Aggregation of Dependent Risks with Specific Marginals by the Family of Koehler-Symanowski Distributions

Many problems in Finance, such as risk management, optimal asset allocation, and derivative pricing, require an understanding of the volatility and correlations of assets returns. In these cases, it may be necessary to represent empirical data with a parametric distribution. In the literature, many distributions can be found to represent univariate data, but few can be extended to multivariate populations. The most widely used multivariate distribution in the aggregation of dependent risks in a portfolio is the Normal distribution, which has the drawbacks of inflexibility and frequent inappropriateness. Here, we consider modelling assets and measuring portfolio risks using the family of Koehler-Symanowski multivariate distributions with specific marginals, as, for example, the generalized lambda distribution. This family of distributions can be defined using the cdf along with interaction terms in the independence case. This family can be derived using a particular transformation of exponential random variables and an independent gamma. This distribution has the advantage of allowing models with complex dependence structures, as we demonstrate with Monte Carlo simulations and the analysis of the risk of a portfolioRisk Management, Monte Carlo Method, Generalized Lambda Distribution, Koehler-Symanowski Distribution

Estimating distribution functions in Johnson translation system by the Starship Procedure with simulated annealing

The computer intensive starship procedure byOwenallows to obtain the best transformation to normality using the global optimization of some measure of non-normality. In this paper, we propose to apply the procedure to estimate a cumulative distribution function in the Johnson translation system by means of the optimization of sampling statistics derived by the minimum distance and non-linear least squares methods. As global optimization method we consider a stochastic optimization method, specifically the simulated annealing, as an alternative to the method proposed by Owen and Li which is based on the Slifker and Shapiro criterion. The application of the starship procedure to a simulated sample shows that the simulated annealing algorithm inserted in the procedure supplies results better than the results obtained with the Slifker and Shapiro criterion. Moreover the problems of convergence that occur with traditional optimization methods are not present

Nonparametric estimation methods for sparse contingency tables

The problems related with multinomial sparse data analysis have been widely underlined in statistical literature in recent years. Concerning the estimation of the mass distribution, it has been widely spread the usage of nonparametric methods, particularly in the framework of ordinal variables. The aim of this paper is to evaluate the performance of kernel estimators in the framework of sparse contingency tables with ordinal variables comparing them with alternative methodologies. Moreover, an approach to estimate the mass distribution nominal variables based on a kernel estimator is proposed. At the end a case study in actuarial field is presented

Copula component analysis for dependence modelling

A copula function can be employed to decompose the information content of a multivariate distribution into marginal and dependence components, with the latter quantified by the mutual information. From this statement, it is possible to state that a link between information and copula theories is valid. On the basis of these results, in the paper we show as it is possibile to use the independent component analysis to estimate the mutual information of a multivariate random sample and, then, to select the model of copula which better interprets the dependence in sample data

GSH dependence modeling with an application to risk management

The generalized secant hyperbolic distribution (GSH) can be used to represent financial data with heavy tails as an alternative to the Student-t, because it guarantees the existence of all moments, also with a high kurtosis value. In order to obtain a multivariate extension of the GSH distribution, in this article we present two approaches to model the dependence, the copula approach and independent component analysis. Since the methodologies considered allow to simulate the GSH dependence, we show also the empirical results obtained in the estimation of risk of a financial portfolio by the Monte Carlo method

Misspecification Testing for the Conditional Distribution Model in GARCH-Type Processes

In this article, we study goodness of fit tests for some distributions of the innovations which are usually adopted to explain the behavior of financial time series. Inference is developed in the context of GARCH-type models. Functional bootstrap tests are employed, assuming that the conditional means and variances of the model are correctly specified. The performances of the functional tests are assessed with a Monte Carlo experiment, based on some of the most common distributions adopted in the financial framework. The results of an application to the series of squared residuals from a PARCH(1,1) model fitted to a series of foreign exchange rates returns are also shown.Bootstrap, Functional tests, GARCH model, Goodness of fit, C12, C15, C52,

Maximum Likelihood Estimation of the APARCH model with Skew Distribution for the Innovation Process

A method normally used in empirical financial studies to estimate the parameters of a general autoregressive conditional heteroskedasticity model is the quasi-maximum likelihood, which maximizes the likelihood function assuming conditional normality, also if it can be a false assumption. When it is possible to assume a nonnormal distribution of errors for this kind of models, it has been shown that there is a loss of efficiency of quasi-maximum likelihood estimators in finite samples with respect to maximum likelihood estimators. In this paper we study, with an empirical application to the daily returns of NASDAQ stock market index, the maximum likelihood estimates of the parameters of the asymmetric power ARCH model, a generalization of the general autoregressive conditional heteroskedasticity model, with skew distributions for the innovation process. The distributions considered are the Student-t, the exponential power and the generalized secant hyperbolic distributions, with reparametrization of the densities which adds inverse scale factors in positive and negative orthants in order to take the skewness into account. For comparison, we have analyzed the daily returns also with the quasi-maximum and the semiparametric maximum likelihood estimation procedures. We have used a quasi-Newton algorithm to optimize the average log-likelihood functions, in which analytical derivatives of the parameters have been obtained by MathStatica, a package of the computer algebra system Mathematica

Simulation and estimation of the Meixner distribution

The Meixner distribution is a special case of the generalized z-distributions. Its properties make it potentially very useful in modeling short-term financial returns. This article proposes an algorithm to simulate the Meixner distribution, and shows how to obtain maximum likelihood estimators of its parameters. A GARCH-type model is then assessed, assuming that the innovation distribution is a standardized Meixner. Goodness-of-fit properties are investigated for some real financial time series, using bootstrap tests based on the empirical process of the residuals

GARCH-type Models with Generalized Secant Hyperbolic Innovations

GARCH-type models have been analyzed assuming various nongaussian distributions of errors. In general, the asymmetric generalized Student-t random variable seems to be the distribution which better captures the nonnormality features of financial data. However, a drawback of this distribution is represented by the technical dificulties due to the evaluation of moments, especially in the case of fractional degrees of freedom. In the paper we propose to model high frequency time series returns using GARCH-type models with a generalized secant hyperbolic (GSH) distribution. The main advantage of the GSH distribution over the Student-t distribution is that all the moments are finite for each value of the shape parameter. The distribution is symmetric with respect to the mean, but we show that it is still possible to obtain the density in a closed form introducing a skewness parameter according to the method proposed by Fernandez and Steel. We use a Monte Carlo experiment to validate this distribution in the context of GARCH models with maximum likelihood estimates of parameters. Finally, we show an application to log returns of a stock index

GARCH-type Models with Generalized Secant Hyperbolic Innovations

GARCH-type models have been analyzed assuming various nongaussian distributions of errors. In general, the asymmetric generalized Student-t random variable seems to be the distribution which better captures the nonnormality features of financial data. However, a drawback of this distribution is represented by the technical dificulties due to the evaluation of moments, especially in the case of fractional degrees of freedom. In this paper we propose to model high frequency time series returns using GARCH-type models with a generalized secant hyperbolic (GSH) distribution. The main advantage of the GSH distribution over the Student-t distribution is that all the moments are finite for each value of the shape parameter. The distribution is symmetric with respect to the mean, but we show that it is still possible to obtain the density in a closed form introducing a skewness parameter according to the method proposed by Fernandez and Steel. We use a Monte Carlo experiment to validate this distribution in the context of GARCH models with maximum likelihood estimates of parameters. Finally, we show an application to log returns of a stock index.