October, 2006
Empirical distributions in finance and economics might show heavy tails, volatility clustering, varying mean returns and multimodality as part of their features. However, most statistical models available in the literature assume some kind of parametric form (clearly neglecting important characteristics of the data) or focus on modeling extreme events (therefore, providing no information about the rest of the distribution). In this paper we develop a Bayesian nonparametric prior for a collection of distributions evolving in discrete time that is dense on the space of absolutely continuous distributions, and therefore allows for the special features mentioned above. The prior is constructed by defining the distribution at any time point as a Dirichlet process mixture of Gaussian distributions, and inducing dependence through the atoms of their stick-breaking decomposition. A general construction, which allows for trends, periodicities and regressors is described, but special emphasis is placed on developing autoregressive processes (AR) for sequences of distributions. The resulting model, labeled Distribution Autoregressive process (DAR) are applied to the estimation of the option-implied risk neutral distribution of the S\&P500 index.
Keywords: Dependent Dirichlet process; Nonparametric Bayes; Random probability measure; Option implied Risk Neutral Distribution.
The manuscript is available in PDF format.