A Recursive Bayesian Solution for the Excess over Threshold Distribution with Stochastic Parameters

In this paper, we propose a new approach for analyzing extreme values that are witnessed in financial markets. Our goal is to compute the predictive distribution of extreme events that are clustered in time and, as opposed to modeling just the maximum of a block of observations, we model the conditional tail for the underlying random process. We apply a stochastic parameterization of the generalized Pareto distribution to model the asymptotic behavior of this conditional tail, or excess distribution. We utilize a Rao-Blackwellized particle filter, which reduces the parameter space, and we derive a concise, recursive solution for the parameters of the distribution. Using the filter, the predictive distribution of the parameters, conditioned on the past data, is computed at each sample-time. We test our model on simulated data which show an improvement over the block-maximum and the maximum likelihood approaches both in parameter estimation and predictive performance.