# Evidence of Microstructure Variables’ Nonlinear Dynamics from Noised High-Frequency Data

Nikolay Andreev and Victor Lapshin

Abstract Research of nonlinear dynamics of finance series has been widely discussed in literature since the 1980s with chaos theory as the theoretical background. Chaos methods have been applied to the S&P 500 stock index, stock returns from the UK and American markets, and portfolio returns. This work reviews modern methods as indicators of nonlinear stochastic behavior and also shows some empirical results for MICEX stock market high-frequency microstructure variables such as stock price and return, price change, spread and relative spread. It also implements recently developed recurrence quantification analysis approaches to visualize patterns and dependency in microstructure data.

Keywords Chaos theory • Correlation integral • Microstructure • Price dynamics • Recurrence plot

JEL Classification C65, G17

Since the nineteenth century, there have been attempts to describe behavior of economic variables via simple linear deterministic systems. Unfortunately, unlike nature phenomena, in many cases financial series could not be reduced to linear dynamic model. Thus, stochastic models proved to be suitable for modeling and prediction. Nevertheless, attempts to find an appropriate deterministic model continued. They had determinism, introduced by Laplace in the early nineteenth century, as a fundamental principle. Poincare (1912) stated that even if all the underlying laws were known, it would still be impossible to predict the state of the system due to error in estimate of the initial condition. But if the system is not too sensitive to the initial data, we can predict future states up to the error of the same

N. Andreev (H) • V. Lapshin

Lomonosov Moscow State University, Moscow, Russia

Financial Engineering and Risk Management Laboratory, National Research University Higher School of Economics, Moscow, Russia e-mail: nandreev@hse. ru

© Springer International Publishing Switzerland 2015 13

A. K. Bera et al. (eds.), Financial Econometrics and Empirical Market Microstructure, DOI 10.1007/978-3-319-09946-0 2

order. This led to the assumption that unpredictable “stochastic” processes could be replaced by fully deterministic but unstable (chaotic) systems. For a detailed review history of nonlinear dynamics research in economics, see Prokhorov (2008).

Particular deterministic processes have been paid a great deal of interest lately due to the quasi-stochastic properties of the generated signals. One of the simple maps producing such effect is the well-known tent map:

{ |

a~1xt-i,0 < xt_i < a,

(1 – a/-1 (1 – xt_i) , a < xt_i < 1,

which has first and second moment properties that are the same as first-order autoregressive process and thus was called ‘white chaos’ by Liu et al. (1992). Some values of parameters cause such processes to behave similar to the i. i.d. series (Sakai and Tokumaru 1980). Thus, the natural question arises if stochastic trajectory can be interpreted as completely deterministic and thus perfectly predictable if the underlying map is completely known. It is necessary to note that predictability is not the main goal and cannot be achieved for microstructure variables, as shown below. The main advantage of the chaotic approach is the possibility to describe the data with a more appropriate model that should be more reliable in times of crisis. This intention is justified by the observed similar properties of microstructure data and characteristics of chaotic natural phenomena, such as earthquakes and avalanches. Therefore it is appropriate to assume that underlying laws of dynamics are similar.

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