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## 数学代写|金融数学代写Financial Mathematics代考|Extraneous Signals: Trading Volume, Volatility, etc.

The trading algorithms that are presented in this chapter thus far focus on the price and return behavior of an equity. In pairs trading, we considered the behavior of price-return of a related equity to decide when to buy one and sell the other. But there is also information, as noted in Section 4.3, present in the trading volume. The model discussed there relates the volume to volatility. If there is no variation in the information flow about an equity, there should not be any correlation between volume traded and the return volatility. If the correlation exists, we could indirectly infer that there is information and we may want to exploit that for trading. There is a vast literature in finance studying the relationship between trading volume and serial correlation in returns. We briefly review some select studies and provide an illustrative example with some possible algorithms.

Blume, Easley and O’Hara (1994) [45] investigate the informational role of volume. If the impounding of information into the stock price is not immediate, volume may provide information about the evolving process of a security’s return. The model discussed in Section $4.3$ by Tauchen and Pitts (1983) [311] and the associated empirical tests clearly document the strong relationship between volume and the absolute return. “But why such a pattern exists or even how volume evolves in markets is not clear.” Recall that the model assumes that the information content affects both the return and the volume specifications; yet the correlation between the return and volume can be shown to be zero. If the information content can be studied through the sequence of security prices and the associated volume, it may provide insights into the inefficiency in the market and how statistical arbitrage rules can exploit the inefficiency especially if there is an empirical lead-lag relationship. The quality of traders information can be captured best by combining price change with volume change. It is shown that the relationship between volume and price is in the form of V-shape thus indicating a non-linear relationship. Both ‘bad’ and ‘good’ information about the stock is likely to result in higher volume of trade.

Campbell, Grossman and Wang (1993) [65] also explore the relationship between volume and returns; volume information is used to distinguish between price movements that occur due to publicly available information and those that reflect changes in expected returns. It is predicted that price changes with high volume tend to be reversed and this relationship may not hold on days with low volume. Llorente, Michaely, Saar and Wang (2002) [246] extend this work by postulating that the market generally consists of liquidity and speculative traders. In periods of high volume but speculative trading, return autocorrelation tends to be positive and if it is liquidity trading, return autocorrelation tends to be negative. In the former case, returns are less likely to exhibit a reversal. Defining the volume turnover $(V)$ as the ratio of number of shares traded to the total number of outstanding shares, the following regression model is fit for each stock:
$$r_{t}=\beta_{0}+\beta_{1} r_{t-1}+\beta_{2} V_{t-1} \cdot r_{t-1}+a_{t} .$$

## 数学代写|金融数学代写Financial Mathematics代考|An Illustrative Example

The methodology discussed in the last section on price-based and volume-based filters on low frequency data can be extended to high frequency data as well. The key issue with the volume data is to come up with proper standardization to identify abnormal level of activity in a high frequency setting. The intra-day seasonality and the overall increase or decrease in inter-day volumes need to be tracked carefully to study and fully exploit the deviations. We highlight the practical issues involved using 30-min price-based data for Treasury yields from June 8, 2006 to August 29, $2013 .$ This can be taken as medium-frequency data. The price bars consist of high, low, open and close prices and the volume traded during the interval are considered. Because of the aggregated nature of the data, the traditional time series methods can be readily used to study the trend in price and volume series. Daily data is just a snapshot at 4 p.m. and the total volume for the day is the sum of all 30 -min volumes since the previous day.

Let $p_{t . m}$ be the log price in the $m$ th interval of the day ‘ $t$ ‘. Note that ‘ $m$ ‘ indexes the 30 -min interval in a day; the value of $m$ can range from 1 to 13. Let $r_{t . m}=p_{t . m}-p_{t . m-1}$ be the return and let $v_{t . m}$ be the log volume in that interval. We define volatility within the $m$ th time unit based on the price bar data as,
$$\hat{\sigma}{t . m}^{2}=0.5\left[\ln \left(H{t . m}\right)-\ln \left(L_{t . m}\right)\right]^{2}-0.386\left[\ln \left(C_{t . m}\right)-\ln \left(\mathrm{O}_{t . m}\right)\right]^{2}$$
similar to a measure defined for the daily data in Section 4.5.

## 数学代写|金融数学代写Financial Mathematics代考|Extraneous Signals: Trading Volume, Volatility, etc.

Blume、Easley 和 O’Hara (1994) [45] 研究了音量的信息作用。如果不能立即将信息纳入股票价格，则交易 量可能会提供有关证券回报演变过程的信息。部分讨论的模型4.3Tauchen 和 Pitts (1983) [311] 以及相关的 实证检验清楚地证明了交易量和绝对回报之间的密切关系。“但为什么存在这种模式，甚至市场交易量如何演变 都不清楚。”回想一下，该模型假设信息内容同时影响回报和数量规格；然而，回报和交易量之间的相关性可以 显示为露。如果可以通过证券价格和相关交易量的顺序来研究信息内容，则可以深入了解市场的低效率，以及 统计套利规则如何利用低效率，尤其是在存在经验领先-滞后关系的情况下。通过将价格变化与交易量变化结合 起来，可以最好地捕捉交易者信息的质量。结果表明，成交量和价格之间的关系呈 $V$ 形，因此表明了一种非线性 关系。关于股票的“坏”和“好”信息都可能导致交易量增加。

Campbell、Grossman 和 Wang (1993) [65］也探讨了交易量和收益之间的关系；交易量信息用于区分由 于公开信息而发生的价格变动和反映预期收益变化的价格变动。据预测，高成交量的价格变化往往会逆转，这 种关系在成交量低的日子可能不成立。Llorente、Michaely、Saar 和 Wang (2002) [246] 通过假设市场通 常由流动性和投机交易者组成来扩展这项工作。在交易量大但投机交易的时期，收益自相关往往是正的，如果 是流动性交易，收益自相关往往是负的。在前一种情况下，回报不太可能出现逆转。定义成交量 $(V)$ 作为交易 股数与流通股总数的比率，以下回归模型适用于每只股票:
$$r_{t}=\beta_{0}+\beta_{1} r_{t-1}+\beta_{2} V_{t-1} \cdot r_{t-1}+a_{t} .$$

## 数学代写|金融数学代写Financial Mathematics代考|An Illustrative Example

$$\hat{\sigma} t . m^{2}=0.5\left[\ln (H t . m)-\ln \left(L_{t . m}\right)\right]^{2}-0.386\left[\ln \left(C_{t . m}\right)-\ln \left(\mathrm{O}_{t . m}\right)\right]^{2}$$

## 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。

Posted on Categories:Financial Mathematics, 金融代写, 金融数学

## 数学代写|金融数学代写Financial Mathematics代考|MT4551 Analysis of Time Aggregated Data

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## 数学代写|金融数学代写Financial Mathematics代考|Realized Volatility and Econometric Models

Estimating volatility using high frequency data has received a great deal of attention as intra-day strategies became more prevalent. In the low frequency (daily) level some estimators based on select prices were presented in Chapter 2. Intra-day dynamics of volatility is of interest to traders and is usually not fully captured by the stock price indicators sampled during the day. If higher frequency data are used to estimate volatility of lower frequency data, it is important to know the model for the return at the lower frequency. To illustrate this, we consider data observed at two time scales; although they are somewhat at low frequency level, the concept can be easily extended to the high frequency context. If $r_{t, i}$ is the $i$ th day return in $t$ th month, the $t$ th month return assuming there ‘ $n$ ‘ trading days, $r_{t}^{m}=\sum_{i=1}^{n} r_{t, i}$. Note that $\sigma_{m}^{2}=\operatorname{Var}\left(r_{t}^{m} \mid F_{t-1}\right)=\sum_{i=1}^{n} \operatorname{Var}\left(r_{t, i} \mid F_{t-1}\right)+2 \sum_{i<j} \operatorname{Cov}\left(r_{t, i}, r_{t, j} \mid F_{t-1}\right)$. If $r_{t, i}$ is a white noise sequence, then
$$\hat{\sigma}{m}^{2}=\frac{n}{n-1} \sum{i=1}^{n}\left(r_{t, i}-\bar{r}{t}\right)^{2} .$$ But we had observed that in the high frequency data, returns do exhibit some serial correlation and so adjusting (4.39) for serial correlation is important to get a more accurate estimate of volatility. A simpler estimate of $\sigma{m}^{2}$ is the so-called realized volatility $\left(\mathrm{RV}{t}\right)$ $$\mathrm{RV}{t}=\sum_{i=1}^{n} r_{t, i}^{2}$$

## 数学代写|金融数学代写Financial Mathematics代考|Volatility and Price Bar Data

The price as assumed earlier is to follow a random walk model. Therefore, price changes (thus returns) over a time interval are distributed with mean zero and variance that is proportional to the length of the interval. Assuming that the prices follow continuous sample paths although the trading is closed for a certain duration and when trading is open, the actual transactions occur at discrete points in time. Treating the trading day as represented in a unit interval $[0,1]$ with $[0, f]$ representing the ‘market close’ time and $[f, 1]$ as ‘open’ time, it is shown that an estimator of volatility
$$\hat{\sigma}{1}^{2}=\frac{\left(O{1}-C_{0}\right)^{2}}{2 f}+\frac{\left(C_{1}-O_{1}\right)^{2}}{2(1-f)},$$
has efficiency two compared to the usual estimator, $\hat{\sigma}{0}^{2}=\left(C{1}-C_{0}\right)^{2}$ based on the closing prices of two successive trading days. This suggests clearly the inclusion of additional data points, such as opening price, which can be quite informative. Garman and Klass (1980) [158] suggest these and other estimators which are superior to the classical estimator of volatility $\hat{\sigma}_{0}^{2}$. It is argued that the high and low prices contain information regarding the volatility during the trading period and a composite estimator that is proposed,
$$\hat{\sigma}{2}^{2}=a \frac{\left(O{1}-C_{0}\right)^{2}}{f}+(1-a) \frac{\left(H_{1}-L_{1}\right)^{2}}{(1-f) \ln 2}$$
with optimal choice of ‘ $a$ ‘ $=0.17$ yields even higher efficiency.

## 数学代写|金融数学代写Financial Mathematics代考|Realized Volatility and Econometric Models

$$\hat{\sigma} m^{2}=\frac{n}{n-1} \sum i=1^{n}\left(r_{t, i}-\bar{r} t\right)^{2} .$$

$$\mathrm{RV} t=\sum_{i=1}^{n} r_{t, i}^{2}$$

## 数学代写|金融数学代写Financial Mathematics代考|Volatility and Price Bar Data

$$\hat{\sigma} 1^{2}=\frac{\left(O 1-C_{0}\right)^{2}}{2 f}+\frac{\left(C_{1}-O_{1}\right)^{2}}{2(1-f)},$$

## 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。