Estimación del VaR mediante un modelo condicional multivariado bajo la hipótesis α-estable sub-Gaussiana: A conditional approach to VaR with multivariate α-stable sub-Gaussian distributions

Ramona Serrano-Bautista; Leovardo Mata-Mata

doi:10.29105/ensayos37.1-2

Authors

Ramona Serrano-Bautista Tecnológico de Monterrey
Leovardo Mata-Mata Tecnológico de Monterrey

DOI:

https://doi.org/10.29105/ensayos37.1-2

Keywords:

α-stable Sub-Gaussian distribution, multivariate stable Sub-Gaussian GARCH model, Value at Risk

Abstract

The purpose of this investigation is to propose a multivariate volatility model that takes into consideration time varying volatility and the property of the α-stable sub-Gaussian distribution to model heavy tails. The principal assumption is that returns follow a sub-Gaussian distribution, which is a particular multivariate stable distribution. The proposed GARCH model is applied to a Value at Risk (VAR) estimation of a portfolio composed by 5 companies listed in the Mexican Stock Exchange Index (IPC) and compared with the one obtained using the normal multivariate distribution, t-Student and Cauchy. In particular, we examine performances during the financial crisis of 2008.

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Author Biographies

Ramona Serrano-Bautista, Tecnológico de Monterrey

Tecnológico de Monterrey, Guadalajara Av. General Ramón Corona 2514 Nuevo México, 45201 Zapopan, Jal., México.

Leovardo Mata-Mata, Tecnológico de Monterrey

Tecnológico de Monterrey, Estado de México.

References

Barndorff-Nielsen, O.E., Mikosch, T. y Resnick, S.I. (2012). Lévy Processes: Theory and Applications, Springer Science & Business Media, New York, NY.

Bollerslev, T. (1986). Generalized autoregressive conditional heteroskedasticity. Journal of Econometrics, 31, 307–327.

Bonato, M. (2012). Modeling fat tails in stock returns: A multivariate stable-GARCH approach. Computational Statistics, 27(3), 499–521.

Byczkowski, T., Nolan, J.P. y Rajput, B. (1993). Approximation of Multidimensional Stable Densities. Journal of Multivariate Analysis, 46(1), 13–31. Available at: http://linkinghub.elsevier.com/retrieve/pii/S0047259X83710444.

Cheng, B., y Rachev, S. (1995). Multivariate stable futures prices. Journal of Mathematical Finance, 5, 133-153.

Curto, J.D., Pinto, J.C. y Tavares, G.N. (2009). Modeling stock markets’ volatility using GARCH models with Normal, Student’s t and stable Paretian distributions. Statistical Papers, 50(2), 311–321.

Engle, R.F. (1982). Autoregressive Conditional Heteroskedasticity with Estimates of the Variance of United Kingdom Inflation. Econometría, 50(4), 987–1007.

Engle, R. (2002). Dynamic Conditional Correlation. Journal of Business & Economic Statistics, 20(3), 339–350. Available at: http://www.tandfonline.com/doi/abs/10.1198/073500102288618487#.VyvCUaCAdvE.mendeley.

Fama, E.F. (1965). The Behavior of Stock-Market Prices. The Journal of Business, 38(1), 34–105.

Feldheim, E. (1937). Etude de la stabilité des lois de probabilité. Ph. D. thesis, Faculté des Sciences de Paris, Paris, France.

Khindanova, I., Rachev, S. y Schwartz, E. (2001). Stable Modeling of Value at Risk. Mathematical and Computer Modelling, 34, 1223–1259.

Kring, S., Rachev, S., Markus, H. y Fabozzi, F. (2009). Estimation of α-Stable Sub-Gaussian Distributions.pdf. In Risk Assessment Decisions in Banking and FinanceDecisions in Banking and Finance. 111–152.

Kupiec, P.H. (1995). Techniques for verifying the accuracy of risk measurement models. Journal of Derivatives, 3(2), 73-84.

Mandelbrot. (1963). The variation of certain speculative prices. The Journal of Business, 36, 394–419.

McCulloch, J.H. (1986). Simple consistent estimators of stable distribution parameters. Communications in Statistics - Simulation and Computation, 15, 1109–1136.

Mcculloch, J.H. (2000). Estimation of the Bivariate Stable Spectral Representation by the Projection Method. Computational Economics, 16(1–2), 47–62.

Mittnik, S., Doganoglu, T. y Chenyao, D. (1999). Computing the probability density function of the stable Paretian distribution. Mathematical and Computer Modelling, 29(10–12), 235–240.

Mittnik, S., Paolella, M.S. y Rachev, S.T. (2002). Stationarity of stable power-GARCH processes. Journal of Econometrics, 106, 97–107.

Mittnik, S. y Rachev, S.T. (1989). Stable distributions for asset returns. Applied Mathematics Letters, 2(3), 301–304.

Mittnik, S. y Rachev, S.T. (1993). Modeling asset returns with alternative stable distributions. Economics Reviews, 12, 261–330.

Mittnik, S., Rachev S., y Paolella, M. (1997). Stable Paretian Modelling in Finance: Some Empirical and Theorical Aspects, in R. Adler, R. E. Feldman and M.S. Taqqu (eds).

Modarres, R., y Nolan, J.P. (1994). A method for simulating stable random vectors. Computational Statistics, 9, 11-19.

Mohammadi, M. (2017). Prediction of -stable GARCH and ARMA-GARCH-M models. Journal of Forecasting, 36, 859–866.

Naka, A. y Oral, E. (2013). With Stable Paretian GARCH. Journal of Business & Economics Research, 11(1), 47–53.

Nolan, J.P. (1997). Numerical calculation of stable densities and distribution functions. Commun. Statist. Stochastic Models, 13, 759-774.

Nolan, J. (1999). Fitting data and assessing goodness-of-fit with stable distributions. Unpublished Manuscript. Washington, DC, 2(1924). Available at: http://academic2.american.edu/~jpnolan/stable/DataAnalysis.pdf.

Nolan, J.P. (2001). Maximum Likelihood Estimation and Diagnostics for Stable Distributions. , (January 2001). Available at: http://link.springer.com/10.1007/978-1-4612-0197-7.

Nolan, J.P. (2013). Multivariate elliptically contoured stable distributions: Theory and estimation. Computational Statistics, 28(5), 2067–2089.

Nolan, J.P., Panorska, A.K. y McCulloch, J.H. (2001). Estimation of stable spectral measures. Mathematical and Computer Modelling, 34(9–11), 1113–1122.

Nolan, J.P., y Rajput, B. (1997). Calculation of multidimensional stable densities. Commun. Statist. Simula, 24, 551-556.

Panorska, A., Mittnik, S., and Rachev, S.T., 1995. Stable GARCH Models for Financial Time Series. Applied Mathematics Letters, 8(5), 33–37.

Press, S.J., (1972). Estimation in Univariate and Multivariate Stable Distributions. Journal of the American Statistical Association, 67(340), 842–846.

Rachev, S.T, y Mittnik, S., (2000). Stable Paretian Models in Finance. New York, NY: Wiley.

Rachev, S. y Han, S., (2000). Portfolio management with stable distributions. Mathematical Methods of Operation Research, 51, 341–352.

Samorodnitsky, G., y Taqqu, M., 1994. Stable Non-Gaussian Random Processes. New York: Chapman and Hall.

Serrano y Mata, (2018). Valor en Riesgo mediante un modelo heterocedástico condicional α-estable. REMEF, 13(1), 1-25.