Long range financial data and model choice

Long range financial data as typified by the daily returns of the Standard and Poor's index exhibit common features such as heavy tails, long range memory of the absolute values and clustering of periods of high and low volatility. These and other features are often referred to as stylized facts and parametric models for such data are required to reproduce them in some sense. Typically this is done by simulating some data sets under the model and demonstrating that the simulations also exhibits the stylized facts. Nevertheless when the parameters of such models are to be estimated recourse is very often taken to likelihood either in the form of maximum likelihood or Bayes. In this paper we expound a method of determining parameter values which depends solely on the ability of the model to reproduce the relevant features of the data set. We introduce a new measure of the volatility of the volatility and show how it can be combined with the distribution of the returns and the autocorrelation of the absolute returns to determine parameter values. We also give a parametric model for such data and show that it can reproduce the required features.