Normal Inverse Gaussian Distribution Research Articles

Modelling Electricity Swaps with Stochastic Forward Premium Models

We present a new model for pricing electricity swaps. Two general factors affect contracts but unique risk elements affect each contract. General factors are average swap prices and deterministic trend-seasonal components, and unique elements are forward premiums. Innovations follow MNIG distributions. We estimate the model with data from the European Energy Exchange. The model outperforms four competitors, both in in-sample valuation and in out-of-sample forecasting, and in fitting the term structure of volatilities by market segments. Competitor models are (i) diffusion spot prices, (ii) jump-diffusion spot prices with time dependent volatility, (iii) HJM-based and (iv) Levy multifactor model with NIG distributions. Value-at-Risk measures based on normality strongly underestimate tail risk but our model gives estimates that are more exact.Keywords: Electricity swaps; Stochastic forward premium; Multivariate Normal Inverse Gaussian distribution; Levy processes

Underlying Assets Distribution in Derivatives: The BRIC Case

This paper addresses one of the main issues regarding numerical derivatives valuation, particularly the search for an alternative to the normality assumption of underlying asset returns, to obtain the price by using numerical techniques. There might be difficulties in making normality assumptions, which could produce over-valuated or sub-valuated prices of derivatives. Under this consideration, the Generalized Hyperbolic family has been proven to be a proper selection to model heavy tailed distribution behavior. The Normal Inverse Gaussian (NIG) distribution is a member flexible enough to model financial returns. NIG distribution can be used to model distribution returns under different states of nature. The indexes of the Brazil, Russia, India and China (BRIC) economies were studied at different time-periods using return data series from 2002 to 2005, 2006 to 2010 and 2011 to 2015, in such a manner to demonstrate with statistical criteria that NIG fits the empirical distribution in the three periods; even throughout economic downturn. This result may be used as an improvement in derivatives valuation with indexes as underlying assets.

SINR Outage Evaluation: Saddle Point Approximation Using Normal Inverse Gaussian Distribution

Signal-to-noise-plus-interference ratio (SINR) outage probability is among one of the key performance metrics of a wireless network. In this paper, we propose a semi-analytical method based on the saddle point approximation (SPA) technique to calculate the SINR outage of a wireless system whose SINR can be modeled in the form $\left({\sum _{i=1}^{M} X_{i}}/ \left(1+\sum _{i=1}^{N} Y_{i}\right) \right)$ where $X_{i}$ denotes the useful signal power and $Y_{i}$ denotes the power of the interference signal. Both $M$ and $N$ can also be random variables. The proposed approach is based on the saddle point approximation to cumulative distribution function as given by Wood-Booth-Butler formula . The approach is applicable whenever the cumulant generating function of the received signal and interference exists, and it allows us to tackle distributions with large skewness and kurtosis with higher accuracy. In this paper, we exploit a four parameter normal-inverse Gaussian (NIG) distribution as a base distribution. Given that the skewness and kurtosis satisfy a specific condition, NIG-based SPA works reliably. When this condition is violated, we recommend SPA based on normal or symmetric NIG distribution, both special cases of NIG distribution, at the expense of reduced accuracy. For the purpose of demonstration, we apply SPA for the SINR outage evaluation of a typical user experiencing a downlink coordinated multi-point transmission from the base stations that are modeled by homogeneous Poisson point process. Numerical results are presented to illustrate the accuracy of the proposed set of approximations.

Discriminating between the normal inverse Gaussian and generalized hyperbolic skew-t distributions with a follow-up the stock exchange data

The statistical methods for the financial returns play a key role in measuring the goodness-of-fit of a given distribution to real data. As is well known, the normal inverse Gaussian (NIG) and generalized hyperbolic skew-t (GHST) distributions have been found to successfully describe the data of the returns from financial market. In this paper, we mainly consider the discrimination between these distributions. It is observed that the maximum likelihood estimators (MLEs) cannot be obtained in closed form. We propose to use the EM algorithm to compute the maximum likelihood estimators. The approximate confidence intervals of the unknown parameters have been constructed. We then perform a number of goodness-of-fit tests to compare the NIG and GHST distributions for the stock exchange data. Moreover, the Vuong type test, based on the Kullback-Leibler information criteria, has been considered to select the most appropriate candidate model. An important implication of the present study is that the GHST distribution function, in contrast to NIG distribution, may describe more appropriate for the proposed data.

Heavy-tailed distribution, GARCH models and the silver returns

After serving as a medium of exchange for the human society, silver is still widely used in our daily life. From the jewellery, electronic and electrical industries as well as medicine, optics, the power industry, automotive industry and many other industries, silver is still playing a very active role. In addition to the industrial usage, silver also serves as an investment tool for many financial institutions. Thus, it is crucial to develop effective quantitative risk management tool for those financial institutions. In this paper, we investigate the conditional heavy tails of daily silver spot returns under the GARCH framework. Our results indicate that that it is important to introduce heavy-tailed distributions to the GARCH framework and the normal reciprocal inverse Gaussian (NRIG) distribution, a newly-developed distribution, has the best empirical performance in capture the daily silver spot returns dynamics.

Conditional Heavy Tails, Volatility Clustering and Asset Prices of the Precious Metal

The world palladium demand has increased steadily and dramatically in the years of 21th century. However, its spot price still has not reached the peak level observed in January 17, 2001. In 2008, a single-day increase of the palladium spot price has exceeded 37%, which witnesses significant risk for investments in the world palladium market. In this paper, we apply the GARCH model with heavy-tailed distributions into the palladium spot returns series for risk management purpose. We compare empirical performance of the Student’s t distribution and the normal reciprocal inverse Gaussian (NRIG) distribution. Our results show the newlydeveloped distribution, the NRIG, cannot outperform the older fashion one, the Student’s t distribution. Nevertheless, our results do demonstrate that it is important to incorporate conditional heavy tails for precious metals’ spot return modelling.

GARCH Models with Fat-Tailed Distributions and the Hong Kong Stock Market Returns

As one of the world’s largest securities markets, the Hong Kong stock market plays a significant role in facilitating the development of Chinese economy. In this paper, we investigate a suite of widely-used models, the GARCH models in risk management of the Hong Kong stock market returns. To account for conditional volatilities, we consider a new type of fat-tailed distribution, the normal reciprocal inverse Gaussian distribution (NRIG), and compare its empirical performance with two other popular types of fat-tailed distribution, the Student’s t distribution and the normal inverse Gaussian distribution (NIG). We show that the NRIG distribution performs slightly better than the other two types of distribution. Also, our results indicate that it is important to introduce both GJR-terms and the NRIG distribution to improve the models’ performance. Our results illustrate that the asymmetric GARCH NRIG model has practical advantages in quantitative risk management, and serves as a very useful tool for industry participants.

Normal Reciprocal Inverse Gaussian Distribution and the Stock Market Returns in Japan

The Tokyo Stock Exchange (TSE) is the fourth largest stock exchange in the world by aggregate market capitalization of its listed companies and largest in East Asia and Asia. It is of great importance for those in charge of managing risk to understand how its market index returns are distributed. The goal of this paper is to examine how various types of heavy-tailed distribution perform in risk management of the N225 Index returns. We compared these heavy-tailed distributions through a variety of criteria. Our results indicate the generalized hyperbolic distribution has the best goodness of fit and generates most suitable risk measures.

Conditional Tail Risk Measures for Skewed Generalised Hyperbolic Family

This paper deals with the estimation of loss severity distribution arising from the historical data on univariate and multivariate losses. We present an innovative theoretical framework where the closed-form expression for the tail conditional expectation (TCE) is derived for the skewed general hyperbolic (GH) family of distributions. Skewed GH family is especially suitable for right-skewed positive random variables such as insurance losses. It allows to capture the asymmetry in the distribution of losses that tends to have a heavy right tail. As opposed to a widely used Value-at-Risk, TCE is a coherent risk measure, which takes into account the expected loss in the tail of the distribution. Our theoretical TCE results are verified for different distributions from the skewed GH family including its general cases: Student-t, variance gamma, normal inverse gaussian and hyperbolic distributions. GH family and its special cases turn out to provide excellent fit to univariate and multivariate data on Danish fire losses. TCE risk measure computed for the skewed family of GH distributions provides conservative estimators of risk, addressing the main challenge faced by insurance companies on how to reliably quantify the risk arising from the loss distribution. We extend our analysis to the multivariate framework when modelling portfolio of losses, allowing multivariate GH distribution to capture the combination of correlated risks and demonstrate how TCE of the portfolio can be decomposed into individual components, representing individual risk in the aggregate (portfolio) loss.

Empirical Performance of GARCH Models with Heavy-Tailed Innovations

We introduce a new type of heavy‐tailed distribution, the normal reciprocal inverse Gaussian distribution (NRIG), to the GARCH and Glosten‐Jagannathan‐Runkle (1993) GARCH models, and compare its empirical performance with two other popular types of heavy‐tailed distribution, the Student's t distribution and the normal inverse Gaussian distribution (NIG), using a variety of asset return series. Our results illustrate that there is no overwhelmingly dominant distribution in fitting the data under the GARCH framework, although the NRIG distribution performs slightly better than the other two types of distribution. For market indexes series, it is important to introduce both GJR‐terms and the NRIG distribution to improve the models’ performance, but it is ambiguous for individual stock prices series. Our results also show the GJR‐GARCH NRIG model has practical advantages in quantitative risk management. Finally, the convergence of numerical solutions in maximum‐likelihood estimation of GARCH and GJR‐GARCH models with the three types of heavy‐tailed distribution is investigated.

Heavy-Tailed Distributions, Volatility Clustering and Asset Prices of the Precious Metal

In the years of 21st century, the world palladium demand has increased steadily and dramatically. However, its spot price still has not reached the peak level observed in January 17, 2001. In 2008, a single-day increase of the palladium spot price has exceeded 37%, which witnesses significant risk for investments in the world palladium market. In this paper, we apply the GARCH model with heavy-tailed distributions into the palladium spot returns series for risk management purpose. We compare empirical performance of the Student’s t distribution and the normal reciprocal inverse Gaussian (NRIG) distribution. Our results show the newly-developed distribution, the NRIG, cannot outperform the older fashion one, the Student’s t distribution. Nevertheless, our results do demonstrate that it is important to incorporate conditional heavy tails for precious metals’ spot return modelling.

GARCH Models with a New Class of Heavy-Tailed Distribution for the Silver Returns

After serving as a medium of exchange for the human society, silver is still widely used in our daily life. From the jewelry, electronic and electrical industries as well as medicine, optics, the power industry, automotive industry and many other industries, silver is still playing a very active role. In addition to the industrial usage, silver also serves as an investment tool for many financial institutions. Thus, it is crucial to develop effective quantitative risk management tool for those financial institutions. In this paper, we investigate the conditional heavy tails of daily silver spot returns under the GARCH framework. Our results indicate that that it is important to introduce heavy-tailed distributions to the GARCH framework and the normal reciprocal inverse Gaussian (NRIG) distribution, a newly-developed distribution, has the best empirical performance in capture the daily silver spot returns dynamics.

A Statistical Analysis of Cryptocurrencies

We analyze statistical properties of the largest cryptocurrencies (determined by market capitalization), of which Bitcoin is the most prominent example. We characterize their exchange rates versus the U.S. Dollar by fitting parametric distributions to them. It is shown that returns are clearly non-normal, however, no single distribution fits well jointly to all the cryptocurrencies analysed. We find that for the most popular currencies, such as Bitcoin and Litecoin, the generalized hyperbolic distribution gives the best fit, while for the smaller cryptocurrencies the normal inverse Gaussian distribution, generalized t distribution, and Laplace distribution give good fits. The results are important for investment and risk management purposes.

A Statistical Analysis of Cryptocurrencies

We show the statistical properties of the most important cryptocurrencies, of which Bitcoin is the most prominent example. We characterize their exchange rates versus the US Dollar by fitting parametric distributions to them. It is shown that returns are clearly non-normal, however, no single distribution fits well jointly to all the cryptocurrencies analysed. We find that for the most popular currencies, such as Bitcoin and Litecoin, the Generalized hyperbolic distribution gives the best fit, whilst for the smaller cryptocurrencies (determined by market capitalization) the Normal inverse Gaussian distribution, Generalized t distribution, and Laplace distributions give good fits. The results are important for investment and risk management purposes.

Heavy-Tailed Distributions and Risk Management of Equity Market Tail Events

Traditional econometric modelling typically follows the idea that market returns follow a normal distribution. However, the concept of tail risk indicates that the distribution of returns is not normal, but skewed and has heavy tails. Thus, a heavy-tailed distribution, which accurately estimates the tail risk, would significantly improve quantitative risk management practice. In this paper, we compare four widely used heavy-tailed distributions using the S&P 500 daily returns. Our results indicate that the Skewed t distribution in Hansen (1994) has the superior empirical performance compared with the Student’s t distribution, the normal reciprocal inverse Gaussian distribution and the generalized hyperbolic distribution. We further showed the Skewed t distribution could generate the VaR estimates closest to the nonparametric historical VaR estimates compared with other heavy-tailed distributions.

A Markov-switching regression model with non-Gaussian innovations: estimation and testing

AbstractIn this paper we propose a very general multivariate Markov-switching regression (MSR) model considering the normal inverse Gaussian (NIG) distribution as conditional form of financial returns and model innovations. It is indeed well-known that the Gaussian distribution is not able to capture many stylized facts of the return series such as skewness, excess kurtosis and heavy tails. Through a large simulation study and an empirical analysis of the US stock market, we show that a NIG-based MSR model allows to adequately account for both skewness and fat tails in the data and, according to model selection criteria, is the best overall model in the majority of the cases considered, even preferred over other popular distributional assumptions such as Student-

Characteristic function-based inference for GARCH models with heavy-tailed innovations

ABSTRACTWe consider estimation and goodness-of-fit tests in GARCH models with innovations following a heavy-tailed and possibly asymmetric distribution. Although the method is fairly general and applies to GARCH models with arbitrary innovation distribution, we consider as special instances the stable Paretian, the variance gamma, and the normal inverse Gaussian distribution. Exploiting the simple structure of the characteristic function of these distributions, we propose minimum distance estimation based on the empirical characteristic function of properly standardized GARCH-residuals. The finite-sample results presented facilitate comparison with existing methods, while the new procedures are also applied to real data from the financial market.

Normal inverse Gaussian 분포에서 모수추정의 보정 방법 연구

금융자산의 수익률 분포를 잘 설명할 수 있는 것으로 알려진 normal inverse Gaussian(NIG)분포는 모수의 조건에 의해 세 배의 초과첨도가 왜도 제곱의 5배보다 커야 하는데, 만약 관측된 초과첨도와 왜도의 관계가 이를 만족하지 못하거나 두 값이 매우 비슷하다면 모수를 안정적으로 추정하기 어렵게 된다. 이 논문에서 우리는 NIG분포의 모수추정에서 발생하는 이러한 문제점을 살펴보고 모의실험을 통해 이를 보정하는 방법을 찾아보았다. KOSPI, S&P500, FTSE와 HANG SENG의 실제 주가지수 자료에 적용하여 보정의 효과를 비교하고 VaR를 이용한 사후검증으로 보정된 추정방법의 성능을 평가해 보았다. 보정 방법을 이용하였을 때, 모수추정의 문제가 있던 구간을 포함한 모든 구간에서 안정적인 모수추정이 가능하였고 VaR를 통한 사후 검증에서도 분포의 성능이 떨어지지 않음을 확인하였다. Numerous studies have shown that normal inverse Gaussian (NIG) distribution adequately fits the empirical return distribution of financial securities. The estimation of parameters can also be done relatively easily, which makes the NIG distribution more useful in financial markets. The maximum likelihood estimation and the method of moments estimation are easy to implement; however, we may encounter a problem in practice when a relationship among the moments is violated. In this paper, we investigate this problem in the parameter estimation and try to find a simple solution through simulations. We examine the effect of our adjusted estimation method with real data: daily log returns of KOSPI, S&P500, FTSE and HANG SENG. We also checked the performance of our method by computing the value at risk of daily log return data. The results show that our method improves the stability of parameter estimation, while it retains a comparable performance in goodness-of-fit.

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.

Option pricing and hedging for optimized Lévy driven stochastic volatility models

This paper pays attention to Ornstein-Uhlenbeck (OU) based stochastic volatility models with marginal law given by Classical Tempered Stable (CTS) distribution and Normal Inverse Gaussian (NIG) distribution, which are subclasses of infinite activity Lévy processes and are compared to finite activity Barndorff-Nielsen and Shephard (BNS) model. They are applied to option pricing and hedging in capturing leptokurtic features in asset returns and clustering effect in volatility that are consistently observed phenomena in underlying asset dynamics. The analytical formula of option pricing can be obtained through use of characteristic functions and Fast Fourier Transform (FFT) technique. Additionally, we introduce two hybrid optimization techniques such as hybrid Particle Swarm optimization (PSO) algorithm and hybrid Differential Evolution (DE) algorithm into parameters calibration schemes to improve the calibration quality for newly constructed models. Finally, we conduct experiments on Chinese emerging option markets to examine the performance of proposed models exploiting hybrid optimization techniques.