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Modeling of financial data: Comparison of the truncated Lévy flight and the ARCH(1) and GARCH(1,1) processes

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  • Mantegna, Rosario N.
  • Stanley, H.Eugene

Abstract

We compare our results on empirical analysis of financial data with simulations of two stochastic models of the dynamics of stock market prices. The two models are (i) the truncated Lévy flight recently introduced by us and (ii) the ARCH(1) and GARCH(1,1) processes. We find that the TLF well describes the scaling and its breakdown observed in empirical data, while it is not able to properly describe the fluctuations of volatility empirically detected. The ARCH(1) and GARCH(1,1) models are able to describe the probability density function of price changes at a given time horizon, but both fail to describe the scaling properties of the PDFs for short-time horizons.

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  • Mantegna, Rosario N. & Stanley, H.Eugene, 1998. "Modeling of financial data: Comparison of the truncated Lévy flight and the ARCH(1) and GARCH(1,1) processes," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 254(1), pages 77-84.
    • Handle: RePEc:eee:phsmap:v:254:y:1998:i:1:p:77-84
    DOI: 10.1016/S0378-4371(98)00020-X
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    1. Marc Potters & Rama Cont & Jean-Philippe Bouchaud, 1996. "Financial markets as adaptative systems," Science & Finance (CFM) working paper archive 500037, Science & Finance, Capital Fund Management.
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    1. Constantinides, A. & Savel’ev, S.E., 2013. "Modelling price dynamics: A hybrid truncated Lévy Flight–GARCH approach," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(9), pages 2072-2078.
    2. Focardi, Sergio & Cincotti, Silvano & Marchesi, Michele, 2002. "Self-organization and market crashes," Journal of Economic Behavior & Organization, Elsevier, vol. 49(2), pages 241-267, October.
    3. Kang, Sang Hoon & Yoon, Seong-Min, 2008. "Long memory features in the high frequency data of the Korean stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(21), pages 5189-5196.
    4. Skjeltorp, Johannes A, 2000. "Scaling in the Norwegian stock market," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 283(3), pages 486-528.
    5. Lehnert, Thorsten & Wolff, Christian C. P., 2004. "Scale-consistent Value-at-Risk," Finance Research Letters, Elsevier, vol. 1(2), pages 127-134, June.
    6. D’Urso, Pierpaolo & Cappelli, Carmela & Di Lallo, Dario & Massari, Riccardo, 2013. "Clustering of financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 392(9), pages 2114-2129.
    7. Stanley, H. Eugene & Plerou, Vasiliki & Gabaix, Xavier, 2008. "A statistical physics view of financial fluctuations: Evidence for scaling and universality," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 387(15), pages 3967-3981.
    8. Wen-Juan Xu & Chen-Yang Zhong & Fei Ren & Tian Qiu & Rong-Da Chen & Yun-Xin He & Li-Xin Zhong, 2020. "Evolutionary dynamics in financial markets with heterogeneities in strategies and risk tolerance," Papers 2010.08962, arXiv.org.
    9. Matsushita, Raul & Da Silva, Sergio & Da Fonseca, Regina & Nagata, Mateus, 2020. "Bypassing the truncation problem of truncated Lévy flights," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 559(C).
    10. Racca, P. & Casarin, R. & Dondio, P. & Squazzoni, F., 2018. "Relating group size and posting activity of an online community of financial investors: Regularities and seasonal patterns," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 493(C), pages 458-466.
    11. Stanley, H.E. & Amaral, L.A.N. & Gabaix, X. & Gopikrishnan, P. & Plerou, V., 2001. "Similarities and differences between physics and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 299(1), pages 1-15.
    12. Wei, Yu, 2012. "Forecasting volatility of fuel oil futures in China: GARCH-type, SV or realized volatility models?," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 391(22), pages 5546-5556.
    13. Demos, Guilherme & Da Silva, Sergio & Matsushita, Raul, 2015. "Some Statistical Properties of the Mini Flash Crashes," MPRA Paper 65473, University Library of Munich, Germany.

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