A Fourier transform method for nonparametric estimation of multivariate volatility

TL;DR

This paper introduces a Fourier transform-based nonparametric method to estimate instantaneous multivariate volatility from asynchronously sampled asset prices, providing consistent and asymptotically normal estimators.

Contribution

It develops a novel Fourier analysis approach for nonparametric estimation of multivariate volatility using irregularly spaced data, linking Fourier transforms of prices and volatility.

Findings

Estimator is consistent in probability uniformly in time

Estimator converges in law to a mixture of Gaussian distributions

Method effectively handles asynchronous sampling

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.