Backward jump continuous-time random walk: An application to market trading

Tomasz Gubiec and Ryszard Kutner

Phys. Rev. E 82, 046119 – Published 27 October 2010

Abstract

The backward jump modification of the continuous-time random walk model or the version of the model driven by the negative feedback was herein derived for spatiotemporal continuum in the context of a share price evolution on a stock exchange. In the frame of the model, we described stochastic evolution of a typical share price on a stock exchange with a moderate liquidity within a high-frequency time scale. The model was validated by satisfactory agreement of the theoretical velocity autocorrelation function with its empirical counterpart obtained for the continuous quotation. This agreement is mainly a result of a sharp backward correlation found and considered in this article. This correlation is a reminiscence of such a bid-ask bounce phenomenon where backward price jump has the same or almost the same length as preceding jump. We suggested that this correlation dominated the dynamics of the stock market with moderate liquidity. Although assumptions of the model were inspired by the market high-frequency empirical data, its potential applications extend beyond the financial market, for instance, to the field covered by the Le Chatelier-Braun principle of contrariness.

Received 28 May 2010

DOI:https://doi.org/10.1103/PhysRevE.82.046119

Authors & Affiliations

Tomasz Gubiec^* and Ryszard Kutner^†

Division of Physics Education, Institute of Experimental Physics, Faculty of Physics, University of Warsaw, Smyczkowa Str. 5/7, PL-02678 Warsaw, Poland

^*tomasz.gubiec@fuw.edu.pl
^†ryszard.kutner@fuw.edu.pl

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Vol. 82, Iss. 4 — October 2010

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Images

Figure 1
Comparison of (a) empirical and (b) theoretical 2D shadow histograms, where larger joint probability is visualized by more intense grayness. The gray scale codes the decimal logarithms of probabilities. These probabilities are shifted by small number $10^{- 5}$ to avoid singularity supplied by log0. The empirical histogram was obtained from empirical time series, for instance, for the PEKAO bank. The theoretical histogram is based on expression (1), where the single-variable distribution $h (r_{n})$ is, however, empirical. The weight $ϵ = 0.198$ was obtained from the fit of expression (1) to the whole empirical histogram. Details of this fit were given in Sec. 4D.Reuse & Permissions
Figure 2
The schematic view of two sides of the order book divided by the bid-ask spread, marked by the shadowed region, shown in three successive snapshot pictures (a), (b), and (c). By means of a typical example, these plots illustrate how the bid-ask bounce, visualized by the position of the tick, is working. Filled circles placed on bars above the abscissa denote the total volume of the sell offers at that price level. If this volume is larger then the bar is longer. The analogous offers below the abscissa denote the complementary buy offers.Reuse & Permissions
Figure 3
The illustration of derivation of (a) the one-step stochastic sharp propagator and (b) the two-step propagator. For both cases, the characteristic sequences of basic probabilities (marked by braces) were visualized. Moreover, the preinitial jump $ξ$ was marked (together with other jumps of prices) by vertical bars having small filled circles on the top.Reuse & Permissions
Figure 4
Comparison of the empirical waiting-time distribution (small black squares) and that theoretical (solid curve) fitted to it. The latter is defined by two exponentials given by the first expression in Eq. (24); the former WTD concerns, for example, the PEKAO bank. For completeness, the dashed curve shows the prediction of WTD given by the single exponential [cf. the first expression in Eq. (22)]. Remarkably, its slope (in the semilogarithmic coordinates) is (to a good approximation) the same for longer time as that of the empirical curve; i.e., it is approximately correct only for the long-term dynamics.Reuse & Permissions
Figure 5
The normalized velocity autocorrelation functions: empirical (small black squares) and theoretical ones given by Eq. (25) (solid curve) and Eq. (23) (dashed curve), respectively, for example, for the PEKAO bank. Although anticorrelation is small (being at most of the order of few percentages) as a result of a small mean value of the bid-as spread, it is sufficiently distinct and significant. Obviously, at $t = 0$ the values of all normalized velocity autocorrelation functions appear above the top of the plot.Reuse & Permissions

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