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# 数学代写|金融数学代写Financial Mathematics代考|MT4551 Extraneous Signals: Trading Volume, Volatility, etc.

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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

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

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