Multivariate stochastic volatility (MSV) models are nonlinear state space models that require either linear approximations or computationally demanding methods for handling the high dimensional integrals arising in the estimation problems of the latent volatilities and model parameters. Markov Chain Monte Carlo (MCMC) methods, which are based on Monte Carlo simulations using special sampling schemes, are by far the most studied method with several extensions and versions in previous stochastic volatility estimation studies. Exact nonlinear filters and particularly numerical integration based methods, such as the method proposed in this paper, were neglected and not studied as extensively as MCMC methods especially in the multivariate settings of stochastic volatility models. Filtering, smoothing, prediction and parameter estimation algorithms based on the sparse grid integration method are developed and proposed for a general MSV model. The proposed algorithms for estimation are compared with an implementation of MCMC based algorithms in a simulation study followed by an illustration of the proposed algorithms on empirical data of foreign exchange rate returns of US dollars and Euro. Results showed that the proposed algorithms based on the sparse grid integration method can be promising alternatives to the MCMC based algorithms especially in practical applications with their appealing characteristics.