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Directed Continuous-Time Random Walk with memory

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  • Jaros{l}aw Klamut
  • Tomasz Gubiec

Abstract

We propose a new Directed Continuous-Time Random Walk (CTRW) model with memory. As CTRW trajectory consists of spatial jumps preceded by waiting times, in Directed CTRW, we consider the case with only positive spatial jumps. Moreover, we consider the memory in the model as each spatial jump depends on the previous one. Our model is motivated by the financial application of the CTRW presented in [Phys. Rev. E 82:046119][Eur. Phys. J. B 90:50]. As CTRW can successfully describe the short term negative autocorrelation of returns in high-frequency financial data (caused by the bid-ask bounce phenomena), we asked ourselves to what extent the observed long-term autocorrelation of absolute values of returns can be explained by the same phenomena. It turned out that the bid-ask bounce can be responsible only for the small fraction of the memory observed in the high-frequency financial data.

Suggested Citation

  • Jaros{l}aw Klamut & Tomasz Gubiec, 2018. "Directed Continuous-Time Random Walk with memory," Papers 1807.01934, arXiv.org.
    • Handle: RePEc:arx:papers:1807.01934
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    References listed on IDEAS

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    1. Scalas, Enrico, 2006. "The application of continuous-time random walks in finance and economics," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 362(2), pages 225-239.
    2. Masoliver, Jaume & Montero, Miquel & Perelló, Josep & Weiss, George H., 2007. "The CTRW in finance: Direct and inverse problems with some generalizations and extensions," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 379(1), pages 151-167.
    3. J. Perello & J. Masoliver & A. Kasprzak & R. Kutner, 2008. "A model for interevent times with long tails and multifractality in human communications: An application to financial trading," Papers 0805.1353, arXiv.org, revised Jul 2008.
    4. Mainardi, Francesco & Raberto, Marco & Gorenflo, Rudolf & Scalas, Enrico, 2000. "Fractional calculus and continuous-time finance II: the waiting-time distribution," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 287(3), pages 468-481.
    5. Tomasz Gubiec & Ryszard Kutner, 2017. "Continuous-Time Random Walk with multi-step memory: an application to market dynamics," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(11), pages 1-15, November.
    6. Miquel Montero, 2011. "Parrondo-like behavior in continuous-time random walks with memory," Papers 1107.2346, arXiv.org, revised Nov 2011.
    7. Enrico Scalas, 2006. "Five Years of Continuous-time Random Walks in Econophysics," Lecture Notes in Economics and Mathematical Systems, in: Akira Namatame & Taisei Kaizouji & Yuuji Aruka (ed.), The Complex Networks of Economic Interactions, pages 3-16, Springer.
    8. Scalas, Enrico & Gorenflo, Rudolf & Mainardi, Francesco, 2000. "Fractional calculus and continuous-time finance," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 284(1), pages 376-384.
    9. R. Cont, 2001. "Empirical properties of asset returns: stylized facts and statistical issues," Quantitative Finance, Taylor & Francis Journals, vol. 1(2), pages 223-236.
    10. A. Kasprzak & R. Kutner & J. Perelló & J. Masoliver, 2010. "Higher-order phase transitions on financial markets," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 76(4), pages 513-527, August.
    11. D. Brockmann & L. Hufnagel & T. Geisel, 2006. "The scaling laws of human travel," Nature, Nature, vol. 439(7075), pages 462-465, January.
    12. Hasbrouck, Joel, 2007. "Empirical Market Microstructure: The Institutions, Economics, and Econometrics of Securities Trading," OUP Catalogue, Oxford University Press, number 9780195301649.
    13. Masoliver, Jaume & Montero, Miquel & Perello, Josep & Weiss, George H., 2006. "The continuous time random walk formalism in financial markets," Journal of Economic Behavior & Organization, Elsevier, vol. 61(4), pages 577-598, December.
    14. Jaume Masoliver & Miquel Montero & George H. Weiss, 2002. "A continuous time random walk model for financial distributions," Papers cond-mat/0210513, arXiv.org.
    15. Raberto, Marco & Scalas, Enrico & Mainardi, Francesco, 2002. "Waiting-times and returns in high-frequency financial data: an empirical study," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 314(1), pages 749-755.
    16. R. Kutner & F. Switała, 2003. "Stochastic simulations of time series within Weierstrass-Mandelbrot walks," Quantitative Finance, Taylor & Francis Journals, vol. 3(3), pages 201-211.
    17. Gençay, Ramazan & Dacorogna, Michel & Muller, Ulrich A. & Pictet, Olivier & Olsen, Richard, 2001. "An Introduction to High-Frequency Finance," Elsevier Monographs, Elsevier, edition 1, number 9780122796715.
    18. Hilfer, R., 2003. "On fractional diffusion and continuous time random walks," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 329(1), pages 35-40.
    19. Ryszard Kutner & Jaume Masoliver, 2017. "The continuous time random walk, still trendy: fifty-year history, state of art and outlook," The European Physical Journal B: Condensed Matter and Complex Systems, Springer;EDP Sciences, vol. 90(3), pages 1-13, March.
    20. Gubiec, T. & Wiliński, M., 2015. "Intra-day variability of the stock market activity versus stationarity of the financial time series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 432(C), pages 216-221.
    21. Repetowicz, Przemysław & Richmond, Peter, 2004. "Modeling share price evolution as a continuous time random walk (CTRW) with non-independent price changes and waiting times," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 108-111.
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