Exact Bayesian moment based inference for the distribution of the small-time movements of an Itô semimartingale

We modify the Gallant and Tauchen (1996) efficient method of moments (EMM) method to perform exact Bayesian inference, where exact means no reliance on asymptotic approximations. We use this modification to evaluate the empirical plausibility of recent predictions from high frequency financial theory regarding the small-time movements of an Itô semimartingale. The theory indicates that the probability distribution of the small moves should be locally stable around the origin. It makes no predictions regarding large rare jumps, which get filtered out. Our exact Bayesian procedure imposes support conditions on parameters as implied by this theory. The empirical application uses S&P Index options extending over a wide range of moneyness, including deep out of the money puts. The evidence is consistent with a locally stable distribution valid over most of the support of the observed data while mildly failing in the extreme tails, about which the theory makes no prediction. We undertake diagnostic checks on all aspects of the procedure. In particular, we evaluate the distributional assumptions regarding a semi-pivotal statistic, and we test by Monte Carlo that the posterior distribution is properly centered with short credibility intervals. Taken together, our results suggest a more important role than previously thought for pure jump-like models with diminished, if not absent, diffusive component.