Abstract
In this paper we will rigorously study some of the properties of continuous time stochastic volatility models. We have five main results: (I) the stochastic volatility class can be linked to Cox process based models of tick-by-tick financial data; (ii) we characterise the moment, autocorrelation function and spectrum of squared returns; (iii) based only on discrete time returns, we give a simple consistent and asymptotically normally distributed estimator of continuous time volatility models without any simulation or discretisation error. Furthermore, we review a new class of Ornstein-Uhlenbeck processes of volatility, introduced in a companion paper, which allows (iv) the discrete time returns to be simulated without any form of discretisation error, (v) explicit modelling of correlation structures and allow analytic calculations of the properties of returns.