Modeling Financial return with Markov Time-Varying Mixed Normal GARCH Model

Subject Areas : Financial Knowledge of Securities Analysis

Shirin Alipour ¹ , Fatemeh Azizzadeh ² , Khosro Manteghi ³

1 - M.Sc in Financial Engineering, Department of Financial Sciences, Kharazmi University. Tehran, Iran
2 - Faculty member at Department of Financial Sciences, Kharazmi University. Tehran, Iran
3 - Faculty member at Department of Financial Sciences, Kharazmi University. Tehran, Iran

Received: 2015-08-23 Accepted : 2015-10-12 Published : 2018-03-21

Keywords: Bayesian inference, Markov process, GARCH composite models, volatility, financial return,

Abstract :

In previous studies, the normal mixture, as well as the Markov process, were used to model the financial return, separately. In this study, the normal mixture model is extended to the Markov mixture of normals. The mixture weights in every state are considered time-varying and as a function of past observations, so the limit of constant weight assumption is removed. The proposed model is estimated using Bayesian inference and a Gibbs sampling algorithm has been created to compute posterior density. The performance of algorithm is tested with simulation, then a two-state Markov time-varying Mixed Normal-GARCH model (MMN) with one and two components in every state, as well as limited cases (mean zero), were compared by comparison of their likelihood function. Finally, the model is applied to S&P500 and TEPIX daily return and results show that MMN models with two components provide better results than MMN model with one component which is so-called Markov switching GARCH model.

References:

*      آل محمد، نفیسه؛ پایان‌نامه با عنوان "مدل‌های آمیخته فرایندهای آرچ با ضریب متغیر نسبت به زمان"،  دانشکده آمار دانشگاه صنعتی امیرکبیر، 1391.

*      پارسیان، احمد (1380)؛ آمار ریاضی: انتشارات دانشگاه شیراز.

*      پاکیزه، کامران (1389)؛ "تلاطم و بازده (شواهدی از بورس اوراق بهادار تهران و بورس های بین‌الملل)"، فصلنامه تحقیقات مدل‌سازی اقتصادی، زمستان 89، شماره 2.

*      حبیبی فرد، نفیسه؛ پایان‌نامه با عنوان "مقایسه مدل گزینی بیزی بر اساس روش MCMC و کاربرد آن در سری‌های زمانی مالی (مدل گارچ) "، دانشکده اقتصاد دانشگاه علامه طباطبایی، 1390.

*      کشاورز حداد، غلامرضا؛ صمدی، باقر (1388)؛ "برآورد و پیش‌بینی تلاطم بازدهی در بازار سهام تهران و مقایسه دقت روش‌ها در تخمین ارزش در معرض خطر: کاربردی از مدل‌های خانواده FI گارچ"، مجله تحقیقات اقتصادی، بهار 88، شماره 86، صفحه 193-235.

*      نیسی، عبدالساده؛ چمنی انباجی، رویا؛ شجاعی منش، لیلی (1391)؛ "سه مدل اساسی در ریاضیات مالی"، مجله مدل‌سازی پیشرفته ریاضی، دوره 2، شماره 1.

*      Alexander, C., and E. Lazar (2004b); “The equity index skew, market crashes and asymmetric normal mixture GARCH.” ISMA Centre Discussion Papers in Finance, pp. 2004-14.

*      Bauwens, Luc; Rombouts, Jeroen V.K. (2005); “Bayesian inference for the mixed conditional heteroskedasticity model”, Les Cahiers du CREF, CREF 05-08.

*      Bauwens, Luc; Preminger, Arie; Rombouts, Jeroen V.K. (2007); “Theory and Inference for a Markov-Switching GARCH Model”, Cahier de recherche/Working Paper 07-33.

*      Brooks, Stephen P.; Roberts, Gareth O. (1998); “Convergence assessment techniques for Markov chain Monte Carlo”, Statistics and Computing, Vol. 8, 319-335.

*      Bollerslev, T. (1987); “A conditionally heteroskedastic time series model for speculative prices and rates of return. ” The review of economics and statistics, pp. 542-547.

*      Chang, George (2006); “Bayesian Markov mixture of normals approach to modeling financial returns”, Studies in Economics and Finance, Vol. 23, No. 2, pp. 141-158.

*      Cont, Ramo (2001); “Empirical properties of asset returns: stylized facts and statistical issues”, Quantitative Finance, Vol. 1, pp.223–236.

*      Geweke, John; Krauseb, Jochen; Amisano, Giovanni (2007); “Hierarchical Markov Normal Mixture Models with Applications to Financial Asset Returns”

*      Haas, Markus; Mittnik, Stefan; Paolellab, Marc S. (2002); “Mixed Normal Conditional Heteroskedasticity”, Center for Financial Studies, No. 10.

*      Haas, Markus; Krauseb, Jochen; Paolellab, Marc S.; Steude, Sven C. (2013); “Time-varying Mixture GARCH Models and Asymmetric Volatility”, QBER discussion paper, No. 2.

*      Tsay, Ruey S. (2010); Analysis of Financial Time Series, Third Edition, John Wiley & Sons, Inc., Hoboken, New Jersey.

*      Wang, K.L., C. Fawson, C.B. Barrett, and J.B. McDonald. (2001); “A flexible parametric GARCH model with an application to exchange rates.” Journal of Applied Econometrics16:521-536.

*      Wang, Jinrui (2014); Modelling Ontario Agricultural Commodity Price Volatility with Mixtures of GARCH Processes, A Thesis presented to The University of Guelph