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# 金融代写|金融数学FINANCIAL MATHEMATICS代写|CRN29082 Time-Varying ARCH Processes (tvARCH)

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## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Time-Varying ARCH Processes (tvARCH)

The underlying assumption of ARCH models is stationarity and with the changing pace of economic conditions, the assumption of stationarity is not appropriate for modeling financial returns over long intervals. We may obtain a better fit by relaxing the assumption of stationarity in all the time series models. It is also appropriate for building models with local variations. Dahlhaus and Subba Rao (2006) [99] generalize the ARCH model with time-varying parameters:
$$\varepsilon_{t}=\sqrt{h_{t}} \cdot a_{t}, \quad h_{t}=w_{0}(t)+\sum_{j=1}^{\infty} w_{j}(t) \varepsilon_{t-j}^{2},$$
which $a_{t}$ ‘s are i.i.d. with mean zero and variance, one. By rescaling the parameters to unit intervals, the tvARCH process can be approximated by a stationary ARCH process. A broad class of models resulting from $(2.47)$ can be stated as:
$$\varepsilon_{t, N}=\sqrt{h_{t, N}} \cdot a_{t}, \quad h_{t, N}=w_{0}\left(\frac{t}{N}\right)+\sum_{j=1}^{p} w_{j}\left(\frac{t}{N}\right) \varepsilon_{t-j, N}^{2}$$
for $t=1,2, \ldots, N$. This model captures the slow decay of the sample autocorrelations in squared returns that is commonly observed in financial data which is attributed to the long memory of the underlying process. But $\operatorname{tvARCH}(p)$ is a non-stationary process that captures the property of long memory.

Fryzlewicz, Sapatinas and Subba Rao (2008) [152] propose a kernel normalizedleast squares estimator which is easy to compute and is shown to have good performance properties. Rewriting (2.48) as
$$\varepsilon_{t, N}^{2}=w_{0}\left(\frac{t}{N}\right)+\sum_{j=1}^{p} w_{j}\left(\frac{t}{N}\right) \varepsilon_{t-j, N}^{2}+\left(a_{t}^{2}-1\right) h_{t, N}^{2}$$
in the autoregressive form, the least squares criterion with the weight function, $k\left(u_{0}, \chi_{k-1, N}\right)$, where $\chi_{k-1, N}^{\prime}=\left(1, \varepsilon_{k-1, N}^{2}, \ldots, \varepsilon_{k-p, N}^{2}\right)$ is
$$L_{t_{0}, N}(\alpha)=\sum_{k=p+1}^{N} \frac{1}{b_{N}} w\left(\frac{t_{0}-k}{b_{N}}\right) \frac{\left(\varepsilon_{k \cdot N}^{2}-\alpha_{0}-\sum_{j=1}^{p} \alpha_{j} \varepsilon_{k-j \cdot N}^{2}\right)^{2}}{k\left(u_{0}, \chi_{k-1 \cdot N}\right)^{2}} .$$

## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Stylized Models for Variance of Asset Returns

Volatility has many important implications in finance. It is one of the most commonly used risk measures and plays a vital role in asset allocation. The estimate of volatility of an asset is obtained using the prices of stock or options or both. Three different measures that are normally studied are stated below:

• Volatility is the conditional standard deviation of daily or low frequency returns.
• Implied volatility is derived from options prices under some assumed relationship between the options and the underlying stock prices.
• Realized volatility is an estimate of daily volatility using high frequency intraday returns.

In this section, we will mainly focus on the first item and the others will be discussed in a later section on high frequency data.

Consider $r_{t}=\ln \left(P_{t}\right)-\ln \left(P_{t-1}\right)$, return of an asset. We observed that ‘ $r_{t}$ ‘ exhibits no serial correlation. This does not imply that the series ‘ $r_{t}$ ‘ consists of independent observations. The plot of $r_{t}^{2}$ given in Figure $2.4$ clearly indicates that volatility tends to cluster over certain time spans. The autocorrelations in Figure $2.4$ confirm that there is some time dependence for the volatility. In fact in some cases, there is some long range dependence that spans up to sixty days.

## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Ljung-Box Omnibus Test

• Ljung-Box Omnibus Test:
$$Q_{k}=T(T+2) \cdot \sum_{j=1}^{k} \frac{\left(\hat{e}^{j}\right)^{2}}{T-j} \sim \chi_{k-m}^{2},$$
where $\hat{e}^{(j)}$ is the $j$ th lag autocorrelation of $r_{t}^{2} ; m$ is the number of independent parameters. Here we test the null hypothesis, $H_{0}: \rho_{1}=\rho_{2}=\cdots \rho_{k}=0$.
• Lagrange Multiplier Test: Regress $r_{t}^{2}$ on $r_{t-1}^{2}, \ldots, r_{t-q}^{2}$ and obtain $R^{2}$ the coefficient of determination; Test the $H_{0}$ : Slope coefficients are all zero, by
$$T \cdot R^{2} \sim \chi_{q}^{2} .$$
These tests were carried out on the exchange rate data; Table $2.3$ has the result of the Ljung-Box Test:
Table 2.3: Ljung-Box Chi-Square Statistics for Exchange Rate Data
\begin{tabular}{ccccc}
$\mathrm{h}$ & 12 & 24 & 36 & 48 \
\hline$Q_{h}$ & $73.2$ & $225.7$ & $463.4$ & $575.7$ \
\hline df & 6 & 18 & 30 & 42 \
\hline$p$-value & $0.000$ & $0.000$ & $0.000$ & $0.000$
\end{tabular}

## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Time-Varying ARCH Processes (tvARCH)

$\mathrm{ARCH}$ 模型的基本假设是平稳性，并且随丰经汶状况的变化步伐，平稳性俷设不适用于长期模拟财 屴回报。我们可以通过放松所有时间序列模型中的平稳性假设来获得更好的拟合。它也适用于构建 具有局部变化的模型。Dahlhaus 和 Subba Rao (2006) [99] 用时悲参数概括了 ARCH 模型:
$$\varepsilon_{t}=\sqrt{h_{t}} \cdot a_{t}, \quad h_{t}=w_{0}(t)+\sum_{j=1}^{\infty} w_{j}(t) \varepsilon_{t-j}^{2},$$

$$\varepsilon_{t, N}=\sqrt{h_{t, N}} \cdot a_{t}, \quad h_{t, N}=w_{0}\left(\frac{t}{N}\right)+\sum_{j=1}^{p} w_{j}\left(\frac{t}{N}\right) \varepsilon_{t-j, N}^{2}$$

Fryzlewicz、Sapatinas 和 Subba Rao (2008) [152] 提出]一种易于计算的核归一化最小二乘估计 器，并被证明具有良好的性能特性。将 (2.48) 重写为
$$\varepsilon_{t, N}^{2}=w_{0}\left(\frac{t}{N}\right)+\sum_{j=1}^{p} w_{j}\left(\frac{t}{N}\right) \varepsilon_{t-j, N}^{2}+\left(a_{t}^{2}-1\right) h_{t, N}^{2}$$

$\chi_{k-1, N}^{\prime}=\left(1, \varepsilon_{k-1, N}^{2}, \ldots, \varepsilon_{k-p, N}^{2}\right)$ 是
$$L_{t_{0} N}(\alpha)=\sum_{k=p+1}^{N} \frac{1}{b_{N}} w\left(\frac{t_{0}-k}{b_{N}}\right) \frac{\left(\varepsilon_{k \cdot N}^{2}-\alpha_{0}-\sum_{j=1}^{p} \alpha_{j} \varepsilon_{k-j \cdot N}^{2}\right)^{2}}{k\left(u_{0}, \chi_{k-1 \cdot N}\right)^{2}} .$$
$\mathrm{~ 王 鬲 代 蜀 | 坓 鬲}$

## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Stylized Models for Variance of Asset Returns

$\mathrm{~ 波 动 性 对 金 融 有 许 多 重 要 的 影 响 。 它 是 最 常 用 的 风 崄 誦}$ 作用。使用股票或期权或两者的价格获得逄产波动率的估计。通常研究的三种不同措施如下所述:

## 金融代写|金融数学FINANCIAL MATHEMATICS代写|Ljung-Box Omnibus Test

Ljung-Box 宗合测跜:
$$Q_{k}=T(T+2) \cdot \sum_{j=1}^{k} \frac{\left(\hat{e}^{j}\right)^{2}}{T-j} \sim \chi_{k-m}^{2},$$

$H_{0}: \rho_{1}=\rho_{2}=\cdots \rho_{k}=0 .$

$$T \cdot R^{2} \sim \chi_{q}^{2}$$

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