The Annals of Statistics

A Fourier transform method for nonparametric estimation of multivariate volatility

Paul Malliavin and Maria Elvira Mancino

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Abstract

We provide a nonparametric method for the computation of instantaneous multivariate volatility for continuous semi-martingales, which is based on Fourier analysis. The co-volatility is reconstructed as a stochastic function of time by establishing a connection between the Fourier transform of the prices process and the Fourier transform of the co-volatility process. A nonparametric estimator is derived given a discrete unevenly spaced and asynchronously sampled observations of the asset price processes. The asymptotic properties of the random estimator are studied: namely, consistency in probability uniformly in time and convergence in law to a mixture of Gaussian distributions.

Article information

Source
Ann. Statist., Volume 37, Number 4 (2009), 1983-2010.

Dates
First available in Project Euclid: 18 June 2009

Permanent link to this document
https://projecteuclid.org/euclid.aos/1245332838

Digital Object Identifier
doi:10.1214/08-AOS633

Mathematical Reviews number (MathSciNet)
MR2533477

Zentralblatt MATH identifier
1168.62030

Subjects
Primary: 62G05: Estimation 62F12: Asymptotic properties of estimators 42A38: Fourier and Fourier-Stieltjes transforms and other transforms of Fourier type
Secondary: 60H10: Stochastic ordinary differential equations [See also 34F05] 62P20: Applications to economics [See also 91Bxx]

Keywords
Continuous semi-martingale instantaneous co-volatility nonparametric estimation Fourier transform high frequency data

Citation

Malliavin, Paul; Mancino, Maria Elvira. A Fourier transform method for nonparametric estimation of multivariate volatility. Ann. Statist. 37 (2009), no. 4, 1983--2010. doi:10.1214/08-AOS633. https://projecteuclid.org/euclid.aos/1245332838


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