This essay collection focuses on the relationship between continuous time models and Autoregressive Conditionally Heteroskedastic (ARCH) models and applications. For the first time, Modelling Stock Market Volatility provides new insights about the links between these two models and new work on practical estimation methods for continuous time models. Featuring the pioneering scholarship of Daniel Nelson, the text presents research about the discrete time model, continuous time limits and optimal filtering of ARCH models, and the specification and estimation of continuous time processes. This work will lead to a rapid growth in their empirical application as they are increasingly subjected to routine specification testing. – Provides for the first time new insights on the links between continuous time and ARCH models- Collects seminal scholarship by some of the most renowned researchers in finance and econometrics- Captures complex arguments underlying the approximation and proper statistical modelling of continuous time volatility dynamics