**6. Methods**

Theoretical researchers mostly rely on the Brownian motion as the proxy for price returns of real financial markets (one of the most famous examples is the work of Black and Scholes (1973)). The analogy between historical price moves and changing coordinates of an ensemble of molecules in thermodynamics is the motive behind this common approach. Osborne (1959) shows in the classical work that the steady-state distribution of log-returns in the stock market is the probability distribution for a particle in Brownian motion. It is important to emphasise that at the telegraph-driven time of Osborne's publication (1959) the structure and the dynamic of the market was very different from the ones typical for our modern digital world. The present-day trading has almost completely moved to the digital online space instead of the physical trading floors where all deals happened more than a half a century ago (see Harris (2003) for the historical endeavour on the evolution of trading and exchanges). In this space, a signal can easily propagate through international borders with the speed of light. It stipulated the majority of trades to happen in a fully automated way. The significantly bigger diversity of the market participants and the ability of the high-frequency trading made the financial stylised facts more important today than ever. The stylised facts describe the deviations of real price returns from the theoretical Brownian motion (see Cont (2001a) for the set of stylised empirical facts). Nevertheless, in our work, we will operate with Brownian motion as the core for the analytical part of the research.

<sup>8</sup> https://github.com/VladUZH/VlPetrov.

The choice of Brownian motion, employed in this work, is justified by two reasons. First, the statistical properties of the number of directional-change intrinsic events, studied in this work, is agnostic to the flow of physical time. Second, the divergence of empirical results from the properties of the selected model, if any, helps in understanding the features of the real markets better.
