In this paper we show that in the lognormal discrete-time stochastic volatility model with predictable conditional expected returns, the conditional expected value of the discounted payoff of a European call option is infinite. Our empirical illustration shows that the characteristics of the predictive distributions of the discounted payoffs, obtained using Monte Carlo methods, do not indicate directly that the expected discounted payoffs are infinite.

This paper presents some new results on exogeneity in models with latent variables. The concept of exogeneity is extended to the class of models with latent variables, in which a subset of parameters and latent variables is of interest. Exogeneity is discussed from the Bayesian point of view. We propose sufficient weak and strong exogeneity conditions in the vector error correction model (VECM) with stochastic volatility (SV) disturbances. Finally, an empirical illustration based on the VECM-SV model for the daily growth rates of two main official Polish exchange rates: USD/PLN and EUR/PLN, as well as EUR/USD from the international Forex market is presented. The exogeneity of the EUR/USD rate is examined. The strong exogeneity hypothesis of the EUR/USD rate is not rejected by the data.

The paper refines Lenk’s concept of improving the performance of the computed harmonic mean estimator (HME) in three directions. First, the adjusted HME is derived from an exact analytical identity. Second, Lenk’s assumption concerning the appropriate subset A of the parameter space is significantly weakened. Third, it is shown that, under certain restrictions imposed on A, a fundamental identity underlying the HME also holds for improper prior densities, which substantially extends applicability of the adjusted HME.

The aim of this paper is to examine the empirical usefulness of two new MSF – Scalar BEKK(1,1) models of n-variate volatility. These models formally belong to the MSV class, but in fact are some hybrids of the simplest MGARCH and MSV specifications. Such hybrid structures have been proposed as feasible (yet non-trivial) tools for analyzing highly dimensional financial data (large n). This research shows Bayesian model comparison for two data sets with n = 2, since in bivariate cases we can obtain Bayes factors against many (even unparsimonious) MGARCH and MSV specifications. Also, for bivariate data, approximate posterior results (based on preliminary estimates of nuisance matrix parameters) are compared to the exact ones in both MSF-SBEKK models. Finally, approximate results are obtained for a large set of returns on equities (n = 34).

The s-period ahead Value-at-Risk (VaR) for a portfolio of dimension n is considered and its Bayesian analysis is discussed. The VaR assessment can be based either on the n-variate predictive distribution of future returns on individual assets, or on the univariate Bayesian model for the portfolio value (or the return on portfolio). In both cases Bayesian VaR takes into account parameter uncertainty and non-linear relationship between ordinary and logarithmic returns. In the case of a large portfolio, the applicability of the n-variate approach to Bayesian VaR depends on the form of the statistical model for asset prices. We use the n-variate type I MSF-SBEKK(1,1) volatility model proposed specially to cope with large n. We compare empirical results obtained using this multivariate approach and the much simpler univariate approach based on modelling volatility of the value of a given portfolio.

The paper aims at comparing forecast ability of VAR/VEC models with a non-changing covariance matrix and two classes of Bayesian Vector Error Correction – Stochastic Volatility (VEC-SV) models, which combine the VEC representation of a VAR structure with stochastic volatility, represented by the Multiplicative Stochastic Factor (MSF) process, the SBEKK form or the MSF-SBEKK specification.

Based on macro-data coming from the Polish economy (time series of unemployment, inflation and interest rates) we evaluate predictive density functions employing of such measures as log predictive density score, continuous rank probability score, energy score, probability integral transform. Each of them takes account of different feature of the obtained predictive density functions.