This thesis focusses on application as well as modifications of sequential Monte Carlo (SMC) utilising the smooth resampling procedure of Pitt and Malik  (smooth bootstrap) as a statistically and computationally efficient method for parameter estimation of discrete and continuous time stochastic processes that have intractable likelihoods; arising in the modelling of volatility, primarily in financial markets but also in other fields. For the models and applications we consider, the likelihoods are intractable arising either from observations of a discrete time process being missing or temporally aggregated, or from discrete observation of a continuous time process. The methods are developed for the discrete time GARCH(1,1) model (Bollerslev ) for conditional heteroscedasticity when there are missing observations or when observation is only through temporal aggregates of the underlying process. The methods to be presented can be generalised to many variants of the GARCH model including the EGARCH(1,1) (Nelson ) and GJR-GARCH(1,1) (Glosten et al. ) for instance. More challenging are the continuous time GARCH model (COGARCH) of Kluppelberg et al.  and the Markov switching (MS-)GARCH model (cf. Bauwens et al. ). The COGARCH process evolves in continuous time but is observed at, typically, irregular time intervals and as a result many events in the underlying process are not directly observed, while the MS-GARCH model allows model parameters to evolve over time according to an unobserved regime process. The same fundamental methods are developed for these two more complicated variants also.