Multivariate Nonparametric Volatility Density Estimation

TL;DR

This paper introduces a new multivariate kernel density estimator for stochastic volatility models, using Fourier deconvolution on discretely observed data, with theoretical bias and variance analysis.

Contribution

It proposes a novel Fourier-type deconvolution kernel estimator for multivariate volatility density based on discrete observations in stochastic volatility models.

Findings

Bias expansion derived for the estimator

Variance bounds established

Method applicable to multivariate stochastic volatility data

Abstract

We consider a continuous-time stochastic volatility model. The model contains a stationary volatility process, the multivariate density of the finite dimensional distributions of which we aim to estimate. We assume that we observe the process at discrete instants in time. The sampling times will be equidistant with vanishing distance. A multivariate Fourier-type deconvolution kernel density estimator based on the logarithm of the squared processes is proposed to estimate the multivariate volatility density. An expansion of the bias and a bound on the variance are derived. Key words: stochastic volatility models, multivariate density estimation, kernel estimator, deconvolution, mixing