Modeling ultra-high-frequency data: the S&P 500 index future

In recent years, marked point processes have found a natural application in the modeling of ultra-high-frequency financial data. The use of these processes does not require the integration of the data which is usually needed by other modeling approaches. Two main classes of marked point processes have so far been proposed to model financial data: the class of the autoregressive conditional duration models of Engle and Russel and the broad class of doubly stochastic Poisson processes with marks. In this paper we show how to model an ultra-high-frequency data set relative to the prices of the future on the S&P 500 index using a particular class of doubly stochastic Poisson process. Our models allow a natural interpretation of the underlying intensity in terms of news reaching the market and does not require the use of ad hoc methods to treat the seasonalities. Filtering and estimation are carried out by Monte Carlo expectation maximization and reversible jump Markov chain Monte Carlo algorithms.