Information Technology and Management Science 93 ISSN 2255-9094 (online) ISSN 2255-9086 (print) December 2018, vol. 21, pp. 93–97 doi: 10.7250/itms-2018-0015 https://itms-journals.rtu.lv ©2018 Oskars Rubenis, Andrejs Matvejevs. This is an open access article licensed under the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0). Increments of Normal Inverse Gaussian Process as Logarithmic Returns of Stock Price Oskars Rubenis 1 , Andrejs Matvejevs 2 1, 2 Riga Technical University, Riga, Latvia Abstract – Normal inverse Gaussian (NIG) distribution is quite a new distribution introduced in 1997. This is distribution, which describes evolution of NIG process. It appears that in many cases NIG distribution describes log-returns of stock prices with a high accuracy. Unlike normal distribution, it has higher kurtosis, which is necessary to fit many historical returns. This gives the opportunity to construct precise algorithms for hedging risks of options. The aim of the present research is to evaluate how well NIG distribution can reproduce stock price dynamics and to illuminate future fields of application. Keywords – Normal inverse Gaussian distribution, normal inverse Gaussian process, log-returns, maximum likelihood estimation. I. INTRODUCTION There has been a long quest for chasing the best distribution to describe returns of stock price time series. NIG distribution was introduced to finance society in 1997. From that moment on it has been exercised to underlie many characteristics of stock price dynamics. With higher kurtoses than Gaussian distribution NIG is applicable to many stock price time series data. We have taken stock market data used in “Irrational Exuberance” [1]. Out of given time series, spot price historical development is loaded into R. From given spot prices, an empirical distribution is constructed. As an approximate distribution NIG is chosen [2]–[8]. The corresponding NIG parameters are evaluated by applying maximum likelihood estimation algorithm. The obtained parameters are used to generate NIG random numbers and NIG process. NIG random values are used to construct distribution, which corresponds to distribution of returns of historical stock prices. Afterwards, simulated distribution is compared with empirical distribution. II. NOMENCLATURE S spot price of stock log logarithm NIG normal inverse Gaussian Cdf cumulative distribution function Qdf quantile density function L likelihood function f probability density function MLE maximum likelihood estimation L-BFGS-B limited-memory Broyden–Fletcher– Goldfarb–Shanno algorithm KS test Kolmogorov-Smirnov test III. EQUATIONS A. Log Returns Logarithmic return at time t: X(t) = log � () (−1) � is chosen to investigate characteristics of stock price dynamics. It will be used to construct empirical histogram. The NIG approximation will be applied. B. NIG Distribution Probability density function of normal inverse Gaussian distribution is defined by the following equation: ()=  1 �� 2 +(−) 2 � � 2 +(−) 2  +(−) , where  1 – the modified Bessel function of third order and index 1;  – a location parameter; Δ – a scale parameter;  – a tail heaviness parameter;  – an asymmetry parameter;  = � 2 − 2 . NIG distribution has higher kurtosis than normal distribution. It is very critical for many cases to accurately describe stock price dynamics [3], [4]. IV. NIG PROCESS NIG process X NIG = �  () , ≥ 0� which starts at 0 and has independent, stationary increments. The entire process is governed by NIG(, , , ) distribution. The aim is to simulate stock price dynamics. The internal properties for time series are described by NIG parameters [2], [5]. A. Maximum Likelihood Estimation As the method to determine the parameters MLE method is chosen. When parameters for NIG are evaluated, it is possible to estimate how well NIG process simulates particular stock price dynamics. The backbone of maximum likelihood estimation can be described in the following way: We have a sample { 1 ,  2 ,…   } with corresponding probability density function (  ; ), where  is a vector of parameters.