### 统计代写|金融统计代写financial statistics代考|Realized measures of integrated volatility

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## 统计代写|金融统计代写financial statistics代考|Realized measures of integrated volatility

Volatility measures variation in the asset prices and thus can be regarded as an indicator of risk. Accurate volatility estimation is very important in both asset allocation and risk management. Since volatility is inherently unobservable, the first two types of parametric models developed to estimate the latent volatility were continuous time (e.g., stochastic volatility) and discrete time models (e.g., ARCH-GARCH models). However, these parametric models have been proven to be misspecified in capturing volatilities implied by option pricing and other financial return variables. With the availability of high-frequency data, a series of nonparametric models have been proposed to examine integrated volatility at intra-day level. Andersen et al. (2001) first introduce a nonparametric volatility measure, termed realized volatility by summing over intra-day squared returns. The authors showed that $R V$ is an error free estimator of integrated volatility in the absence of noise and jumps. When the sampling frequency of the data is relatively high, microstructure noise creates a bias in the volatility estimation procedure. Zhang et al. (2005, 2006) and Kalnina and Linton (2008) solve this problem with microstructure noise robust estimators based on subsampling with multiple time scales. Barndorff-Nielsen et al. $(2008,2011)$, on the other hand, use kernelbased estimators to account for the microstructure noise in finely sampled data. When estimating integrated volatility in the presence of jumps within the underlying price process, jump components should be separated from the quadratic variation. Barndorff-Nielsen et al. (2003) and BarndorffNielsen and Shephard (2004) provide asymptotically unbiased integrated volatility estimators, the $B P V \mathrm{~s}$ and $T P V \mathrm{~s}$, which are robust to the presence of jumps. Aït-Sahalia et al. (2009) propose a threshold method to identify and truncate jumps and further develop a consistent nonparametric jump robust estimator of the integrated volatility. Corsi et al. (2010) introduce threshold bipower variation (TBPV) by combining the concepts from BarndorffNielsen et al. (2003) and Mancini (2009). Jacod et al. (2014) estimate local volatility by using the empirical characteristic function of the return and then remove bias due to jump variation. When combining both jumps and microstructure noise in the price process, Fan and Wang (2007) propose a wavelet-based multiscale approach to estimate integrated volatility.

## 统计代写|金融统计代写financial statistics代考|Realized bipower variation

Realized volatility or $R V$ as developed in Andersen et al. (2001) is one of the first empirical measures that used high-frequency intra-day returns to compute daily return variability without having to explicitly model the intra-day data. The authors show that under suitable conditions $R V$ is an unbiased and highly efficient estimator of $Q V$ as in (4). By extension it can be shown that in the absence of jumps or when jumps populate the data infrequently, $R V$ converges in probability to $I V$ as $M \rightarrow \infty$. It should also be noted that $R V$ has been used widely as part of the HAR-RV forecasting models. Here
$$R V_{t, M}=\sum_{j=1}^{M-1}\left(X_{t+(j+1) / M}-X_{t+j / M}\right)^{2}$$

In Barndorff-Nielsen and Shephard (2004), the authors demonstrate that they could untangle the continuous component of quadratic variation from its discontinuous component (jumps). This led them to develop $B P V$, one of the first asymptotically unbiased estimators of $I V$ which was robust to the presence of price jumps. It takes the following form
$$B P V_{t, M}=\left(\mu_{1}\right)^{-2} \sum_{j=2}^{M-1}\left|\Delta_{j} X | \Delta_{j-1} X\right|$$
where $\Delta_{H} X$ is the same as in $(10)$ and $\mu_{1}=2^{\frac{1}{2}} \frac{\Gamma(1)}{\Gamma\left(\frac{1}{2}\right)}$.

The $B P V$ does not allow the consistency of the $I V$ estimate to be impacted by finite activity jumps. However, it is subject to finite sample jump distortions or upward bias. To counter this problem, $B P V$ is generalized to $T P V$ in BarndorffNielsen and Shephard (2004), by utilizing products of the (lower order) power of three adjacent intra-day returns. Theoretically speaking, although $T P V$ is more efficient, it is also more vulnerable to microstructure noise of the highfrequency return data compared to $B P V . T P V$ can be given as
$$T P V_{t, M}=\left(\mu_{\frac{2}{3}}\right)^{-3} \sum_{j=3}^{M-1}\left|\Delta_{j} X\right|^{2 / 3}\left|\Delta_{j-1} X\right|^{2 / 3}\left|\Delta_{j-2} X\right|^{2 / 3}$$
where $\Delta_{j} X$ is the same as in $(10)$ and $\mu_{\frac{2}{3}}=2^{\frac{1}{3}} \frac{\Gamma\left(\frac{5}{6}\right)}{\Gamma\left(\frac{1}{2}\right)}$.

## 统计代写|金融统计代写financial statistics代考|Two-scale realized volatility

It is found that when the sampling interval of the asset prices is small, microstructure noise issues become more prominent and $R V$ ceases to function as a robust volatility estimator. Due to the bias introduced by the market microstructure noise in the finely sampled data, initially longer time horizons are preferred by econometricians. It is found that ignoring microstructure noise works well for intervals more than $10 \mathrm{~min}$. However, sampling over lower frequencies does not quantify and correct the noise effect on volatility estimation. As a solution, two-scale realized volatility $(T S R V$ ) is introduced in Zhang et al. (2005) by combining estimators obtained over two time scales, $a v g$ and $M$. It forms an unbiased and consistent, microstructure noise robust estimator of $I V$ in the absence of jumps. It takes the following form
$$T S R V_{t, M}=[X, X]^{\alpha v g}-\frac{1}{K}[X, X]^{M}$$
where
$$\begin{gathered} {[X, X]^{m_{i}}=\sum_{j=1}^{m_{i}-1}\left(X_{t+((j+1) K+i) / M}-X_{t+(j K+i) / M}\right)^{2}, i=1, \ldots, K \text { and } m_{i}=\frac{M}{K}} \ {[X, X]^{a v g}=\frac{1}{K} \sum_{i=1}^{K}[X, X]^{m_{i}}} \ {[X, X]^{M}=\sum_{j=1}^{M-1}\left(X_{t+(j+1) / M}-X_{t+j / M}\right)^{2}} \end{gathered}$$ $K=c M^{2 / 3}$ is the number of subsamples, $\frac{M}{K}$ is subsample size, $c>0$ is a constant, and $M$ is the number of equi-spaced intra-daily observations.

## 统计代写|金融统计代写financial statistics代考|Setup

Ĵ吨=∑s≤吨Δ是s

Ĵ吨=∑j=1ñ吨Xj

[是,是]吨d=∑j=1ñ吨Xj2

X=是+ε

X吨+j/米=是吨+j/米+ε吨+j/米,吨=0,…,吨 和 j=1,…,米

ΔjX=X吨+(j+1)/米−X吨+j/米

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