Multivariate Time-Changed Brownian Motion: The Expectation–Maximization Estimation Method

Abstract

The main topics covered in this chapter are:a description of a method to estimate the parameters of models based on the multivariate time-changed Brownian motion;a review of the expectation–maximization (EM) maximum likelihood estimation (MLE) method to estimate the parameters of multivariate generalized hyperbolic distributions;an extension of the EM-based MLE algorithm to normal mean–variance mixture distributions in which only the characteristic function of the mixing distribution is known in closed form, while the density function is not;an error analysis of the estimation method applied to the multivariate normal tempered stable case;an empirical test showing the model performance on a five-and a 30-dimensional series of index (stock) returns;how to evaluate well-known risk measures (i.e., value-at-risk and average value-at-risk) under this framework;how to backtest the value-at-risk by taking into account the number of exceedances.