Citation: Liang, Z.; Weng, F.; Ma, Y.;
Xu, Y.; Zhu, M.; Yang, C.
Measurement and Analysis of High
Frequency Assert Volatility Based on
Functional Data Analysis.
Mathematics 2022, 10, 1140. https://
doi.org/10.3390/math10071140
Academic Editors: Sławomir
Nowaczyk, Rita P. Ribeiro and
Grzegorz Nalepa
Received: 1 March 2022
Accepted: 31 March 2022
Published: 1 April 2022
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Article
Measurement and Analysis of High Frequency Assert Volatility
Based on Functional Data Analysis
Zhenjie Liang
1,†
, Futian Weng
2,3,4,†
, Yuanting Ma
5
, Yan Xu
6,7
, Miao Zhu
8,
* and Cai Yang
9,
*
1
School of Economics, Xiamen University, Xiamen 361005, China; liangzhenjie@stu.xmu.edu.cn
2
School of Medicine, Xiamen University, Xiamen 361005, China; wengfutian@stu.xmu.edu.cn
3
National Institute for Data Science in Health and Medicine, Xiamen University, Xiamen 361005, China
4
Data Mining Research Center, Xiamen University, Xiamen 361005, China
5
School of Economics and Management, East China Jiaotong University, Nanchang 330013, China;
yuanting_ma@scuec.edu.cn
6
School of Mathematical Sciences, Ocean University of China, Qingdao 266100, China; yan_xu@dufe.edu.cn
7
National Economic Engineering Laboratory, Dongbei University of Finance and Economics,
Dalian 116025, China
8
School of Statistics, Huaqiao University, Xiamen 361005, China
9
School of Business Administration, Hunan University, Changsha 410082, China
* Correspondence: s104064881@m104.nthu.edu.tw (M.Z.); yangcaier@hnu.edu.cn (C.Y.)
† These authors contributed equally to this work.
Abstract:
Information and communication technology have enabled the collection of high-frequency
financial asset time series data. However, the high spatial and temporal resolution nature of these
data makes it challenging to compare financial asset characteristics patterns and identify the risk.
To address this challenge, a method for realized volatility calculation based on the functional data
analysis (FDA) method is proposed. A time–price functional curve is constructed by the functional
data analysis method to calculate the realized volatility as the curvature integral of the time–price
functional curve. This method could effectively eliminate the interference of market microstructure
noise, which could not only allow capital asset price to be decomposed into a continuous term and
a noise term by asymptotic convergence, but also could decouple the noise from the discrete-time
series. Additionally, it could obtain the value of volatility at any given time, which is no concern
about correlations between repeated, mixed frequencies and unequal intervals sampling problems
and relaxes the structural constraints and distribution setting of data acquisition. To demonstrate
our methods, we analyze a per-second level financial asset dataset. Additionally, sensitivity analysis
on the selection of the no equally spaced sample is conducted, and we further add noise to ensure
the robustness of our methods and discuss their implications in practice, especially being conducive
to more micro analysis of the volatility of the financial market and understanding the rapidly
changing changes.
Keywords: functional data analysis; high frequency data; Bernstein basis function; curvature
MSC: 91G70
1. Introduction
In recent years, with the rapid and convenient acquisition of high-frequency data
of asset returns, scholars and investors have given an increasing amount of attention
to volatility modeling [
1
,
2
]. ARCH class model, SV class model, and realized volatility
class model are used to calculate volatility [
3
–
5
]. However, these three class models
characterize volatility indirectly through rate of return and cannot depict dynamic changes
at the intraday level. In particular, modeling intraday volatility usually uses continuous
time stochastic process methods [
6
–
8
], which assume that volatility is generated by a
potentially unknown diffusion process. However, these methods are unable to describe
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