Planning for future movements in asset prices and understanding the variation in the return on assets are key to the successful management of investment portfolios. This thesis investigates issues related to modelling both asset return volatility and the large movements in asset prices that may be induced by the events in the general economy, as random processes, with the implications for risk compensation and the prediction thereof being a particular focus. Exploiting modern numerical Bayesian tools, a state space framework is used to conduct all inference, with the thesis making three novel contributions to the empirical finance literature. First, observable measures of physical and option-implied volatility on the S&P 500 market index are combined to conduct inference about the latent spot market volatility, with a dynamic structure specified for the variance risk premia factored into option prices. The pooling of dual sources of information, along with the use of a dynamic model for the risk premia, produces insights into the workings of the U.S. markets, plus yields accurate forecasts of several key variables, including over the recent period of stock market turmoil. Second, a new continuous time asset pricing model allowing for dynamics in, and interactions between, the occurrences of price and volatility jumps is proposed. Various hypotheses about the nature of extreme movements in both S&P 500 returns and the volatility of the index are analyzed, within a state space model in which the usual returns measure is supplemented by direct measures of physical volatility and price jumps. The empirical results emphasize the importance of modelling both types of jumps, with the link between the intensity of volatility jumps and certain key extreme events in the economy being drawn. Finally, an empirical exploration of an alternative framework for the statistical evaluation of price jumps is conducted, with the aim of comparing the resultant measures of return variance and jumps with those induced by more conventional methods. The empirical analysis sheds light on the potential impact of the method of measurement construction on inference about the asset pricing process, and ultimately any financial decisions based on such inference.