A Fourier transform method for nonparametric estimation of multivariate volatility

A Fourier transform method for nonparametric estimation of multivariate volatility

Paul Malliavin and Maria Elvira Mancino

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

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.

Primary Subjects: 62G05, 62F12, 42A38

Secondary Subjects: 60H10, 62P20

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

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Links and Identifiers

Permanent link to this document: http://projecteuclid.org/euclid.aos/1245332838
Digital Object Identifier: doi:10.1214/08-AOS633
Zentralblatt MATH identifier: 05582016
Mathematical Reviews number (MathSciNet): MR2533477

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