Crises de liquidité endogènes dans les marchés financiers

   A market is a place where buyers meet sellers to execute transactions. The products they can buy and sell are very diversied: shares of stocks, indices, foreign exchange, derivative products (futures, options, etc.), bonds, commodities, cryptocurrency. Even though each product has its own specicity, dynamics is mainly determined by the way agents interact. Such dynamics is characterised by being very rich and complex due to the heterogeneity of market participants. Indeed, the spectrum of the time horizon of their strategies is very wide: from seconds or less for high-frequency agents to several years for low-frequency ones. A presentation of the market ecology will help us to understand and study the dynamics of the market. A full classification of the market participants would be beyond the scope of this work. Nevertheless , we can identify the following main types of agents:
• Market makers: they are liquidity providers. Tipically high-frequency agents, they post passive orders at the buy and sell side, then, they wait for other participants of the market to come and take their orders. As their name implies, they make the market by intermediating between slower buyers and sellers that would not be able to meet on the market at the same time. Essential to markets’ ecology, some of them are paid by the exchange to ensure sufficient liquidity. Otherwise, they make money from the difference between their buying and selling quotes and are then very profitable when the price is mean-reverting. Their risk comes mainly from their inventory: if the price trends in the opposite direction of their position they lose a lot of money. They therefore adjust their price quotations to manage their risk as well as possible. The capacity of their strategies is low, but they trade very frequently, which makes them profitable on average.
• Long investors: they take a position on the market and re-balance their portfolio from time to time. The time horizon of their strategy is very long, around several years, and their assets under management are huge. Thus, it can take them days to buy/sell their market positions. Because of the long time horizon, executing their transactions does not require sophisticated optimization techniques. They can use different types of strategies (momentum, ETF, fundamental etc.). They are usually institutional investors or pension fund managers.
• Brokers: they give an access to the market to those who cannot. Their clients cannot trade themselves on the market due to a technological gap, nor the best strategy to execute large orders. Their goal is to minimize the transactions costs by splitting smartly the orders of their clients.
• Banks: they oer derivative products, that we can interpret as insurance contracts, to their clients and manage their risks. They have teams working on elaborating such products that will suit the needs of their clients. Nevertheless, they can also have teams that do market-making, long term investment and brokerage.
• Other participants: other participants can play in between all the possible strategies, on both short or long time scale. Although technological advances have reshaped the market by providing new opportunities,the way agents trade and strategies are much the same. We can therefore say that this classication is stable over time. In what follows, we will keep in mind that market participants are highly heterogeneous. Electronic markets and the limit order book Historically, agents were physically present in the stock exchange. Such vivid scenes were well described by Emile Zola in his novel « L’Argent ». Some movies also represent the market at this stage, in the « Wolf of Wall Street » of Martin Scorsese for example. As previously mentionned, the emergence of new technologies have deeply reshaped the markets. First,the market switched to Over The Counter (OTC) markets where people negotiated products via telephones. Then, the development of the internet enabled news possibilities: trading on fully electronic markets. These markets are managed by a computer that matches the orders of agents, under systematic rules. Even though some OTC markets remain, the majority are electronic nowadays. Electronic markets are believed to be an optimal setting to allow fair access to the market. It aims at a better spread of the information to everyone and reduces insider trading. On the other hand, all the trades and intentions of trades of agents are stored and the regulation uses this data to detect frauds and illegal actions. Market regulators are therefore encouraging platforms to upgrade to such automated operation. From our point of view, electronic markets have made it possible to record an incredible amount of data. By dissecting this real treasure, we have the opportunity to understand the functioning of markets in detail. Since 2007, the emergence of Multilateral Trading Facilities has enabled agents to buy the same contracts on dierent electronic exchanges. Since then, electronic markets have become fragmented. We can cite « Chi-X » or « Turquoise » as competitors of the classical exchanges Euronext, NYSE, LSE etc. Oering dierent types of fees, their creation was made to reduce trading costs for market participants. From our perspective, it may complicate our modeling as we are interested in the aggregated market. By aggregate, we mean the sum of what happens on all the exchanges where we can buy the same asset. Nevertheless, we first need to understand what electronic markets are all about. We chose to focus on a particular type of electronic markets: a double auction one. It offers the possibility for an agent to trade passively or aggressively. With the first option, an agent can post a buy or sell order, which consists in a number of shares at a given price and wait for a counterpart. This passive order is called a limit order. Agents can cancel their passive orders if no counterpart has yet been found. The second option, called a market order, consists of matching an existing limit order. All these limit orders are stored in the limit order book. A schematic describing it is displayed on Fig. 1.1. It represents the interaction between supply and demand on nancial markets, essential to understand the price formation process. We refer to bid (resp. ask) side for the buy (resp. sell) side in the following. An important notion emerges from the order book: the liquidity. It refers to the quantity of shares available in the order book at a given time. Thus, posting a limit order in the limit order book provides liquidity while posting a market order takes liquidity. Note that some exchanges provide to their clients other fancy types of orders. More difficult to model, we restrict ourselves to exchanges and assets where only these three orders are available. Concerning market fragmentation, we assume that either we have access to the aggregated order book and if not that, the one we have is representative of the  aggregated one.

   After brievy describing markets’ ecology, we ask ourselves the key question: why are nancial markets so prone to crashes? Indeed, in the past decades and century, nancial markets have episodically sufered from a lack of stability. Important price drops, crashes happen too frequently, shocking people’s mind and sometimes leading to terrible economic outcomes. Historical review of major crashes will certainly give us insights about the mechanisms of crises. The 1920th decade was characterized by a huge growth of the economic production, built on post-war optimism. During that period, stock prices dimbed too fast and became too high relative to the underlying economic production, which caused overproduction. This mechanism created a speculative bubble that was stopped by the very infamous « Black Thursday » on October 24th 1929. On this very day, the Dow Jones sufered of a 22.6% drop at noon, followed by two consecutive drops the Monday and Tuesday after, starting then the Great Depression. The analysis of what happened this day is very important to understand the mechanism of financial crises. A great panic took the New York Stock Exchange and the liquidity of buyers i.e. the number of buyers vanished: market participants were too afraid to buy at any price. This mechanism made the price crash strongly. Unfortunately, this crisis spread and hit the economic sector creating recession, abnormally high unemployment, etc. The damages lasted several years, worldwide, and it took years for the economy to recover. More recently, in October 1987, the US stock market sufered from another crash of a huge intensity: the price of the Dow Jones dropped of 22.6%. To prevent the financial crisis to propagate into the real economy, the Fed injected liquidities in the financial sector. This action avoided the emergence of a global crisis but the crash spread to other foreign markets with huge drops on indices. But thanks to the Fed, this crash, called « Black Monday », was just a « liquidity crisis » and not an economic one. It had another interesting consequence: in 1998, the US government introduced a system of circuit-breakers to the stock markets. It functions as follow: when the price moves more than a given threshold, market activity halts. Unfortunately, these mechanisms could not fully prevent crises from occurring as the 21st century has already known a few, we can mention the nancial crises of 2008 caused by sub-prime mortgage backed securities, the Flash Crash on the S&P500 of 2010 [4] and the Covid-19 crisis we are currently suering from. The origins of crises vary, depending on the economic context, nevertheless, the crash itself seems universal. Panic takes the buyers leading to liquidity dry out that makes the price crash.

  The Efficient Markets Theory was rst criticized by the economist Robert Shiller. His works have shown that the volatility of the market is too high to be explained by rational visions of the future. Well established in the literature [9] , this point is now considered common fact. Agents are not rational and behave due to their emotions, fears and not complex future expectations. This means that Eq. 1.9 is not well suited to describe the price dynamics with Ft containing the exogenous news. Previously, we gave examples of news that have a strong effect on the price. We would like to reverse this analysis: considering large price changes, are they triggered by some news? A recent study by Joulin et al [10] answered this question. They measured the typical volatility σ on one minute and selected price changes bigger than 4σ. In a gaussian world, those changes could not exist. But there nancial markets are full of them. Then the authors tried to link these large price changes with external news. They have drawn the following conclusions: only 5% of such price changes are triggered by external news, meaning that the other 95% are endogenous i.e. created by the market itself. The behavior of the volatility around a piece of news or an endogenous event is different. Indeed, while in both cases it jumps right at the event and then relaxes to ito its preceding value, the way it relaxes varies. In the first case the relaxation is a power law in time with exponent −1, whereas in the second case, the exponent is −0.5. Note that, in both cases, the market takes some time to digest the information but it takes more time for endogenous events. They lead to a persistent high volatility regime that cannot be explained by any rational theory, providing another evidence that justies Schiller’s statement: agents in financial markets are far from rational. We also note that modeling this high ratio of endogeneity will require to introduce feedback in the dynamics in order to model agents’ behavior.