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Remarks on the possible universal mechanism of the non-linear long-term autocorrelations in financial time-series

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  • Kutner, Ryszard
  • Świtała, Filip

Abstract

The paper consists of two parts: (i) the empirical one where the non-linear, long-term autocorrelations present in high-frequency data extracting from the Warsaw Stock Exchange were analyzed and (ii) theoretical one where predictions of our model (Quantitative Finance 3 (2003) 201; Physica A (2003); Chem. Phys. 284 (2002) 481; Phys. Comm. 147 (2002) 565; Physica A 264 (1999) 84; Physica A 264 (1999) 107; Lecture Notes in Computer Science 2657 (2003) 407; Eur. Phys. J. B 33 (2003) 495) were shown and discussed. This model introduces the possibility that the Weierstrass (hierarchical) random walk can be occasionally intermitted by momentary localizations; the localizations themselves are again described by the Weierstrass process. In other words, this combined walk is a kind of the non-separable, generalized continuous-time random walk formalism. To adapt the model to the description of empirical data recorded at time horizon Δt=1min, we applied a discretization procedure into the continuous-time series produced by the model. We observed that such a procedure generates the non-linear, long-term autocorrelations even in the Gaussian regime, as turning points of the random walk trajectory are, most often, incommensurable with discretization time-step. These autocorrelations appear to be similar to those observed in the financial time series (Physica A 287 (2000) 396; Physica A 299 (2001) 1; Physica A 299 (2001) 16; Physica A 299 (2001) 28), although single steps of the walker within continuous time are, by definition, uncorrelated. Our approach suggests a surprising origin of the non-linear, long-term autocorrelations alternative to the one proposed very recently (cf. Phys. Rev. E 67 (2003) 021112 and refs. therein) although both approaches involve related variants of the well-known CTRW formalism applied in yet many different branches of knowledge (Phys. Rep. 158 (1987) 263; Phys. Rep. 195 (1990) 127; in: A. Bunde, S. Havlin (Eds.), Fractals in Science, Springer, Berlin, 1995, pp. 119–161).

Suggested Citation

  • Kutner, Ryszard & Świtała, Filip, 2004. "Remarks on the possible universal mechanism of the non-linear long-term autocorrelations in financial time-series," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 344(1), pages 244-251.
    • Handle: RePEc:eee:phsmap:v:344:y:2004:i:1:p:244-251
    DOI: 10.1016/j.physa.2004.06.126
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    References listed on IDEAS

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    1. Metzler, Ralf & Klafter, Joseph, 2000. "Boundary value problems for fractional diffusion equations," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 278(1), pages 107-125.
    2. Kehr, K.W. & Kutner, R., 1982. "Random walk on a random walk," Physica A: Statistical Mechanics and its Applications, Elsevier, vol. 110(3), pages 535-549.
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