Asymptotic results for the Fourier estimator of the integrated quarticity

  • Giulia LivieriEmail author
  • Maria Elvira Mancino
  • Stefano Marmi


In this paper, we prove a central limit theorem for an estimator of the integrated quarticity based on Fourier analysis, strictly related to the one proposed in Mancino and Sanfelici (Quant Finance 12: 607–622, 2012). Also, a consistency result is derived. We show that the estimator reaches the parametric rate \(\rho (n)^{1/2}\), where \(\rho (n)\) is the discretization mesh and n the number of points of such discretization. The optimal variance is obtained, with a suitable choice of the number of frequencies employed to compute the Fourier coefficients of the volatility, while the limiting distribution has a bias. As a by-product, thanks to the Fourier methodology, we obtain consistent estimators of any even power of the volatility function as well as an estimator of the spot quarticity. We assess the finite-sample performance of the Fourier quarticity estimator in a numerical simulation.


(Powers of) volatility estimation Quarticity Central limit theorem Fourier analysis High-frequency data. 

JEL Classification

C13 C14 C58 G10 



We thank two anonymous Referees for their helpful review and recommendations and suggestions. We are indebted to Aleksey Kolokolov for providing the MATLAB code relative to the Efficient Multipower estimator. We are also grateful to the participants of the Quantitative Finance Workshop 2019 hosted by ETH in Zürich and of the Second Italian Meeting on Probability and Mathematical Statistics 2019.


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Copyright information

© Associazione per la Matematica Applicata alle Scienze Economiche e Sociali (AMASES) 2019

Authors and Affiliations

  • Giulia Livieri
    • 1
    Email author
  • Maria Elvira Mancino
    • 2
  • Stefano Marmi
    • 1
  1. 1.Scuola Normale SuperiorePisaItaly
  2. 2.Department Economics and ManagementUniversity of FirenzeFirenzeItaly

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