GARCH-type volatility models based on Brownian inverse Gaussian intra-day return processes

ABSTRACT GARCH models are useful to estimate daily volatility in financial return series. When intra-day return data are available realized volatility may be used for the same purpose. We formulate a new model that is a hybrid between a traditional GARCH model and a stochastic volatility model. This model postulates that over each trading day anew the intra-day return process follows a Brownian motion with drift and volatility that are inherited from the previous day in typical GARCH fashion but are also subject to random inverse Gaussian (IG) distributed news noise impacts that arrive after market closure on the previous day. We call it the BIG–GARCH model and derive the likelihood function needed to fit it. It turns out that this uses the daily returns and realized volatilities as sufficient statistics. To make this possible we introduce a number of new distributions related to the IG-distribution and also derive diagnostics that may be used to check the quality of fit. The new model produces two volatility measures, called the expected volatility and the actual volatility. We show that the latter is close to the realized volatility. The model is illustrated by fitting it to IBM daily and intra-day returns.