Generalized Hyperbolic Distribution Research Articles

Expectation and Optimal Allocations in Existential Contests of Finite, Heavy-Tail-Distributed Outcomes

Financial time series and other human-driven, non-natural processes are known to exhibit fat-tailed outcome distributions. That is, such processes demonstrate a greater tendency for extreme outcomes than the normal distribution or other natural distributional processes would predict. We examine the mathematical expectation, or simply “expectation”, traditionally the probability-weighted outcome, regarded since the seventeenth century as the mathematical definition of “expectation”. However, when considering the “expectation” of an individual confronted with a finite sequence of outcomes, particularly existential outcomes (e.g., a trader with a limited time to perform or lose his position in a trading operation), we find this individual “expects” the median terminal outcome over those finite trials, with the classical seventeenth-century definition being the asymptotic limit as trials increase. Since such finite-sequence “expectations” often differ in values from the classic one, so do the optimal allocations (e.g., growth-optimal). We examine these for fat-tailed distributions. The focus is on implementation, and the techniques described can be applied to all distributional forms. We make no assertion that the empirical data for any financial time series comports to the generalized hyperbolic distribution (GHD), which we will use as a proxy of any heavy-tailed distribution herein. Rather, we have selected the GHD to highlight the process for determining expectation and other important time-dependent metrics in existential contests, using the GHD as a generic proxy for the specific distributional form an implementor of the presented technique might ascribe to the empirical data.

Value at Risk and Expected Shortfall for Normal Weighted Inverse Gaussian Distributions

Value at Risk (VaR) and Expected Shortfall (ES) is commonly used measures of potential risk for losses in financial markets. In literature VaR and ES for the Normal Inverse Gaussian (NIG) distribution, a special case of Generalized Hyperbolic Distribution (GHD), is frequently used. There are however, Normal Inverse Gaussian related distributions, which are also special cases of GHD that can also be used. The objective of this paper is to calculate VaR for Normal Weighted Inverse Gaussian (NWIG) distributions. The Expectation-Maximization (EM) algorithm has been used to obtain the Maximum Likelihood (ML) estimates of the proposed models for the Range Resource Corporation (RRC) financial data. We used Kupiec likelihood ratio (LR) for backtesting of VaR. Kolmogorov-Smirnov test and Anderson-Darling test have been used for goodness of fit test. Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC) and Log-likelihood have been used for model selection. The results clearly show that the NWIG distributions are good alternatives to NIG for determining VaR and ES.

Value at Risk and Expected Shortfall for Normal Variance Mean Mixtures of Finite Weighted Inverse Gaussian Distributions

The Normal Inverse Gaussian (NIG) distribution, a special case of the Generalized Hyperbolic Distribution (GHD) has been frequently used for financial modelling and risk measures. In this work, we consider other normal Variance mean mixtures based on finite mixtures of special cases for Generalised Inverse Gaussian as mixing distributions. The Expectation-Maximization (EM) algorithm has been used to obtain the Maximum Likelihood (ML) estimates of the proposed models for some financial data. We estimate Value at risk (VaR) and Expected Shortfall (ES) for the fitted models. The Kupiec likelihood ratio (LR) has been applied for backtesting of VaR. Akaike Information Creterion (AIC), Bayesian Information Creterion (BIC) and Log-likelihood have been used for model selection. The results clearly show that the proposed models are good alternatives to NIG for determining VaR and ES.

Two Factor Stochastic Mortality Modeling with Generalized Hyperbolic Distribution

In this paper, we reconsider the two-factor stochastic mortality model introduced by Cairns, Blake and Dowd (2006) (CBD). The error terms in the CBD model are assumed to form a two-dimensional random walk. We first use the Doornik and Hansen (2008) multivariate normality test to show that the underlying normality assumption does not hold for the considered data set. Ainou (2011) proposed independent univariate normal inverse Gaussian Levy processes to model the error terms in the CBD model. We generalize this idea by introducing a possible dependency between the 2-dimensional random variables, using a bivariate Generalized Hyperbolic distribution. We propose four non-Gaussian, fat-tailed distributions: Student's t, normal inverse Gaussian, hyperbolic and generalized hyperbolic distributions. Our empirical analysis shows some preferences for using the new suggested model, based on Akaike's information criterion, the Bayesian information criterion and likelihood ratio test, as our in-sample model selection criteria, as well as mean absolute percentage error for our out-of-sample projection errors.

Model-Based Clustering, Classification, and Discriminant Analysis Using the Generalized Hyperbolic Distribution: MixGHD R package

The MixGHD package for R performs model-based clustering, classification, and discriminant analysis using the generalized hyperbolic distribution (GHD). This approach is suitable for data that can be considered a realization of a (multivariate) continuous random variable. The GHD has the advantage of being flexible due to skewness, concentration, and index parameters; as such, clustering methods that use this distribution are capable of estimating clusters characterized by different shapes. The package provides five different models all based on the GHD, an efficient routine for discriminant analysis, and a function to measure cluster agreement. This paper is split into three parts: the first is devoted to the formulation of each method, extending them for classification and discriminant analysis applications, the second focuses on the algorithms, and the third shows the use of the package on real datasets.

A Special Weight for Inverse Gaussian Mixing Distribution in Normal Variance Mean Mixture with Application

Normal Variance-Mean Mixture (NVMM) provides a general framework for deriving models with desirable properties for modelling financial market variables such as exchange rates, equity prices, and interest rates measured over short time intervals, i.e. daily or weekly. Such data sets are characterized by non-normality and are usually skewed, fat-tailed and exhibit excess kurtosis. The Generalised Hyperbolic distribution (GHD) introduced by Barndorff-Nielsen (1977) which act as Normal variance-mean mixtures with Generalised Inverse Gaussian (GIG) mixing distribution nest a number of special and limiting case distributions. The Normal Inverse Gaussian (NIG) distribution is obtained when the Inverse Gaussian is the mixing distribution, i.e., the index parameter of the GIG is . The NIG is very popular because of its analytical tractability. In the mixing mechanism, the mixing distribution characterizes the prior information of the random variable of the conditional distribution. Therefore, considering finite mixture models is one way of extending the work. The GIG is a three parameter distribution denoted by and nest several special and limiting cases. When , we have which is called an Inverse Gaussian (IG) distribution. When , , , we have , and distributions respectively. These distributions are related to and are called weighted inverse Gaussian distributions. In this work, we consider a finite mixture of and and show that the mixture is also a weighted Inverse Gaussian distribution and use it to construct a NVMM. Due to the complexity of the likelihood, direct maximization is difficult. An EM type algorithm is provided for the Maximum Likelihood estimation of the parameters of the proposed model. We adopt an iterative scheme which is not based on explicit solution to the normal equations. This subtle approach reduces the computational difficulty of solving the complicated quantities involved directly to designing an iterative scheme based on a representation of the normal equation. The algorithm is easily programmable and we obtained a monotonic convergence for the data sets used.

Portfolio optimization under the generalized hyperbolic distribution: optimal allocation, performance and tail behavior

In this paper, we analyze the asset allocation problem under the generalized hyperbolic (GH) distribution of returns and exponential utility. We provide closed-form expressions to compute the optimal portfolio weights; and we introduce two new measures, associated with a more general mean-risk trade-off, that allow us to express the optimal solution as an affine combination of two efficient portfolios: one minimizing risk and the other maximizing mean given a particular level of risk. Also, we prove that optimal portfolio performance is not monotonic in tail behavior since it increases when tails become lighter or heavier with respect to a particular threshold; however, distributions with heavier tails produce more conservative allocations in terms of the weight given to the minimum-risk portfolio increments. Finally, the practical relevance of our paper show that tail behavior greatly affects portfolio construction and performance, and that including non-normality features of short-term asset returns, through a GH distribution, has the potential to significantly improve the investor's certainty equivalent excess return.

Volatility Modelling of the BRICS Stock Markets

Volatility modelling is a key factor in equity markets for risk and portfolio management. This paper focuses on the use of a univariate generalized autoregressive conditional heteroscedasticity (GARCH) models for modelling volatility of the BRICS (Brazil, Russia, India, China and South Africa) stock markets. The study was conducted under the assumptions of seven error distributions that include the normal, skewed-normal, Student’s t, skewed-Student’s t, generalized error distribution (GED), skewed-GED and the generalized hyperbolic (GHYP) distribution. It was observed that using an ARMA(1, 1)-GARCH(1, 1) model, volatilities of the Brazilian Bovespa and the Russian IMOEX markets can both be well characterized (or described) by a heavy-tailed Student’s t distribution, while the Indian NIFTY market’s volatility is best characterized by the generalized hyperbolic (GHYP) distribution. Also, the Chinese SHCOMP and South African JALSH markets’ volatilities are best described by the skew-GED and skew-Student’s t distribution, respectively. The study further observed that the persistence of volatility in the BRICS markets does not follow the same hierarchical pattern under the error distributions, except under the skew-Student’s t and GHYP distributions where the pattern is the same. Under these two assumptions, i.e. the skew-Student’s t and GHYP, in a descending hierarchical order of magnitudes, volatility with persistence is highest in the Chinese market, followed by the South African market, then the Russian, Indian and Brazilian markets, respectively. However, under each of the five non-Gaussian error distributions, the Chinese market is the most volatile, while the least volatile is the Brazilian market.

Nonlinear regression using order statistics from the multivariate generalized hyperbolic distributions

In this paper, by considering an -dimensional random vector from the multivariate generalized hyperbolic (GH) distribution, we derive the joint distribution of Y and the order statistics from We show that this joint distribution is indeed a mixture of the selection GH distributions. The results were applied to some regression functions via the order statistics and established with a real data set.

Tail risk measures and risk allocation for the class of multivariate normal mean–variance mixture distributions

The Conditional Tail Expectation (CTE), also known as the Expected Shortfall and Tail-VaR, has received much attention as a preferred risk measure in finance and insurance applications. A related risk management exercise is to allocate the amount of the CTE computed for the aggregate or portfolio risk into individual risk units, a procedure known as the CTE allocation. In this paper we derive analytic formulas of the CTE and its allocation for the class of multivariate normal mean–variance mixture (NMVM) distributions, which is known to be extremely flexible and contains many well-known special cases as its members. We also develop the closed-form expression of the conditional tail variance (CTV) for the NMVM class, an alternative risk measure proposed in the literature to supplement the CTE by capturing the tail variability of the underlying distribution. To illustrate our findings, we focus on the multivariate Generalized Hyperbolic Distribution (GHD) family which is a popular subclass of the NMVM in connection with Lévy processes and contains some common distributions for financial modelling. In addition, we also consider the multivariate slash distribution which is not a member of GHD family but still belongs to the NMVM class. Our result is an extension of the recent contribution of Ignatieva and Landsman (2015).

Selected singular-generalized-hyperbolic distributions with applications to order statistics and reliability

In this paper, the truncated version of the selected multivariate generalized-hyperbolic distributions is introduced. Considering special truncations, the joint distribution of the consecutive order statistics from the multivariate generalized-hyperbolic (GH) distribution is derived. It is shown that this joint distribution can be expressed as mixtures of the truncated selected-GH distributions. All of these truncated distributions are expressed as the selected singular-GH distributions. These results are used to obtain some expressions for the reliability measures such as mean residual life time, mean inactivity time and regression mean residual life for k-out-of-n systems.

Using the Generalized Pareto and Pearson Type-IV Distributions to Measure Value-At-Risk for the Daily South African Mining Index

Risk management tools, such as value-at-risk (VaR) and expected shortfall (conditional value-at-risk) are highly dependent on an appropriate set of underlying distributional assumptions being made. Identifying a distribution that best captures all aspects of financial data sets may benefit both investors and risk managers. In this study, we compare the relative performance of the GARCH-generalized Pareto distribution and the GARCH-Pearson type-IV models in estimating value-at-risk of the South African mining index returns. VaR and backtesting are performed via Kupiec likelihood ratio test. The proposed models capture some key stylized facts associated with daily index returns; e.g. heavy tails (non-normality), asymmetry, volatility clustering and their leverage effect. The advantage of the proposed models lies in their ability to capture conditional heteroscedasticity in the returns through the GARCH framework and at the same time model their heavy tail behaviour through the generalized Pareto distribution (GPD) and the Pearson type-IV distribution. The main findings indicate that the GARCH-GPD and GARCH-Pearson type-IV models give better results when compared with generalized hyperbolic distributions (GHDs), thereby providing a good alternative candidate for modelling financial returns. The accuracy of the volatility model is essential in forecasting volatility of future returns in which the predictability of volatility plays an integral role in risk management and portfolio management.

A Stein characterisation of the generalized hyperbolic distribution

The generalized hyperbolic (GH) distributions form a five parameter family of probability distributions that includes many standard distributions as special or limiting cases, such as the generalized inverse Gaussian distribution, Student’s t -distribution and the variance-gamma distribution, and thus the normal, gamma and Laplace distributions. In this paper, we consider the GH distribution in the context of Stein’s method. In particular, we obtain a Stein characterisation of the GH distribution that leads to a Stein equation for the GH distribution. This Stein equation reduces to the Stein equations from the current literature for the aforementioned distributions that arise as limiting cases of the GH superclass.

Probability Density Functions of Vorticities in Turbulent Channels with Effects of Blowing and Suction

Abstract This paper presents direct numerical simulation (DNS) result of the Navier–Stokes equations for turbulent channel flows with blowing and suction effects. The friction Reynolds number is R e τ = 394 ${\rm{R}}{{\rm{e}}_\tau} = 394$ and a range of blowing and suction conditions is covered with different perturbation strengths, i. e. A = 0.05 , $A = 0.05, $ 0.1, 0.2. While the mean velocity profile has been severely altered, the probability density function (PDF) for (spanwise) vorticity – depending on wall distance ( y + ) $({y^ +})$ and blowing/suction strength (A) – satisfies the generalized hyperbolic distribution (GHD) of Birnir [The Kolmogorov-Obukhov statistical theory of turbulence, J. Nonlinear Sci. (2013a), doi: 10.1007/s00332-012-9164–z; The Kolmogorov-Obukhov theory of turbulence, Springer, New York, 2013b] in the bulk of the flow. The latter leads to accurate descriptions of all PDFs (at y + = 40 , ${y^ +} = 40, $ 200, 390 and A = 0.05 , $A = 0.05, $ 0.2, for instance) with only four parameters. The result indicates that GHD is a general tool to quantify PDF for turbulent flows under various wall surface conditions.

Optimal Portfolio Selection Based on Expected Shortfall Under Generalized Hyperbolic Distribution

This paper discusses optimal portfolio selection problems under Expected Shortfall as the risk measure. We employ multivariate Generalized Hyperbolic distribution as the joint distribution for the risk factors of underlying portfolio assets, which include stocks, currencies and bonds. Working under this distribution, we find the optimal portfolio strategy.

Nonparametric Risk Management with Generalized Hyperbolic Distributions

In this paper we propose the GHADA risk management model that is based on the generalized hyperbolic (GH) distribution and on a nonparametric adaptive methodology. Compared to the normal distribution, the GH distribution possesses semi-heavy tails and represents the financial risk factors more appropriately. The nonparametric adaptive methodology has the desirable property of estimating homogeneous volatility in a short time interval. For DEM/USD exchange rate data and a German bank portfolio data the proposed GHADA model provides more accurate value at risk calculation than the traditional model based on the normal distribution. All calculations and simulations are done with XploRe.