Estimation of the quadratic variation of log prices based on the Itô semi-martingale

Erlin Guo; Patrick Ling; Erlin Guo; Patrick Ling

doi:10.3934/era.2024038

Electronic Research Archive

2024, Volume 32, Issue 2: 799-811. doi: 10.3934/era.2024038

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Research article

Estimation of the quadratic variation of log prices based on the Itô semi-martingale

Erlin Guo ^{1
,
,},
Patrick Ling ²

1.
School of Mathematics and Statistics, Xuzhou University of Technology, Xuzhou 221018, China
2.
Department of Mathematics, Utah Valley University, Orem, USA

Received: 02 October 2023 Revised: 23 December 2023 Accepted: 03 January 2024 Published: 12 January 2024

As the availability of high-frequency data becomes more widespread, it has become very popular to model random fluctuations of some econometric variables over time using Itô semi-martingale. An emblematic problem is to estimate the quadratic variation, i.e., the integrated volatility of log prices, using noisy high frequency data with endogenous time and jumps. We propose a methodology that combines the multiple sub-grids and thresholds. First, the sub-sample is used to reduce the effect of the noise. Then, the threshold method is used to get rid of the effect of jumps. Finally, the multiple sub-grids method is used to increase the convergence rate. The asymptotic properties, such as consistency and asymptotic normality, are investigated. Simulation is also included to illustrate the performance of the proposed procedure.
- multiple sub-grids,
- threshold,
- endogenous,
- jumps,
- semi-martingale
Citation: Erlin Guo, Patrick Ling. Estimation of the quadratic variation of log prices based on the Itô semi-martingale[J]. Electronic Research Archive, 2024, 32(2): 799-811. doi: 10.3934/era.2024038

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Abstract

As the availability of high-frequency data becomes more widespread, it has become very popular to model random fluctuations of some econometric variables over time using Itô semi-martingale. An emblematic problem is to estimate the quadratic variation, i.e., the integrated volatility of log prices, using noisy high frequency data with endogenous time and jumps. We propose a methodology that combines the multiple sub-grids and thresholds. First, the sub-sample is used to reduce the effect of the noise. Then, the threshold method is used to get rid of the effect of jumps. Finally, the multiple sub-grids method is used to increase the convergence rate. The asymptotic properties, such as consistency and asymptotic normality, are investigated. Simulation is also included to illustrate the performance of the proposed procedure.

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