Multivariate Normal Inverse Gaussian Research Articles

Infinite Mixtures of Multivariate Normal-Inverse Gaussian Distributions for Clustering of Skewed Data

Mixtures of multivariate normal inverse Gaussian (MNIG) distributions can be used to cluster data that exhibit features such as skewness and heavy tails. For cluster analysis, using a traditional finite mixture model framework, the number of components either needs to be known a priori or needs to be estimated a posteriori using some model selection criterion after deriving results for a range of possible number of components. However, different model selection criteria can sometimes result in different numbers of components yielding uncertainty. Here, an infinite mixture model framework, also known as Dirichlet process mixture model, is proposed for the mixtures of MNIG distributions. This Dirichlet process mixture model approach allows the number of components to grow or decay freely from 1 to \(\infty\) (in practice from 1 to N) and the number of components is inferred along with the parameter estimates in a Bayesian framework, thus alleviating the need for model selection criteria. We run our algorithm on simulated as well as real benchmark datasets and compare with other clustering approaches. The proposed method provides competitive results for both simulations and real data.

Multivariate Distribution in the Stock Markets of Brazil, Russia, India, and China

The purpose of this article is to analyze the dependence between Brazil, Russia, India, and China (BRIC) stock markets, adjusting the multivariate Normal Inverse Gaussian probability distribution (NIG) in 2010–2019 on data yields. Using the estimated parameters, a robust estimator of the correlation matrix is calculated, and evidence is found of the degree of integration in BRIC financial markets during the period 2000–2019. In addition, it is found that the Value at Risk presents a better performance when using the NIG distribution versus multivariate generalized autoregressive conditional heteroscedastic models.

ON SPREAD OPTION PRICING USING TWO-DIMENSIONAL FOURIER TRANSFORM

Spread options are multi-asset options with payoffs dependent on the difference of two underlying financial variables. In most cases, analytically closed form solutions for pricing such payoffs are not available, and the application of numerical pricing methods turns out to be nontrivial. We consider several such nontrivial cases and explore the performance of the highly efficient numerical technique of Hurd & Zhou[(2010) A Fourier transform method for spread option pricing, SIAM J. Financial Math. 1(1), 142–157], comparing this with Monte Carlo simulation and the lower bound approximation formula of Caldana & Fusai[(2013) A general closed-form spread option pricing formula, Journal of Banking & Finance 37, 4893–4906]. We show that the former is in essence an application of the two-dimensional Parseval’s Identity. As application examples, we price spread options in a model where asset prices are driven by a multivariate normal inverse Gaussian (NIG) process, in a three-factor stochastic volatility model, as well as in examples of models driven by other popular multivariate Lévy processes such as the variance Gamma process, and discuss the price sensitivity with respect to volatility. We also consider examples in the fixed-income market, specifically, on cross-currency interest rate spreads and on LIBOR/OIS spreads.

On Numerical Methods for Spread Options

Spread options are multi-asset options whose payoffs depend on the difference of two underlying financial variables. In most cases, analytically closed form solutions for pricing such payoffs are not available, and the application of numerical pricing methods turns out to be non-trivial. We consider several such non-trivial cases and explore the performance of the highly efficient numerical technique of Hurd and Zhou (2010), comparing this with Monte Carlo simulation and the lower bound approximation formula of Caldana and Fusai (2013). We show that the former is in essence an application of the two–dimensional Parseval Identity. As application examples, we price spread options in a model where asset prices are driven by a multivariate normal inverse Gaussian (NIG) process, in a threefactor stochastic volatility model, as well as in examples of models driven by other popular multivariate Levy processes such as the variance Gamma process, and discuss the price sensitivity with respect to volatility. We also consider examples in the fixed–income market, specifically, on cross–currency interest rate spreads and on LIBOR/OIS spreads. In terms of FFT computation, we have used the FFTW library (see Frigo and Johnson (2010)) and we document appropriate usage of this library to reconcile it with the MATLAB ifft2 counterpart.

Learning-based EM algorithm for normal-inverse Gaussian mixture model with application to extrasolar planets

ABSTRACTKarlis and Santourian [14] proposed a model-based clustering algorithm, the expectation–maximization (EM) algorithm, to fit the mixture of multivariate normal-inverse Gaussian (NIG) distribution. However, the EM algorithm for the mixture of multivariate NIG requires a set of initial values to begin the iterative process, and the number of components has to be given a priori. In this paper, we present a learning-based EM algorithm: its aim is to overcome the aforementioned weaknesses of Karlis and Santourian's EM algorithm [14]. The proposed learning-based EM algorithm was first inspired by Yang et al. [24]: the process of how they perform self-clustering was then simulated. Numerical experiments showed promising results compared to Karlis and Santourian's EM algorithm. Moreover, the methodology is applicable to the analysis of extrasolar planets. Our analysis provides an understanding of the clustering results in the ln P−ln M and ln P−e spaces, where M is the planetary mass, P is the orbital period and e is orbital eccentricity. Our identified groups interpret two phenomena: (1) the characteristics of two clusters in ln P−ln M space might be related to the tidal and disc interactions (see [9]); and (2) there are two clusters in ln P−e space.

A Feynman–Kac-type formula for Lévy processes with discontinuous killing rates

The challenge to fruitfully merge state-of-the-art techniques from mathematical finance and numerical analysis has inspired researchers to develop fast deterministic option pricing methods. As a result, highly efficient algorithms to compute option prices in Levy models by solving partial integro-differential equations have been developed. In order to provide a solid mathematical foundation for these methods, we derive a Feynman–Kac representation of variational solutions to partial integro-differential equations that characterize conditional expectations of functionals of killed time-inhomogeneous Levy processes. We allow a wide range of underlying stochastic processes, comprising processes with Brownian part as well as a broad class of pure jump processes such as generalized hyperbolic, multivariate normal inverse Gaussian, tempered stable, and $\alpha$ -semistable Levy processes. By virtue of our mild regularity assumptions as to the killing rate and the initial condition of the partial integro-differential equation, our results provide a rigorous basis for numerous applications in financial mathematics and in probability theory. We implement a Galerkin scheme to solve the corresponding pricing equation numerically and illustrate the effect of a killing rate.

Integrability of multivariate subordinated Lévy processes in Hilbert space

We investigate multivariate subordination of Lévy processes which was first introduced by Barndorff-Nielsen et al. [O.E. Barndorff-Nielsen, F.E. Benth, and A. Veraart, Modelling electricity forward markets by ambit fields, J. Adv. Appl. Probab. (2010)], in a Hilbert space valued setting which has been introduced in Pérez-Abreu and Rocha-Arteaga [V. Pérez-Abreu and A. Rocha-Arteaga, Covariance-parameter Lévy processes in the space of trace-class operators, Infin. Dimens. Anal. Quantum Probab. Related Top. 8(1) (2005), pp. 33–54]. The processes are explicitly characterized and conditions for integrability and martingale properties are derived under various assumptions of the Lévy process and subordinator. As an application of our theory we construct explicitly some Hilbert space valued versions of Lévy processes which are popular in the univariate and multivariate case. In particular, we define a normal inverse Gaussian Lévy process in Hilbert space. The resulting process has the property that at each time all its finite dimensional projections are multivariate normal inverse Gaussian distributed as introduced in Rydberg [T. Rydberg, The normal inverse Gaussian Lévy process: Simulation and approximation, Commun. Stat. Stochastic Models 13 (1997), pp. 887–910].

Variational Bayes approximations for clustering via mixtures of normal inverse Gaussian distributions

Parameter estimation for model-based clustering using a finite mixture of normal inverse Gaussian (NIG) distributions is achieved through variational Bayes approximations. Univariate NIG mixtures and multivariate NIG mixtures are considered. The use of variational Bayes approximations here is a substantial departure from the traditional EM approach and alleviates some of the associated computational complexities and uncertainties. Our variational algorithm is applied to simulated and real data. The paper concludes with discussion and suggestions for future work.

Multivariate Shrinkage for Image Denoising in Shearlet Domain

A new multivariate threshold function for image denoising in the shearlet transfrom is proposed. The new threshod exploits a multivariate normal inverse gaussian probability density function to model neighboring shearlet coefficients. Under this prior, a multivariate Bayesian shearlet estimator is derived by using the maximum a posteriori rule. Experimental results show that the new method achieves state-of-art performance in terms of peak signal-to-noise ratio (PSNR), structural similarity (SSIM) index and visual quality than existing shearlet-based image denoising methods.

Hidden Markov Models with Nonelliptically Contoured State Densities

Hidden Markov models (HMMs) are a popular approach for modeling sequential data comprising continuous attributes. In such applications, the observation emission densities of the HMM hidden states are typically modeled by means of elliptically contoured distributions, usually multivariate Gaussian or Student's-t densities. However, elliptically contoured distributions cannot sufficiently model heavy-tailed or skewed populations which are typical in many fields, such as the financial and the communication signal processing domain.Employing finite mixtures of such elliptically contoured distributions to model the HMM state densities is a common approach for the amelioration of these issues.Nevertheless, the nature of the modeled data often requires postulation of a large number of mixture components for each HMM state, which might have a negative effect on both model efficiency and the training data set's size required to avoid overfitting. To resolve these issues, in this paper, we advocate for the utilization ofa nonelliptically contoured distribution, the multivariate normal inverse Gaussian (MNIG) distribution, for modeling the observation densities of HMMs. As we experimentally demonstrate, our selection allows for more effective modeling of skewed and heavy-tailed populations in a simple and computationally efficient manner.

Pricing of basket options using univariate normal inverse Gaussian approximations

In this paper we study the approximation of a sum of assets having marginal log-returns being multivariate normal inverse Gaussian distributed. We analyse the choice of a univariate exponential NIG distribution, where the approximation is based on matching of moments. Probability densities and European basket call option prices of the two-asset and univariate approximations are studied and analysed in two cases, each case consisting of nine scenarios of different volatilities and correlations, to assess the accuracy of the approximation. We find that the sum can be well approximated, failing, however, to match the tails for some extreme parameter choices. The approximated option prices are close to the true ones, although becoming significantly underestimated for far out-of-the-money call options. Copyright © 2010 John Wiley & Sons, Ltd.

Closed-form valuations of basket options using a multivariate normal inverse Gaussian model

This paper uses a multivariate normal inverse Gaussian model to develop closed-form pricing formulas for both geometric and arithmetic basket options. For geometric basket options, an exact analytical solution is possible; for arithmetic basket options, the formula is an approximation. The model is based on a jump-driven financial process, which is known empirically to be more realistic than a geometric Brownian motion. By comparing our results to Monte Carlo experiments, we confirm the internal consistency of our formulas. The “Greeks” can be derived from the closed-form formulas in a straightforward manner.

Risk estimation using the multivariate normal inverse Gaussian distribution

Appropriate modeling of time-varying dependencies is very important for quantifying financial risk, such as the risk associated with a portfolio of financial assets. Most of the papers analyzing financial returns have focused on the univariate case. The few that are concerned with their multivariate extensions are mainly based on the multivariate normal assumption. The idea of this paper is to use the multivariate normal inverse Gaussian (MNIG) distribution as the conditional distribution for a multivariate GARCH model. The MNIG distribution belongs to a very flexible family of distributions that captures heavy tails and skewness in the distribution of individual stock returns, as well as the asymmetry in the dependence between stocks observed in financial time series data. The usefulness of the MNIG GARCH model is highlighted through a value-at-risk (VAR) application on a portfolio of European, American and Japanese equities. Backtesting shows that for a one-day holding period this model outperforms a Gaussian GARCH model and a Student’s t GARCH model. Moreover, it is slightly better than a skew Student’s t GARCH model.

EM-estimation and modeling of heavy-tailed processes with the multivariate normal inverse Gaussian distribution

The heavy-tailed multivariate normal inverse Gaussian (MNIG) distribution is a recent variance-mean mixture of a multivariate Gaussian with a univariate inverse Gaussian distribution. Due to the complexity of the likelihood function, parameter estimation by direct maximization is exceedingly difficult. To overcome this problem, we propose a fast and accurate multivariate expectation-maximization (EM) algorithm for maximum likelihood estimation of the scalar, vector, and matrix parameters of the MNIG distribution. Important fundamental and attractive properties of the MNIG as a modeling tool for multivariate heavy-tailed processes are discussed. The modeling strength of the MNIG, and the feasibility of the proposed EM parameter estimation algorithm, are demonstrated by fitting the MNIG to real world hydrophone data, to wideband synthetic aperture sonar data, and to multichannel radar sea clutter data.