The multivariate normal inverse gaussian mnig is a recent variancemean mixture of a multivariate gaussian distribution with an inverse gaussian mixing distribution. The family of nig distributions is a special case of the generalised hyperbolic distributions barndor. Thus, an interesting feature with the nig modelling framework is that it essentially implies a certain behavior for the volatility, although this is in principle unobserved. May 19, 2015 normal inverse gaussian distribution s. Unfortunately, different books use different conventions on how to parameterize the various. Additional functionality is provided for the hyperbolic distribution, normal inverse. The normalinverse gaussian distribution nig is a continuous probability distribution that is defined as the normal variancemean mixture where the mixing density is the inverse gaussian distribution. The parameter is the mean or expectation of the distribution and also its median and mode. Comparison to the normal, weibull and inverse normal. The normal inverse gaussian distribution for synthetic cdo.
The benefits are that the mixture component can be skewed and that it has fatter tails than the multivariate gaussian distribution yu, 2011. Sep 19, 2008 to achieve this we choose to work with the normal inverse gaussian distribution, which can accommodate both of these features. In probability theory, the inverse gaussian distribution is a twoparameter family of continuous. It is used extensively in geostatistics, statistical linguistics, finance, etc. Normal inverse gaussian process subordination a b s t r a c t a fractional normal inverse gaussian fnig process is a fractional brownian motion subordinated to an inverse gaussian process.
In probability theory and statistics, the normal inverse wishart distribution or gaussian inverse wishart distribution is a multivariate fourparameter family of continuous probability distributions. Continuous univariate normal distribution, also known as gaussian distribution. The main question this thesis answers is whether a normal inverse gaussian distribution performs. In terms of the regress documentation \\muab\ and \bs\ and \l\ is not a parameter in that distribution. Some of the recent works model the behaviour of stock price movements using the levy process especially the normal inverse gaussian nig distribution to model stock returns for example, bolviken and benth, 2000, carr and wu, 2004. The use of conjugate priors allows all the results to be derived in closed form. Conjugate bayesian analysis of the gaussian distribution kevin p. This collection of mfiles supplements this toolbox with the most important functionalities for the nig distribution. We present a modification of the lhp model replacing the student t distribution with the normal inverse gaussian nig distribution. It is the conjugate prior of a multivariate normal distribution with unknown mean and covariance m. Let a scalar quantity u i g, termed the mixing component, be inverse gaussian distributed u i g. Also, the cumulative distribution function cdf of the single parameter inverse gaussian distribution is related to the standard normal distribution by. An alternative is to use a multivariate normal inverse gaussian distribution karlis and santourian, 2008. Python normal inverse gaussian distribution in statistics.
Normal inverse gaussian nig distribution updated version. The appeal of the nig class of distributions is that it is characterized by the first four moments. Generalized inverse gaussian distribution wikipedia. It is the conjugate prior of a multivariate normal distribution with. Comparison tests on several standard cds index portfolios show that the nig distribution has better tail characteristics than the normal and it is much more efficient for large scale computations than the multivariate student t. It is also convenient to provide unity as default for both mean and scale. Modelling the volatility of financial assets using the. The icdf is the reverse of the cumulative distribution function cdf, which is the area that is associated with a value. Inverse normal inverse gaussian distribution scipy v1. The result is an approximate version of now in fractions, which is given by where, converting the result into. The authors propose the class of normal inverse gaussian nig distributions to approximate an unknown riskneutral density. Modelbased clustering with the multivariate normal inverse. The distribution determines an homogeneous levy process, and this process is representable through subordination of brownian motion by the inverse gaussian process.
In probability theory, the inverse gaussian distribution also known as the wald distribution is a twoparameter family of continuous probability distributions with support on 0. Methods and formulas for inverse cumulative distribution. Fitting the normal inverse gaussian distribution to the s. A random variable with a gaussian distribution is said to be normally distributed and is called a normal deviate. In probability theory and statistics, the normalinversewishart distribution or gaussianinversewishart distribution is a multivariate fourparameter family of continuous probability distributions. Normal inverse gaussian distributions and stochastic. With the aim of modelling key stylized features of observational series from finance and turbulence a number of stochastic processes with normal inverse gaussian marginals and various types of dependence structures are discussed. Valuation of insurance products using a normal inverse. Clustering with the multivariate normal inverse gaussian. Statistics and probability letters fractional normal. The inverse gaussian is a skew ed, twoparameter continuous distribution whose density is sim ilar to the gamma distribution with greater skewness and a sharper peak. The nig process can be constructed either as a process with nig increments or, alternatively, via random time change of brownian motion using the inverse gaussian process to determine time. Generating random numbers based on an inverse gaussian. Apr 11, 2007 the normal inverse gaussian distribution is defined as a variance.
This article proposes the normal inverse gaussian nig distribution as a more tractable alternative. These are the moments that are important to many risk management applications. Probability density function the general formula for the probability density function of the normal distribution is \ fx \fracex \mu22\sigma2 \sigma\sqrt2\pi \ where. The normal inverse gaussian henceforth nig distribution is characterized via a normal inverse gaussian mixing distribution. Conjugate bayesian analysis of the gaussian distribution. Processes of normal inverse gaussian type springerlink. In our opinion, the normal inverse gaussian nig distribution is an appropriate distribution to solve the problem. The distribution is also called normalinverse gaussian distribution, and normal inverse distribution. Normal inverse gaussian distribution file exchange matlab. Note that xt is a variance mixture of normal distribution with mean zero. In this article, the normal inverse gaussian distribution model nigdm is extended to a new extended normal inverse gaussian distribution enigdm and its derivate models find many applications. The normal inverse gaussian distribution can be generalised with a fifth parame ter to the socalled generalized inverse gaussian distributions.
Formally stated, let y be a random variable that follows an inverse gaussian probability law ig discussed in seshadri 1993. The distribution is also called normal inverse gaussian distribution, and normal inverse distribution. The acronym ppf stands for percent point function, which is another name for the quantile function in 20. The normal inverse gaussian distribution and the pricing. This paper shows how the fnig process emerges naturally as the limit of a random walk with correlated jumps separated by i. The inverse gaussian distribution was first studied in relation to brownian motion. A quasimonte carlo algorithm for the normal inverse gaussian distribution and valuation of financial derivatives. The student tdistribution has more plausible tail behavior, but because it is not stable under convolution, it becomes computationally very unwieldy as the number of variables increases. Inverse gaussian distribution wald distribution statistics how to. The probability density function for norminvgauss is. It was rediscovered and popularised by ole barndorffnielsen, who called it the generalized inverse gaussian distribution. In the last two decades data with heavy tails have been collected in several. This collection is an update of a flawed older version. High accurate simple approximation of normal distribution.
The gaussian or normal distribution is one of the most widely used in statistics. A number of models have been proposed for such heavy tailed random processes. For all continuous distributions, the icdf exists and is unique if 0 density function pdf is positive for the entire real number line for example, the normal pdf, the icdf is not defined for. Fatter tails of the t distribution allows for outliers. To achieve this we choose to work with the normal inverse gaussian distribution, which can accommodate both of these features. The acronym ppf stands for percent point function, which is another name for the quantile function. I can easily generate a random number along a gaussian normal probability distribution.
Skewt distributions incorporate both skew and heavy tails. Pdf the normal inverse gaussian nig distribution is a recent flexible closed form distribution that may be applied as a model of heavytailed. You know, generate a uniformly distributed random number, pass it to the inverse cdf of the desired distribution, and voila. Pdf we propose the class of normal inverse gaussian nig distributions to approximate an unknown risk neutral density. The nig distribution is completely specified by four real valued parameters that have natural interpretations in terms of the shape of the resulting probability density function. The cumulative distribution function is defined as where is the standard normal probability density function defined as follows. Normal inverse gaussian pdf, cdf, inverse cdf matlab release compatibility. Normal inverse gaussian model jonsson 2010 major reference. Pdf the normal inverse gaussian distribution and the. The nig distribution was noted by blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by ole barndorffnielsen, in the next year barndorffnielsen. Eberlein and keller 6 used a subfamily called the hyperbolic distributions to study. Ornsteinuhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more dimensions are considered in particular, and some.
The nig distribution was noted by blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by ole barndorffnielsen. The normal inverse gaussian distribution and the pricing of derivatives article pdf available in the journal of derivatives 163 august 2007 with 700 reads how we measure reads. May 01, 2017 anyway, i guess this function has many uses, but my main personal interest has always been in generating pseudo random numbers that follow a normal gaussian distribution. This distribution was introduced in the finance literature recently and used together with garch models in, for example, barndorffnielsen 1997, andersson 2001, and jensen and lunde 2001. The distribution will be initialized with the default random number generator. The scale scale keyword specifies the standard deviation. Pdf on extended normal inverse gaussian distribution. Density, distribution and quantile functions and random number generation are provided for the hyperbolic distribution, the generalized hyperbolic distribution, the generalized inverse gaussian distribution and the skewlaplace distribution. Modelbased clustering with the multivariate normal. I can easily generate a random number along a gaussiannormal probability distribution. Pdf the normal inverse gaussian distribution and the pricing of. From and it can be concluded thatthen, the process applied to is repeated to convert coefficients of into fractions.
In the field of finance, distributions of logarithmic asset returns can often be fitted extremely well by normal inverse gaussian distributions barndorffnielsen. Inverse normal inverse gaussian distribution the standard form involves the shape parameter \\mu\ in most definitions, \l0. The normal inverse gaussian nig distribution is a recent flexible closed form distribution that may be applied as a model of heavytailed processes. Statistics and probability letters fractional normal inverse. American option pricing using garch models and the normal. Normal inverse gaussian distribution file exchange. A monte carlo method for the normal inverse gaussian. The nig distribution was noted by blaesild in 1977 as a subclass of the generalised hyperbolic distribution discovered by ole barndorffnielsen, in the next year barndorffnielsen published the. Normal inverse gaussian distributions and stochastic volatility. A random variable x follows a normal, inverse gaussian distribution with parameter vector cl, 6, cy,0, in symbolic. The nig distribution was noted by blaesild in 1977 as a subclass of the generalised hyperbolic di. Ornsteinuhlenbeck type processes, superpositions of such processes and stochastic volatility models in one and more.
Estimating its parameters using bayesian inference and conjugate priors is also widely used. The normal inverse gaussian distribution is defined as a variancemean mixture of a normal distribution with the inverse gaussian as the mixing distribution. At the moment, the normal inverse gaussian nig distribution is not included in the statistics toolbox. Pdf introduction to the inverse gaussian distribution. A normal inverse gaussian continuous random variable. The family of nig distributions is a special case of the generalized hyperbolic distributions barndor.
Do october 10, 2008 a vectorvalued random variable x x1 xn t is said to have a multivariate normal or gaussian distribution with mean. Normalinversewishart distribution wikimili, the free. Inverse gaussian distribution definition in plain english. Calculates the nig pdf, cdf and inverse cdf with four given parameters alpha, beta, mu, delta. The multivariate normal inverse gaussian distribution. Normalinverse gaussian distribution wikimili, the free. In a riskneutral setting the application in a bs setting for the valuation of insurance products is tested. The normal inverse gaussian distribution is defined as a variance. The equation for the standard normal distribution is. The normal inverse gaussian distribution nig is a continuous probability distribution that is defined as the normal variancemean mixture where the mixing density is the inverse gaussian distribution. Normsinv mentioned in a comment is the inverse of the cdf of the standard normal distribution. The invniginitisl function produces an appropriate initial guess for the newtons method.
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