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Studies in Nonlinear Dynamics & Econometrics

Ed. by Mizrach, Bruce

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Volume 17, Issue 2


Empirical analysis of ARMA-GARCH models in market risk estimation on high-frequency US data

Alexander Beck / Young Shin Aaron Kim / Svetlozar Rachev / Michael Feindt / Frank Fabozzi
Published Online: 2013-04-11 | DOI: https://doi.org/10.1515/snde-2012-0033


In this paper, we examine the S&P 500 index log-returns on short intraday time scales with three different ARMA-GARCH models. In order to forecast market risk, we describe the innovation process with tempered stable distributions which we compare to commonly used methods in financial modeling. Value-at-risk backtests are provided where we find that models based on the tempered stable innovation assumption significantly outperform traditional models in forecasting risk on short time-scales. In addition to value-at-risk, the idiosyncratic differences in average value-at-risk are compared between the models.

This article offers supplementary material which is provided at the end of the article.

Keywords: tempered stable distribution; ARMA-GARCH model; average value-at-risk (AVaR); high-frequency


  • Andersen, T. G., and T. Bollerslev. 1997. “Intraday Periodicity and Volatility Persistence in financial markets.” Journal of Empirical Finance 4(2–3): 115–158.Google Scholar

  • Basel Committee on Banking Supervision. 2006. International convergence of capital measurements and capital standards.Google Scholar

  • Berkowitz, J. 2001. “Testing Density Forecasts, with Applications to Risk Management.” Journal of Business and Economic Statistics 19(4): 465–474.Google Scholar

  • Bollerslev, T. 1986. “Generalized Autoregressive Conditional Heteroskedasticity.” Journal of Econometrics 31(3): 307–327.CrossrefWeb of ScienceGoogle Scholar

  • Bollerslev, T., R. Y. Chou, and K. F. Kroner. 1992. “Arch Modeling in Finance: A Review of the Theory and Empirical Evidence.” Journal of Econometrics 52: 5–59.CrossrefGoogle Scholar

  • Bollerslev, T., J. Litvinova, and G. E. Tauchen. 2006. “Leverage and Volatility Feedback Effects in High-Frequency Data.” Journal of Financial Econmetrics 4(3): 354–384.Google Scholar

  • Boyarchenko, S., and S. Levendorskĭ. 2000. “Option Pricing for Truncated Levy processes.” International Journal of Theoretical and Applied Finance 3: 546–552.Google Scholar

  • Carr, P., H. Geman, D. B. Madan, and M. Yor. 2002. “The Fine Structure of Asset Returns: An Empirical Investigation.” Journal of Business 75(2): 305–332.CrossrefGoogle Scholar

  • Christoffersen, P. F. 1998. “Evaluating Interval Forecasts.” International Economic Review 39(4): 841–862.Google Scholar

  • Engle, R. F. 1982. “Autoregressive Conditional Heteroscedasticity with Estimates of the Variance of United Kingdom inflation.” Econometrica 50(4): 987–1007.CrossrefGoogle Scholar

  • Kim, Y., S. T. Rachev, D. M. Chung, and M. L. Bianchi. 2009. “The Modified Tempered Stable Distribution, GARCH-Models and Option Pricing.” Probability and Mathematical Statistics 29: 91–117.Google Scholar

  • Kim, Y. S., S. T. Rachev, M. L. Bianchi, and F. J. Fabozzi. 2010a. “Computing VaR and AVaR in Infinitely Divisible Distributions.” Probability and Mathematical Statistics 30: 223–245.Google Scholar

  • Kim, Y. S., S. T. Rachev, M. L. Bianchi, and F. J. Fabozzi. 2010b. “Tempered Stable and Tempered Infinitely Divisible GARCH Models.” Journal of Banking and Finance 34(9): 2096–2109.Web of ScienceGoogle Scholar

  • Kim, Y. S., S. T. Rachev, M. L. Bianchi, I. Mitov, and F. J. Fabozzi. 2011. “Time Series Analysis for Financial Market Meltdowns.” Journal of Banking & Finance 35: 1879–1891.Web of ScienceGoogle Scholar

  • Koponen, I. 1995. “Analytic Approach to the Problem of Convergence of Truncated L\u0115vy Flights Towards the Gaussian Stochastic Process.” Phys. Rev. E 52(1): 1197–1199.Google Scholar

  • Kupiec, P. H. 1995. “Techniques for Verifying the Accuracy of Risk Measurement Models.” The Journal of Derivatives 3(2): 73–84.CrossrefGoogle Scholar

  • Lee, J., T. S. Kim, and H. K. Lee. 2011. “Return-Volatility Relationship in High Frequency Data: Multiscale Horizon Dependency.” Studies in Nonlinear Dynamics & Econometrics 15(1). Available at: URL http://www.bepress.com/snde/vol15/iss1/art3.

  • Mackenzie, M. 2009. “High-Frequency Trading Under Scrutiny.” Financial Times, July 28.Google Scholar

  • Mandelbrot, B. 1963. “The Variation of Certain Speculative Prices.” Journal of Business 36: 394–419.CrossrefGoogle Scholar

  • Rachev, S. T., C. Menn, and F. J. Fabozzi. 2005. “Fat-Tailed and Skewed Asset Return Distributions.” Wiley Finance.Google Scholar

  • Sun, W., S. Rachev, Y. Chen, and F. J. Fabozzi. 2008a. “Measuring Intra-Daily Market Risk: A Neural Network Approach.” Technical report, Karlsruhe Institute of Technology (KIT).Google Scholar

  • Sun, W., S. Z. Rachev, and F. J. Fabozzi. 2008b. “Long-Range Dependence, Fractal Processes, and intra-daily data.” In: Handbook on Information Technology in Finance, edited by P. Bernus, J. Baewics, G. Schmidt, M. Shaw, D. Seese, C. Weinhardt, and F. Schlottmann, 543–585. Tiergartenstrasse, Heidelberg: Springer Berlin Heidelberg.Google Scholar

About the article

Corresponding author: Alexander Beck, Karlsruhe Institute of Technology, Karlsruher Str 88, 76139 Karlsruhe, Germany

Published Online: 2013-04-11

High-frequency in the context of this paper is used to address time-scales ranging from 75 s to 5 min in order to clearly distinguish it from time-series based on daily recorded data. It is not to be confused with high-frequency in the context of sub-millisecond trading. Very high-frequency time-series exhibit phenomena such as long-range dependency. Those effects cannot be captured with the ARMA-GARCH approach and as a consequence, different models, such as fractional Brownian motion or neural network approaches, have to be employed [see Sun, Rachev, and Fabozzi (2008)].

Moreover, many studies have shown that the normal distribution hypothesis for most financial assets traded is not an adequate model to describe observed return distributions. This was first reported by Mandelbrot (1963). A summary of studies is provided in Rachev, Menn, and Fabozzi (2005).

Some dates are excluded from the study where either the data were completely missing or the dataset is affected by large gaps. The exclusion criteria for 1 day are fulfilled if either more than 3% of the values are missing or if gaps occur which exceed 1% of the number of values for 1 day.

N=10 is chosen such that a time horizon of two business weeks is used to forecast the volatility. τ=0.1 has been chosen such that the 10th days till has a non-vanishing influence on the volatility.

The period has been chosen under the constraint that the dataset contains no missing days.

In order to conserve space, we restricted the analysis to only two time-series on different time-scales.

The CLR test is an enhancement of the Kupiec test [see Kupiec (1995)].

Citation Information: Studies in Nonlinear Dynamics and Econometrics, Volume 17, Issue 2, Pages 167–177, ISSN (Online) 1558-3708, ISSN (Print) 1081-1826, DOI: https://doi.org/10.1515/snde-2012-0033.

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