A new approach to state filtering and parameter estimation for a class of stochastic volatility models for which the likelihood function is unknown is considered. The alpha-stable stochastic volatility model provides a flexible framework for modeling asymmetry and heavy tails, which is useful when modeling financial returns. However, a problem posed by the alpha-stable distribution is the lack of a closed form for the probability density function, which prevents its direct application to standard filtering and estimation techniques such as sequential Monte Carlo (SMC) and Markov chain Monte Carlo (MCMC). To circumvent this difficulty, researchers have recently developed various approximate Bayesian computation (ABC) methods, which require only that one is able to simulate data from the model.

To obtain filtered volatility estimates, we develop a novel ABC based auxiliary particle filter (APF-ABC). The algorithm we develop can be easily applied to many state space models for which the likelihood function is intractable or computationally expensive. APF-ABC improves on the accuracy through better proposal distributions in cases where the optimal importance density of the filter is unavailable.

Further, a new particle based MCMC (PMCMC) method is proposed for parameter estimation in this class of volatility models. PMCMC methods combine SMC with MCMC to produce samples from the joint stationary distribution of the latent states and parameters. If full conditional distributions for all parameters are available then the particle Gibbs sampler is typically adopted; otherwise, the particle marginal Metropolis-Hastings can be used for posterior estimation. Although, several ABC based extensions of PMCMC have been proposed for the symmetric alpha-stable stochastic volatility model, all have used the particle marginal Metropolis-Hastings algorithm due to the inability to obtain full conditional distributions for all parameters in the model. However, the availability of full conditional distributions for a subset of the parameters raises the natural question of whether it is possible to estimate some of the parameters using their full conditionals, while others using a Metropolis-Hastings step. The algorithm that is proposed addresses this exact question. It is shown through a simulation study, that such a strategy can lead to increases in efficiency in the estimation process. Moreover, in contrast to previous works, this thesis studies the asymmetric alpha-stable stochastic volatility model.