BAYESIAN CLUSTERING OF SIMILAR MULTIVARIATE GARCH MODELS

We consider the estimation of a large number of GARCH models, say of the order of several hundreds. Especially in the multivariate case, the number of parameters is extremely large. To reduce this number and render estimation feasible, we regroup the series in a small number of clusters. Within a cluster, the series share the same model and the same parameters. Each cluster should therefore contain similar series. What makes the problem interesting is that we do not know a piori which series belongs to which cluster. The overall model is therefore a finite mixture of distributions, where the weights of the components are unknown parameters and each component distribution has its own conditional mean and variance specification. Inference is done by the Bayesian approach, using data augmentation techniques. Illustrations are provided.Large financial systems, Multivariate GARCH, Clustering, Bayesian methods, Gibbs sampling, Finite mixture distributions

On marginal likelihood computation in change-point models

Change-point models are useful for modeling time series subject to structural breaks. For interpretation and forecasting, it is essential to estimate correctly the number of change points in this class of models. In Bayesian inference, the number of change points is typically chosen by the marginal likelihood criterion, computed by Chib's method. This method requires to select a value in the parameter space at which the computation is done. We explain in detail how to perform Bayesian inference for a change-point dynamic regression model and how to compute its marginal likelihood. Motivated by our results from three empirical illustrations, a simulation study shows that Chib's method is robust with respect to the choice of the parameter value used in the computations, among posterior mean, mode and quartiles. Furthermore, the performance of the Bayesian information criterion, which is based on maximum likelihood estimates, in selecting the correct model is comparable to that of the marginal likelihood.BIC, change-point model, Chib's method, marginal likelihood

Dynamic Optimal Portfolio Selection in a VaR Framework

We propose a dynamic portfolio selection model that maximizes expected returns subject to a Value-at-Risk constraint. The model allows for time varying skewness and kurtosis of portfolio distributions estimating the model parameters by weighted maximum likelihood in a increasing window setup. We determine the best daily investment recommendations in terms of percentage to borrow or lend and the optimal weights of the assets in the risky portfolio. Two empirical applications illustrate in an out-of-sample context which models are preferred from a statistical and economic point of view. We find that the APARCH(1,1) model outperforms the GARCH(1,1) model. A sensitivity analysis with respect to the distributional innovation hypothesis shows that in general the skewed-t is preferred to the normal and Student-t.Portfolio Selection; Value-at-Risk; Skewed-t distribution; Weighted Maximum Likelihood.

Multivariate Option Pricing with Time Varying Volatility and Correlations

In recent years multivariate models for asset returns have received much attention, in particular this is the case for models with time varying volatility. In this paper we consider models of this class and examine their potential when it comes to option pricing. Specifically, we derive the risk neutral dynamics for a general class of multivariate heteroskedastic models, and we provide a feasible way to price options in this framework. Our framework can be used irrespective of the assumed underlying distribution and dynamics, and it nests several important special cases. We provide an application to options on the minimum of two indices. Our results show that not only is correlation important for these options but so is allowing this correlation to be dynamic. Moreover, we show that for the general model exposure to correlation risk carries an important premium, and when this is neglected option prices are estimated with errors. Finally, we show that when neglecting the non-Gaussian features of the data, option prices are also estimated with large errors.Multivariate risk premia, option pricing, GARCH models

Density and Hazard Rate Estimation for Censored and ?-mixing Data Using Gamma Kernels

In this paper we consider the nonparametric estimation for a density and hazard rate function for right censored ?-mixing survival time data using kernel smoothing techniques. Since survival times are positive with potentially a high concentration at zero, one has to take into account the bias problems when the functions are estimated in the boundary region. In this paper, gamma kernel estimators of the density and the hazard rate function are proposed. The estimators use adaptive weights depending on the point in which we estimate the function, and they are robust to the boundary bias problem. For both estimators, the mean squared error properties, including the rate of convergence, the almost sure consistency and the asymptotic normality are investigated. The results of a simulation demonstrate the excellent performance of the proposed estimators.Gamma kernel, Kaplan Meier, density and hazard function, mean integrated squared error, consistency, asymptotic normality.

Bayesian inference for the mixed conditional heteroskedasticity model

We estimate by Bayesian inference the mixed conditional heteroskedasticity model of (Haas, Mittnik, and Paolella 2004a). We construct a Gibbs sampler algorithm to compute posterior and predictive densities. The number of mixture components is selected by the marginal likelihood criterion. We apply the model to the SP500 daily returns.Finite mixture, ML estimation, bayesian inference, value at risk.

Nonparametric density estimation for multivariate bounded data.

We propose a new nonparametric estimator for the density function of multivariate bounded data. As frequently observed in practice, the variables may be partially bounded (e.g., nonnegative) or completely bounded (e.g., in the unit interval). In addition, the variables may have a point mass. We reduce the conditions on the underlying density to a minimum by proposing a nonparametric approach. By using a gamma, a beta, or a local linear kernel (also called boundary kernels), in a product kernel, the suggested estimator becomes simple in implementation and robust to the well known boundary bias problem. We investigate the mean integrated squared error properties, including the rate of convergence, uniform strong consistency and asymptotic normality. We establish consistency of the least squares cross-validation method to select optimal bandwidth parameters. A detailed simulation study investigates the performance of the estimators. Applications using lottery and corporate finance data are provided.Asymmetric kernels, multivariate boundary bias, nonparametric multivariate density estimation, asymptotic properties, bandwidth selection, least squares cross-validation.

Mixed Exponential Power Asymmetric Conditional Heteroskedasticity

To match the stylized facts of high frequency financial time series precisely and parsimoniously, this paper presents a finite mixture of conditional exponential power distributions where each component exhibits asymmetric conditional heteroskedasticity. We provide stationarity conditions and unconditional moments to the fourth order. We apply this new class to Dow Jones index returns. We find that a two-component mixed exponential power distribution dominates mixed normal distributions with more components, and more parameters, both in-sample and out-of-sample. In contrast to mixed normal distributions, all the conditional variance processes become stationarity. This happens because the mixed exponential power distribution allows for component-specific shape parameters so that it can better capture the tail behaviour. Therefore, the more general new class has attractive features over mixed normal distributions in our application: Less components are necessary and the conditional variances in the components are stationarity processes. Results on NASDAQ index returns are similar.Finite mixtures, exponential power distributions, conditional heteroskedasticity, asymmetry, heavy tails, value at risk

Bayesian Option Pricing Using Mixed Normal Heteroskedasticity Models

While stochastic volatility models improve on the option pricing error when compared to the Black-Scholes-Merton model, mispricings remain. This paper uses mixed normal heteroskedasticity models to price options. Our model allows for significant negative skewness and time varying higher order moments of the risk neutral distribution. Parameter inference using Gibbs sampling is explained and we detail how to compute risk neutral predictive densities taking into account parameter uncertainty. When forecasting out-of-sample options on the S&P 500 index, substantial improvements are found compared to a benchmark model in terms of dollar losses and the ability to explain the smirk in implied volatilities. Les modèles à volatilité stochastique apportent des améliorations en ce qui a trait à l’erreur d’établissement des prix des options comparativement au modèle de Black-Scholes-Merton. Toutefois, la fixation incorrecte des prix persiste. Le présent document a recours à des modèles mixtes avec hétéroscédasticité normale pour fixer les prix des options. Notre modèle permet de tenir compte de l’asymétrie négative importante et des moments d’ordre élevé variant dans le temps liés à la distribution du risque nul. Nous expliquons l’inférence des paramètres selon l’échantillonnage de Gibbs et détaillons la façon de traiter les densités prédictives de risque neutre en prenant en considération l’incertitude des paramètres. Dans le cas des prévisions concernant les options hors-échantillonnage sur l’indice S&P 500, nous constatons des améliorations importantes, par rapport à un modèle de référence, en termes de pertes exprimées en dollars et de capacité d’expliquer l’ironie des volatilités implicites.Bayesian inference, option pricing, finite mixture models, out-of-sample prediction, GARCH models, Inférence bayésienne, fixation du prix des options, modèles à mélanges finis, prédiction hors-échantillon, modèles GARCH.

Bayesian option pricing using mixed normal heteroskedasticity models

Bayesian inference, option pricing, finite mixture models, out-of-sample prediction, GARCH models