Int. J. Dynam. Control DOI 10.1007/s40435-017-0351-5 Volatility modeling with COGARCH(1,1) driven by Meixner-Lévy process: an application to Tokyo stock exchange market (Nikkei225) Yavuz Yıldırım 1 · Gazanfer Ünal 1 Received: 21 July 2017 / Revised: 1 September 2017 / Accepted: 6 September 2017 © Springer-Verlag GmbH Germany 2017 Abstract In this paper, the authors propose a new outlook to model Tokyo stock exchange market’s volatility (Nikkei225) in the concept of continuous-time GARCH(1,1) driven by Meixner Lévy Process for the first time. GARCH(1,1) model is estimated as an appropriate discrete-time model for the timeseries, then Meixner Lévy process driven continuous- time model is developed to analyze the volatility charac- teristics of standardized log returns of Nikkei225. Pseudo Maximum Likelihood (PML), is employed as the parameter estimation process for the Meixner-COGARCH(1,1) model. The data covers the period from 2005 to 2013. The empir- ical results show that continuous-time GARCH(1,1) driven by a Meixner Lévy process successfully captures volatility clustering and heavy-tail behaviour of Nikkei225. Keywords Lévy process · Meixner process · Tokyo stock exchange market · GARCH · COGARCH · Pseudo maximum likelihood method · Nikkei225 1 Introduction Volatility is considered as an important concept for many eco- nomic and financial applications. Volatility is widely used for asset pricing, risk management assessment, general portfolio management and also volatility is a key parameter for pricing financial derivatives. Volatility changes can have a positive or negative potential impact on investments. A good estimation B Gazanfer Ünal gunal@yeditepe.edu.tr Yavuz Yıldırım yavuz.yildirim@oxfordbrookes.net 1 Department of International Finance, Yeditepe University, Istanbul, Turkey of the volatility can therefore provide a substantial advantage to the investors and financial institutions. There are handful of models in the literature which deals with the volatility modeling of financial time series. Discrete-time GARCH models have become very pop- ular in financial economics because of their capability of capturing heavy-taillness, correlation and persistency. How- ever, it is unable to capture the jumps in volatility. To resolve the limitations of the discrete-time models, there have been many studies that consider continuous-time stochastic pro- cess to model volatility [2, 7–9, 19, 24]. In 2004, Klüppelberg, Lindner, and Maller (KLM) [24] introduced the COGARCH model as a continuous-time analogue to the enormously influ- ential and successful discrete-time GARCH model. Like the GARCH model, the COGARCH is based on a single source of random variability on a single background driving Lévy process, and generalises the essential features of the discrete- time GARCH process in a natural way. It is well-known that the log-returns of most financial assets have an actual kurtosis that is higher than the nor- mal distribution. In this paper we therefore propose a model which is based on the Meixner distribution. The Meixner distribution belongs to the class of the infinitely divisible dis- tributions which rise to a Meixner Lévy process. The Meixner process is very flexible, has a simple structure and leads to analytically and numerically tractable formulas. It was intro- duced by Schoutens and Teugels in 1998 [35] and originates from the theory of orthogonal polynomials and was proposed to serve as a model of financial data by Grigelionis [17]. The advantage of the Meixner model over the other Lévy models is that all crucial formulas are explicitly given, so that it is not depending on computationally demanding numerical inver- sion procedures. This numerical advantage can be important, when a big number of prices or related quantities has to be computed simultaneously. 123