A fractionally integrated ECOGARCH process

In this paper we introduce a fractionally integrated exponential continuous time GARCH(p,d,q) process. It is defined in such a way that it is a continuous time extension of the discrete time FIEGARCH(p,d,q) process. We investigate stationarity and moment properties of the new model. It is also shown that the long memory effect introduced in the log-volatility propagates to the volatility process

Empirical Study of Intraday Option Price Changes using extended Count Regression Models

In this paper we model absolute price changes of an option on the XETRA DAX index based on quote-by-quote data from the EUREX exchange. In contrast to other authors, we focus on a parameter-driven model for this purpose and use a Poisson Generalized Linear Model (GLM) with a latent AR(1) process in the mean, which accounts for autocorrelation and overdispersion in the data. Parameter estimation is carried out by Markov Chain Monte Carlo methods using the WinBUGS software. In a Bayesian context, we prove the superiority of this modelling approach compared to an ordinary Poisson-GLM and to a complex Poisson-GLM with heterogeneous variance structure (but without taking into account any autocorrelations) by using the deviance information criterion (DIC) as proposed by Spiegelhalter et al. (2002). We include a broad range of explanatory variables into our regression modelling for which we also consider interaction effects: While, according to our modelling results, the price development of the underlying, the intrinsic value of the option at the time of the trade, the number of new quotations between two price changes, the time between two price changes and the Bid-Ask spread have significant effects on the size of the price changes, this is not the case for the remaining time to maturity of the option. By giving possible interpretations of our modelling results we also provide an empirical contribution to the understanding of the microstructure of option markets

Introducing and evaluating a Gibbs sampler for spatial Poisson regression models

In this paper we present a Gibbs sampler for a Poisson model including spatial effects. Frühwirth-Schnatter und Wagner (2004b) show that by data augmentation via the introduction of two sequences of latent variables a Poisson regression model can be transformed into a normal linear model. We show how this methodology can be extended to spatial Poisson regression models and give details of the resulting Gibbs sampler. In particular, the influence of model parameterisation and different update strategies on the mixing of the MCMC chains are discussed. The developed Gibbs samplers are analysed in two simulation studies and appliedto model the expected number of claims for policyholders of a German car insurance data set. In general, both large and small simulated spatial effects are estimated accurately by the Gibbs samplers and reasonable low autocorrelations are obtained when the data variability is rather large. However, for data with very low heterogeneity, the autocorrelations resulting from the Gibbs samplers are very high, withdrawing the computational advantage over a Metropolis Hastings independence sampler which exhibits very low autocorrelations in all settings

Spatial modelling of claim frequency and claim size in insurance

In this paper models for claim frequency and claim size in non-life insurance are considered. Both covariates and spatial random e ects are included allowing the modelling of a spatial dependency pattern. We assume a Poisson model for the number of claims, while claim size is modelled using a Gamma distribution. However, in contrast to the usual compound Poisson model going back to Lundberg (1903), we allow for dependencies between claim size and claim frequency. Both models for the individual and average claim sizes of a policyholder are considered. A fully Bayesian approach is followed, parameters are estimated using Markov Chain Monte Carlo (MCMC). The issue of model comparison is thoroughly addressed. Besides the deviance information criterion suggested by Spiegelhalter et al. (2002), the predictive model choice criterion (Gelfand and Ghosh (1998)) and proper scoring rules (Gneiting and Raftery (2005)) based on the posterior predictive distribution are investigated. We give an application to a comprehensive data set from a German car insurance company. The inclusion of spatial e ects significantly improves the models for both claim frequency and claim size and also leads to more accurate predictions of the total claim sizes. Further we quantify the significant number of claims e ects on claim size