Continuous Time Behavior of Non Linear Arch Models

Abstract

This chapter presents convergence results for the A-PARCH model (1.6) that was originally proposed by Ding et al. (1993). We remind that in addition to be a particular convenient tool to model volatility asymmetries, such a model imposes a sort of Box-Cox power transformation to the conditional standard deviation. According to this model, the ‘volatility concept’ is thus not imposed a priori by the modeler, but it has to be estimated from data. By assuming that such a transformation is the same at every sampling frequency, we derive continuous time results for model (1.6). Such results are useful for three main reasons: (1) they help formulating continuous time models that are flexible with respect to the choice of the volatility concept (see chapter 5); (2) they provide a simple identification device through which estimating the correlation process between a continuous time asset price process and its instantaneous volatility; (3) they help understanding the role played by the volatility concept in determining the long run behavior of the error process of the model.