**1. Introduction**

Flash crashes in financial markets can be defined as extreme changes in the price of one or multiple assets within a short interval of time. These have become increasingly relevant for practitioners and, in particular, market makers whilst being increasingly studied and reported in the quantitative finance literature.

The most notorious flash crash is likely that of 6 May 2010, which involved the major U.S. stock indices (S&P, DJIA, and NASDAQ composite) and caused a ≈9% drop in the DJIA in the 36 min it lasted for. This event led to a variety of empirical and theoretical papers trying to understand the event and its causes, with the aim to shed light on other black swan events too (up/down crashes). High frequency traders are at the center of interest in a large portion of this literature; hence, we report a brief summary of their role in markets and its regulatory concerns.

It has been shown that HFT market players contribute to price efficiency and tighter spreads, thereby improving the price discovery process. These players and electronic trading as a whole have become increasingly dominant in recent years to the point of constituting a large portion of the traded volume in financial markets. On the other hand, some characteristics of HFT players have caused other market players to raise concerns, as the run to incredibly fast execution leaves many behind and allows HFTs to front run other players [1]. The ability of HFTs to process information faster than other players leads to adverse selection and its fixed cost to a size advantage for larger players, which might hurt the overall welfare of market participants [2]. It can now perhaps be argued that the run to faster execution is going beyond price efficiency, which benefits investors and toward an unstable price process driven by competition between large firms. This is supported by a

**Citation:** Turiel, J.D.; Aste, T. Heterogeneous Criticality in High Frequency Finance: A Phase Transition in Flash Crashes. *Entropy* **2022**, *24*, 257. https://doi.org/ 10.3390/e24020257

Academic Editors: H. Eugene Stanley, Ryszard Kutner and Christophe Schinckus

Received: 4 January 2022 Accepted: 4 February 2022 Published: 10 February 2022

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**Copyright:** © 2022 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https:// creativecommons.org/licenses/by/ 4.0/).

large body of literature on flash crashes, which places HFTs at the center of some disruptive systemic events, as discussed below.

The SEC's report on the flash crash of May 6th [3] finds that most market participants automatically halted their trading due to hard risk constraints triggered by the sudden price change, while some HFT firms kept trading, as it was deemed still profitable by their algorithms. These absorbed most of the original large sell order, but once they reached inventory or loss constraints, they started selling too. This increased the selling pressure in the market, and some works hold that it caused HFTs to trade with each other repeatedly ("hot potato phenomenon"), thereby increasing the traded volume (but not the real liquidity). This apparent increase in liquidity in the form of high trading volume caused large sell orders to ge<sup>t</sup> executed faster [4]. This chain of events highlights that the phenomenon has a dangerous positive feedback loop.

The results in [5] show through simulations how reducing either (or both) the number of HFT players or the size of the large sell order greatly reduced the size of the drawdown. Further, other works find that black swan phenomena of duration <1.5 s are about ten times more frequent than longer ones, and their return distribution deviates from the canonical power law distribution of returns. The authors sugges<sup>t</sup> a phase transition to an all-machine environment at ∼1 s, as human reaction time is in the order of seconds. The authors also investigate the time scales via additional simulations to show the rise in extreme events and their magnitude around ∼1 s in what they define as the all-machine phase [6,7]. Findings along those lines, on the distribution of high frequency black swan events deviating from the canonical return distributions, were also recently published by the authors of this work [8].

From the review above, we see that crashes of different sizes seem to involve a selfperpetuating cycle [5] with positive feedback loops.

This type of self-excited process is also investigated in [9] for the liquidity and information dependence between two sample assets, showing how liquidity shocks to an asset can propagate to related ones (and by extension to the wider market).

The frequency and size (in terms of number of securities involved) of simultaneous-like crashes in HFT is also investigated in the literature. For instance, the works by Lillo and coauthors [10,11] investigate the dynamics of simultaneous flash crashes, and motivate their importance by showing the growth in the number of mini crashes in recent years. Further, they show how the number of simultaneously crashing securities has grown over the last 10 years, thereby highlighting the increasing systemic relevance of this phenomenon.

We recognize that systemic risk is traditionally defined as "the risk of a cascading failure in the financial sector" [12]. In this work, we do not investigate interbank connectivity, but rather the connectivity of trading patterns across financial assets which can lead to breakdowns or temporary dysfunctions in financial markets, as per the definition of systemic risk in [13]. We phrase the concept in a slightly different manner in the context of our work as follows. In this paper, we define systemic risk as the risk component of an event (say a flash crash) that is given by the interconnectedness of assets, likely as a result of correlated actions and arbitrage between market participants. This causes isolated events to spread in the market and affect more assets, thereby increasing their impact and relevance for all market participants. A related concept is that of "synchronization" which is the systemic and concentration aspect that arises from the alignment and interdependence of actions between market players (on a single asset) rather than across assets.

Our phrasing of the concept of systemic risk from [13] highlights our microstructural investigation of the trading dynamics which lead to dysfunctions and disruptions in the orderly functioning of financial markets. Indeed, crashes can be just due to microstructural dynamics, but as price efficiency deteriorates and volatility spikes, investors shy away from financial markets. Financial markets allow investors to provide companies in the real economy with capital, and their dysfunction can turn mere trading issues into real economic panic and crisis. Therefore, even high frequency black swan events can have

dramatic effects on the real economy, as proven multiple times in recent history, which ties our interpretation of systemic risk back to Ref. [12] as well.

The systemic risk posed by HFTs was investigated in the literature in the last decade. The work by Paulin et al. [14] simulates flash crashes through agent-based modeling and highlights the importance of market structures in the systemic propagation of extreme events. The works by Abreu and Brunnermeier [15] and Bhojraj et al. [16] investigate the risks of synchronization between arbitrageurs in financial markets and acknowledge its existence. Other works investigate the systemic risk of HFT dynamics. Jain et al. [17] investigate how low-latency HFT trading can worsen extreme systemic events in financial markets and argue for the need to incorporate correlation and market structure in regulating these risks. The work by Harris [18] discusses many mechanisms, among which systemic risks originating from order routing and self-reinforcing mechanisms which cause crashes. The review by De Gruyter [19] summarizes the systemic aspects of HFTs and market structure, such as position correlation and herd behavior, adverse selection in orders and crowding, as well as negative contribution to price discovery at times.

Co-crashes are becoming more frequent and systemic. It is, therefore, important to investigate their structure. In particular, it is relevant to understand which stocks are central to larger systemic events as well as the contagion structure between stocks in the market. This is a central theme in market stability for regulators as well as in risk managemen<sup>t</sup> for market makers.

The present work joins the two themes of flash crashes and systemic risk by delving deeper into the dynamics of simultaneous flash crashes of different sizes throughout 300 liquid stocks traded on the NASDAQ. We investigate the empirical distribution of crash sizes and the structure of these events in the market. We also investigate whether larger systemic events involve highly unstable stocks (which crash often) or stocks that are more stable in their price dynamics, ye<sup>t</sup> more influential to trigger larger systemic events when subject to liquidity shocks. We apply tools from statistical physics to show the difference between crashes which involve a small or large number of assets. We uncover a phase transition occurring when the crash size exceeds five stocks. Implications for systemic risk in high frequency markets are discussed from both a trading and regulatory perspective.
