.

## 统计代写|风险建模代写Financial risk modeling代考|Dynamic Asset-Allocation

Malliavin 和 Mancino 提出了一种不同的方法(2002)，这是为多变量分析而明确设想的。该方法基于傅立叶分析，不依赖任何数据同步过程，而是使用所有可用数据。因此，从从业者的角度来看，傅里叶估计器很容易实现，因为它不需要任何同步方法或采样方案的选择。

## 统计代写|风险建模代写Financial risk modeling代考|Some properties of the Fourier estimator

Malliavin 和 Mancino 提出了用于估计共挥发性的傅里叶方法(2002)考虑到在将二次协变定理应用于真实收益数据时在多元设置中出现的困难，因为不同资产的观察价格不同步。实际上，二次协变公式是

dpj(吨)=∑一世=1dσ一世j(吨)d在一世(吨)+bj(吨)d吨j=1,…,ķ

Malliavin 和 Mancino (2009) 的主要结果涉及傅里叶变换Σ∗对数返回的傅里叶变换dp. 更准确地说，证明了以下结果：计算傅里叶变换dpj为了j=1,…,ķ, 为任何整数定义和经过
F(dpj)(和)=12圆周率∫02圆周率和−一世和吨dpj(吨)

F(Σ一世j)(和)=12圆周率∫02圆周率和−一世和吨Σ一世j(吨)d吨

F(Σ一世j)(和)=林ñ→∞2圆周率2ñ+1∑|s|≤ñF(dp一世)(s)F(dpj)(和−s)

## 统计代写|风险建模代写Financial risk modeling代考|Forecasting and asset allocation

ω吨′μ吨+(1−ω吨′1ķ)RF=μp,在哪里ω吨是一个ķ-投资组合权重的向量，μp是投资组合的目标预期回报，并且1ķ是一个ķ×1的向量。该程序的解决方案是
ω吨=(μp−RF)披吨−1(μ吨−RF1ķ)(μ吨−RF1ķ)′披吨−1(μ吨−RR′).

## 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。