Convergence rate for integrated self-weighted volatility by using intraday high-frequency data with noise

1.   Introduction

1.1.   Related studies

With the rapid development of economics, finance and electronic technology, high-frequency financial data are becoming increasingly available and need to be analyzed. In the USA, one-second records are available from the Trade and Quote (TAQ) database, whereas in China, the tick-by-tick stock transactions can be obtained from the private databases of some fund management companies. Such data are widely used and gamer much attention, particularly for applications related to the market prices of stocks, currencies, risk analysis, portfolio management and other financial instruments ^[1,2,3,4]. However, it is still a big challenge for data analysts to use such data in finance. Most of the studies on using high-frequency data, by far, have focused on the estimation of volatility, which is a key aspect of risk analysis, portfolio management and so on ^[5,6,7,8].

Theoretically, if the latent log price $X = (X_t)$ follows an Itô process

where $b$ and $\sigma$ are locally bounded optional processes and $W$ is a standard Brownian motion. Then, the integrated volatility is the quadratic variation of $X$, that is, $< X, X > _t = \int_0^t\sigma_s^2ds$.

The most commonly used estimator of the integrated volatility is the realized volatility, which is based on discrete-time observations ^[1,9]. Realized volatility uses intraday high-frequency data to directly measure the volatility in a general semi-martingale model setting. On the one hand, a few works in the literature claim that the diffusion process is unwise. These scholars think that the release of significant "news" in an efficient market will induce discontinuities or jumps in the price process. In this case, the realized volatility will not work as the jumps will be included in the limit. To deal with the jumps, two widely used methods are available, i.e., the bipower method ^[10] and the threshold estimator ^[11,12]. On the other hand, when it comes to the practical side, observed high-frequency financial data are often contaminated by market microstructure noise. Microstructure noise is induced by various frictions in the trading process, such as the asymmetric information of traders, the discreteness of price change, bid-ask spreads and/or rounding errors. Empirical evidence has shown a visible noise accumulation effect at high frequencies, and there has been a number of studies on estimating quantities of interest with prices observed to have microstructure noise. These include the following: the two-time scale method ^[13,14]; the multi-scale method ^[15]; the kernel method ^[16]; the pre-averaging method ^[17]; the quasi-maximum likelihood method ^[18,19]; auto-covariance and auto-correlation cases including irregular observation times ^[20]; dependent noise with irregular tick observation times ^[21]; multiple transaction cases with noise ^[22,23,24]. Based on the likelihood of an MA (Moving Average) model, Da and Xiu ^[25] proposed a simple volatility estimator that assumes that the observed transaction price follows a continuous-time contaminated Itô semi-martingale.

Owing to the rapid development of computer science technology, estimating the integrated volatility matrix of a large number of assets is a challenging problem. The ARVM estimator is proposed based on the use of contaminated high-frequency data under the sparse integrated volatility matrix ^[26]. The convergence rate of a large volatility matrix estimator using contaminated high-frequency data is given separately in ^[27] and ^[28]. By applying principal component analysis to the spatial Kendall's tau matrix instead of the sample covariance matrix, a high-dimensional factor analysis without moment constraints is considered ^[29]. Since the integrated volatility matrix often has entries with a wide range of variability, an adaptive thresholding estimator and the convergence rate of the estimator are shown under sparse conditions ^[30]. A non-parametric estimator for a single asset is proposed in^[31], and a threshold estimator for multiple assets is considered in ^[32] for integrated volatility in the presence of both jumps and microstructure noise. Regarding the integrated volatility matrix of a large number of assets, a thresholding estimator has been constructed in the presence of both jumps and microstructure noise, as presented in ^[33].

Here, based on the observed intraday high-frequency data with microstructure noise, we perform statistical inference on the integrated self-weighted volatility, given as

where $X_t$ is an Itô process, $f: \mathbb{R}^2\to\mathbb{R}$ is some real Lipschitz function and $\sigma_t^2$ is the spot volatility of $X$. The study of integrated self-weighted volatility $ISV$ is important for both pricing and hedging purposes in financial econometrics. In the current paper, we focus on the latent price asset as a continuous Itô process; the estimation of the jump-diffusion process will be dealt with in future work.

1.2.   Contributions and organization

The integrated self-weighted volatility $ISV = \int_0^1f(X_t)\sigma_t^2dt$ is a generalization of the integrated volatility $IV = \int_0^1\sigma_t^2dt$. The purpose of introducing a general function $f$ is as follows. Theoretically or practically, to make the processes of setting the interest rate or financial asset pricing more realistic or extend the range of the price from $(0, \infty)$ to the whole real line, we transform the interest rate or price process and then analyze the transformed process. Specifically, after taking the logarithm, the price process can yield negative values ^[34,35]. For example, the convergence rates of a large volatility matrix estimator in the presence of jumps, microstructure noise and asynchronization are given in ^[33]. Leveraging a variety of factor models, a pre-averaging-based large covariance matrix estimator using high-frequency transaction prices has been constructed ^[36]. Suppose that one is interested in the integrated volatility of the transformed price processes of ${X_t}$. On one hand, the microstructure noise associated with the log price process becomes different when one uses the observed log price and the latent log price, i.e., $\log{\left(X(t^n_i)+\varepsilon(t^n_i)\right)}-\log{\left(X(t^n_i)\right)}$. Unfortunately, the microstructure noise associated with the log price process is not centered at zero conditional on $X$. But this is a necessary condition for consistency of the estimator of integrated volatility for the log price process in many studies ^[37,38,39]. Note that the integrated volatility of $\log{X_t}$ is of the form $\int^1_0f(X_t)\sigma^2_tdt$. On the other hand, the self-weighted estimator can potentially balance the correlation between trading price and volume. For these reasons, we chose to estimate $ISV$ with $f(x) = 1/x^2$. So developing inference schemes for $ISV$ provides a more flexible tool.

The rest of the paper is structured as follows. In Section 2, some assumptions of models are given, and the main results, i.e., the law of large numbers and central limit theory, are presented in Section 3. All of the proofs of the theorems are shown in Section 4. Section 5 provides several experimental evaluations and an application to demonstrate the effectiveness of the proposed method. Some conclusions and future research are proposed in Section 6.

2.   Assumptions of models

Let $X = (X_t)$ be the true log price at time $t$, and let it satisfy (1.1). To state our main results, we use the following assumptions.

Assumption 1. Processes $b$ and $\sigma$ are adapted to $\{\mathcal{F}_t\}$, i.e., the natural filtration generated by $X$; moreover, process $b$ is locally bounded; process $\sigma$ is a càdlàg; for any $t > s > 0$, $W_t-W_s$ is independent of $\mathcal{F}_s$.

Assumption 2. Assume that for some constant $L$, $\kappa > 0$ and $\alpha\in(0, 1]$, $f(x)$ satisfies that

for any $x, y\in\mathbb{R}$.

However, when it comes to the practical side, observed high-frequency financial data are often contaminated by market microstructure noise. We cannot directly observe $X_t$; instead, we observe $Y_t$, where

Let us make the following assumptions about $\varepsilon$.

Assumption 3. The microstructure noise $\varepsilon$ is an $i.i.d.$ process given X with

for any $p > 0$.

To establish the central limit theorem, a standard structural assumption on the volatility process $\sigma$ is needed.

Assumption 4. The volatility functions $\{\sigma_t, t\geq 0\}$ satisfy the equation

where $\tilde{b}$ and $\tilde{\sigma}$ are adapted càdlàg processes with $\tilde{b}$ being predictable and locally bounded, and $\tilde{W}$ is a standard Brownian motion.

3.   Results

Before displaying the main results, we briefly introduce the estimator of $ISV$. Let us divide the interval $[0, 1]$ into $m$ equal sub-intervals, set $K = \lfloor\frac{n}{m}\rfloor$ and denote

In order to eliminate the effect of microstructure noise, we select $l$ such that $l\rightarrow \infty$ and $\frac{l}{n}\rightarrow 0$ as $n\rightarrow \infty$. For each $i\geq l$, we denote

and then use the following $\hat{ISV}_n$ to estimate $ISV$:

where $\hat{\eta^2} = \frac{1}{2n}\sum_{r = 1}^m\sum_{k = 1}^K(Y_{\tau_{r}^k}-Y_{\tau_{r}^{k-1}})^2$ is the variance estimator of microstructure noise which will be used to reduce the impact of microstructure noise on integrated self-weighted volatility and yield a better estimator.

Theorem 1. Under Assumptions 1–3, and given

the sequence of random variables $(\frac{1}{m^{\alpha/2}}+\frac{1}{l^{\alpha/2}}+\frac{l^{\max(\frac{\alpha}{2}, \frac{1-\alpha}{2})}}{n^{\alpha/2}})^{-1}(\hat{ISV}_n-ISV)$ is tight.

To deduce the central limit theorem, the concept of stable convergence is also needed. Let us state it briefly. A sequence of random variables $X_n$ converges stably in law to a random variable X defined on the appropriate extension of the original probability space, written as $X_n\overset{\mathcal{S}}\to X$, if and only if for any set $A\in\mathcal{F}_1$ and real number $x$, we have

Hence, stable convergence is slightly stronger than convergence in law.

Theorem 2. Assume that Assumptions 1–4 hold, and $\frac{\sqrt{m}}{l^{\alpha/2}}\rightarrow 0$ and $\frac{\sqrt{m}l^{\max\{\frac{\alpha}{2}, \frac{1-\alpha}{2}\}}}{n^{\alpha/2}}\rightarrow 0$. Then, we have

where $B$ is a standard Brownian motion defined on an extension of the original space, and it is independent of $\mathcal{F}$.

Remark 1. In Theorem 1, it is worthy of notice that the upper bound is no more than $\frac{1}{m^{\alpha/2}}$, where $\alpha\in(0, 1]$. However, in Theorem 2, the upper bound is no more than $\frac{1}{m^{1/2}}$, which is faster convergence rate than $\frac{1}{m^{\alpha/2}}$. The difference between Theorems 1 and 2 is derived from the perspectives of two aspects:

(1). The estimator we studied in Theorem 2 is derived from the "pre-averaging" technique which essentially tries to "clean" the contaminated data by smoothing first and then applying the usual statistical procedures, while the estimator of Theorem 1 is derived from the two-time-scale technique which applies the usual statistical procedure to the raw contaminated data first, and then corrects the bias caused by the microstructure noise.

(2). An extra bias appears due to the presence of $f$ in the derivation of our asymptotic normality. This is because we need to open an appropriate window with length $l$ to the left of $\tau_{r-1}^k$ in order to smooth away the noise in the contaminated data.

To perform statistical inference for the suggested estimator, the asymptotic conditional variance $\int_0^1 f^2(X_t)\sigma_t^4dt$ needs to be estimated. Using Theorem 1, we can give an estimator for the asymptotic conditional variance:

Remark 2. The authors of ^[13] have shown that

Thus, we obtain the consistent estimators as follows:

From these equations, and by using a procedure similar to those for Theorems 1 and 2, we obtain a consistent estimator of $\int_0^1f^2(X_s)\sigma_s^4ds$:

Consequently, a studentized version of the central limit theorem is as follows.

Theorem 3. Assume that the conditions in Theorem 2 are satisfied; then, we have

where $\mathcal{N}(0, 1)$ is a standard normal variable independent of $\mathcal{F}$.

4.   Proofs of theorems

Because of the local boundedness of $b$ and $\sigma$, we can replace the local boundedness assumptions with boundedness through a standard localization course. As a consequence, process $Y$ has uniformly bounded moments up to any powers on the interval [0, 1], i.e., for some constant $C > 0$,

Lemma 1. For any $\beta > 0$, we have

Proof. (1) By the definitions of $\overline{Y}_{\tau_{r-1}^k}$ and $\overline{X}_{\tau_{r-1}^k}$, we have, for some given r,

The last inequality follows from Assumption 3.

(2) We write

(3) Following from Burkholder-David-Gundy's inequality and the boundedness of $b$ and $\sigma$, we can get (4.4) and (4.5). □

Lemma 2. For any $\beta > 0$, we have

Proof. Using the Lipschitz property of $f$, one has

Similar to Lemma 1, we can also get

Denote that

□

Lemma 3.

Proof. (1) For each $k = 1, \cdots, K$, we obtain

Thus, by Itô isometry, we have

where $C\frac{1}{m}$ is derived from the drift term.

Using the Lipschitz property of $f$ and Hölder's inequality, we have

(2) By applying Hölder's inequality, we have

(3) By applying Hölder's inequality, we have

(4) By ^[13], it has been proved that $\frac{1}{n}\sum_{r = 1}^{m-1}\sum_{k = 1}^{K}(Y_{\tau_{r}^{k+1}}-Y_{\tau_r^k})^2$ is the variance estimator of microstructure noise. So we use $\frac{m}{n}\sum_{r = 1}^{m-1}\sum_{k = 1}^{K}(Y_{\tau_{r}^{k+1}}-Y_{\tau_r^k})^2f(\overline{Y}_{\tau_r^k})$ to reduce the impact of microstructure noise on integrated self-weighted volatility; this yields (4.11).

(5) Since $\varepsilon(\cdot)$ is mutually independent given $\sigma(X)$, we have

It follows from Hölder's inequality that

Moreover, because $f$ satisfies the $\alpha-$Lipschitz Assumption 2, there exists a finite constant $C$ such that $|f(x)|\leq C(1+|x|^{2\kappa+\alpha})$ for all $x\in \mathbb{R}$. This fact together with the uniform boundedness of any moments of random variables $X(t)$ on $[0, 1]$ yields

Then, $E(S_5)^2 \leq C\frac{1}{K}$. Hence,

□

Proof of Theorem 1. We write

From Lemma 3, we can get

Because $\frac{m}{n} = o(\frac{1}{\sqrt{K}})$ and $\frac{1}{\sqrt{K}} = o(\frac{1}{K^{\alpha/2}})$, we always have that

which finishes the proof of Theorem 1. □

Next, let us denote that

where $U_{2r} = \frac{1}{K}\sum_{k = 1}^K (X_{\tau_{2r}^k}-X_{\tau_{2r-1}^k})^2f(X_{\tau_{2r-1}^1})$ and $U_{2r+1} = \frac{1}{K}\sum_{k = 1}^K (X_{\tau_{2r+1}^k}-X_{\tau_{2r}^k})^2f(X_{\tau_{2r}^1})$. □

Proposition 1.

Proof. Let $\sigma(X_{\tau_{2r+1}^1})$ and $\sigma(X_{\tau_{2r+2}^1})$ denote the natural filtration respectively generated by $X_{\tau_{2r+1}^1}$ and $X_{\tau_{2r+2}^1}$; then, $U_{2r}$ is $\sigma(X_{\tau_{2r+1}^1})$-adapted and $U_{2r+1}$ is $\sigma(X_{\tau_{2r+2}^1})$-adapted. By the product formula and the definition of $\sigma$, for any given $s < t$, we have

Hence,

with $|R_{1m}|\leq \sum_{r = 1}^{[m/2]}|f(X_{\tau_{2r-1}^1})|C(\frac{1}{m})^2$, i.e.,

Similarly, we have

with $|R_{2m}|\leq \sum_{r = 1}^{[m/2]}|f(X_{\tau_{2r}^1})|C(\frac{1}{m})^2$, and

Therefore,

□

Proposition 2.

Proof. Now, let us study the focus of this paper, i.e., $L_2$.

Up to a smaller order,

Let $L_{21} = (U_{2r}-E[U_{2r}|\sigma(X_{\tau_{2r}^1})])$, $L_{22} = (U_{2r+1}-E[U_{2r+1}|\sigma(X_{\tau_{2r}^1})])$ and $L_{23} = (E[U_{2(r+1)}|\sigma(X_{\tau_{2r+2}^1})]-E[U_{2(r+1)}|\sigma(X_{\tau_{2r+2}^1})])$. Then,

Now the summands in the sums of $L_{21}$, $L_{22}$ and $L_{23}$ are sequences of martingale differences with respect to $\sigma(X_{\tau_{2r+2}^1})$. To us, the remaining work is to find the limit of the conditional variance of $\tilde{L}_2$, i.e., the limit of $\sum_{r = 1}^{[m/2]}E[(L_{21}+L_{22}+L_{23})^2|\sigma(X_{\tau_{2r}^1})]$.

(Ⅰ) By the definition of conditional expectation, we have

So,

where,

Using the Itô product formula, we get

Meanwhile, for $k < j$,

then, one gets

Thus, we obtain that

(Ⅱ) Denote $L_{22}^k = (X_{\tau_{2r+1}^k}-X_{\tau_{2r}^k})^2-E[(X_{\tau_{2r+1}^k}-X_{\tau_{2r}^k})^2|\sigma(X_{\tau_{2r}^1})]$; then,

Now, for $k < j$, one obtains

thus,

(Ⅲ) Let

then,

By the property of the function $f$, we have

Now

so, we have

(Ⅳ) By $L_{21}^k$ and $L_{22}^k$, we have that

Given

and

we can obtain

(Ⅴ) By applying the following equation:

we can get that

(Ⅵ) Similar to (4.21), we have

Combining (4.17)–(4.22), we have

□

Proof of Theorem 2. Let

By Theorem 1, we have

So it is enough to prove the result for $\hat{I}_n$. Now, we have

Thus, in view of Proposition 1, we have that

Combining Propositions 1 and 2, we have

□

Proof of Theorem 3. It suffices to prove the consistency of $\hat{C}_n$ with $\int_0^1 f^2(X_t)\sigma_t^4dt$, which is implied by the proof of Theorem 2. □

5.   Simulations and applications

5.1.   A simulation study

In order to evaluate the performance of the estimator, we consider three sample sizes: $n = 4680$, $7800$ and $23400$ (which correspond to sampling every 5 seconds, 3 seconds and 1 second) within one trading day ($T = 1$), the latent value is drawn from the geometric Brownian process with drift, namely,

where $W_{s}$ is a standard Brownian motion. The microstructure noise $\varepsilon_i$'s are independent and identically distributed, i.e., $N(0, \omega)$. The variance of microstructure noise was chosen to match the size of the integrated self-weighted volatility. Let $t_i$ be equally spaced in [0, 1] with $t_i-t_{i-1} = \frac{1}{n}$, $m = c_m\lfloor n^{\frac{1}{3}}\rfloor$ and $l = \lfloor n^{\frac{1}{3}}\rfloor$ as suggested in Remark 2.

We constructed the observed data according to the definition of $X(t_i)$. The same procedure was repeated 1000 times, and our results, in the form of relative biases, standard errors and mean square errors, are displayed in Table 1. Through the different choices of grid point $c_m\in [2,4]$ with a fixed step length of 0.2, we calculated the sensitivity of the coverage probabilities(CPs) and confidence intervals(CIs) in Tables 2 and 3. Additionally, the corresponding histogram and QQ-plot are displayed in Figure 1, which verify our central limit theorem. We make the following observations from the simulation results.

(1) In all cases of Table 1, as $n$ increases, all of the biases, standard errors and mean square errors of the estimator tend to decrease; moreover, the estimator is robust against different levels of applied theoretical volatility and variance of microstructure noise. This is in line with our theoretical results.

(2) Tables 2 and 3 show the CPs and CIs at $95\%$ and $90\%$ levels, respectively. Both cases reveal that the CPs and CIs are not sensitive to the choice of $c_m$.

(3) We demonstrate the central limit theorem in Figure 1 by displaying the histogram and QQ-plot for n = 23400, $\sigma^2 = 0.02$, $c_m = 3$.

5.2.   An application

In order to investigate the influence of the estimation on the integrated self-weighted volatility in high-frequency data, we collected the intraday transaction prices of 10 stocks from the TAQ database from December 1, 2020 to December 30, 2020. The numbers of observations of the original tick-by-tick data sets for GOOGLE, DELL, JBGS, AAPL, BDN, CACC, EBAY, FF, HMDT and ICE, respectively, were 23,853, 24,478, 20,696, 52,644, 22,701, 23,895, 25,338, 20,265, 18,736 and 24,306. The one-second data sets were constructed by setting the closing price of every one-second interval as the price for the corresponding second. Hence, for each stock, we have approximately second-by-second observations. We constructed an equally spaced data set, because regular spacing is assumed in Theorems 1 and 2. We adopted $c_m = 2$, $m = 60$ and $l = 30$ in the empirical data. Because there is no real integrated volatility for comparison, we calculated the standard deviation of the integrated self-weighted volatility estimators for different periods for 10 stocks in the empirical study. The results are displayed in Table 4.

6.   Conclusions

In this paper, we have proposed a consistent estimator of the integrated self-weighted volatility and demonstrated the central limit theorem of the estimator based on the high-frequency data in the presence of microstructure noise. A studentized version of the proposed estimator has been given. The estimator can potentially be applied to a general semi-martingale and the results can be employed to deal with the statistical inference of volatility.

In our future work, a jump-diffusion model will be considered, where we estimate the integrated self-weighted volatility or integrated self-weighted cross-volatility. Furthermore, we will extend the work to include estimation for multiple-transaction cases and time-endogenous cases under the conditions of more generalized settings than described in this paper.

Use of AI tools declaration

The authors declare they have not used artificial intelligence tools in the creation of this article.

Acknowledgments

This paper was supported by Jiangsu Province (BY2022768), the Qing Lan Project of Jiangsu Province, and NSF China (12071408, 12201235).

Conflict of interest

The authors declare that there are no conflicts of interest regarding the publication of this paper.