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# 数学代写|金融数学代写Financial Mathematics代考|MATH3090 Applications in Finance

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## 数学代写|金融数学代写Financial Mathematics代考|Applications in Finance

There is a vast literature on commonality in finance and the summary presented here is based on a few select studies. In addition to price and return variables of stocks, the commonality is studied on liquidity and turnover measures as well, which we define below
\begin{aligned} \mathrm{Liq}{d} &=-\log \left(1+\frac{\left|r{d}\right|}{P_{d}}\right) \ \mathrm{TO}{d} &=\log \left(1+\frac{v{d}}{\mathrm{NSH}}\right) . \end{aligned}
Here $r_{d}$ is the return on day ‘ $d$ ‘, $P_{d}$ is the price, $v_{d}$ is the volume traded and NSH is the number of outstanding shares at the beginning of the year. The turnover measure is adjusted by a moving average based on ‘ $K$ ‘ days past ‘ $d$ ‘. As outlined in Chordia, Roll and Subrahmanyam (2000) [79] the commonality in liquidity can come from several sources:

• Traders manage their inventory in response to market-wide price swings, thus variation in volume traded can lead to changes in liquidity, which are measured by bid-ask spread, effective spread, quoted depth, etc., besides (3.34).
• Volatility of the stock and presence of informed traders who may have better knowledge about the market-movements.
• Trading cost that includes price impact as a major component.
The commonality is simply inferred from the coefficients of the regression model
$$\mathrm{DL}{j, t}=\alpha{j}+\beta_{j}^{\prime} X_{t}+\epsilon_{j, t},$$
where $\mathrm{DL}$ is the percentage difference in a liquidity measure and where $X_{t}$ can include the corresponding market, industry variables. Generally the $\beta_{j}$ ‘s tend to be positive and a fair number of them are significant which are meant to indicate commonality. Some general observations from related studies are worth noting.
• Commonality in liquidity is high during periods of high market volatility and during high market-wide trading activity; volatility effect is asymmetric; the commonality is more pronounced when market performance is in decline.

## 数学代写|金融数学代写Financial Mathematics代考|Multivariate GARCH Models

It is well known that financial volatilities across assets and markets move together over time. It is of interest to study how a shock in one market increases the volatility on another market and in general, how correlations among asset returns change over time. For example, in the computation of hedge-ratio which is the slope of the regression line of spot returns on the futures return, both the covariance between spot and future returns and the variance of future returns can change over time. It must be noted that variances and covariances are generally sensitive to mean shifts. In addition, unlike modeling the mean behavior where a fewer number of quantities are involved, modeling the variance-covariance behavior involves a substantially larger number of quantities and hence various models that account for some redundancy are proposed in the literature. We cover only a select few in this section. Interested readers should refer to the excellent survey paper by Bauwens, Laurent and Rombouts (2006) [34].
The regression model (3.1), to begin with, can be written as
$$Y_{t}=\mu_{t}+\epsilon_{t},$$
where $\mu_{t}=C X_{t}$ and $\epsilon_{t} \sim N\left(0, \Sigma_{\epsilon \epsilon}\right)$. As $\Sigma_{\epsilon \epsilon}$ is taken to be positive definite, it can be written as $\Sigma_{\epsilon \epsilon}=H^{1 / 2} \cdot H^{1 / 2}$, where $H^{1 / 2}$ is the positive square root of the $\Sigma_{\epsilon \epsilon}$ matrix. Thus $\epsilon_{t}=H^{1 / 2} a_{t}$ where $a_{t} \sim N\left(0, I_{m}\right)$. When the covariance matrix, $\Sigma_{\epsilon \epsilon}$ changes over time, it can be characterized by
$$\epsilon_{t}=H_{t}^{1 / 2} a_{t},$$
where $H_{t}$ is the conditional variance-covariance matrix.

## 数学代写|金融数学代写FINANCIAL MATHEMATICS代 考|APPLICATIONS IN FINANCE

$$\operatorname{Liq} d=-\log \left(1+\frac{|r d|}{P_{d}}\right) \operatorname{TO} d=\log \left(1+\frac{v d}{\mathrm{NSH}}\right) .$$

79

$\$ \$$\ \$$

## 数学代写|金融数学代写FINANCIAL MATHEMATICS代 考|MULTIVARIATE GARCH MODELS

34

$$Y_{t}=\mu_{t}+\epsilon_{t},$$

$$\epsilon_{t}=H_{t}^{1 / 2} a_{t}$$

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