Behind the price: on the role of agent's reflexivity in financial market microstructure
aa r X i v : . [ q -f i n . T R ] A ug Behind the price: on the role of agent’s reflexivityin financial market microstructure ∗ Paolo Barucca and Fabrizio Lillo
1. Department of Banking and Finance, University of Zurich, Switzerland2. Scuola Normale Superiore di Pisa, Pisa, Italy
August 24, 2017
Abstract
In this chapter we review some recent results on the dynamics of priceformation in financial markets and its relations with the efficient markethypothesis. Specifically, we present the limit order book mechanism formarkets and we introduce the concepts of market impact and order flow,presenting their recently discovered empirical properties and discussingsome possible interpretation in terms of agent’s strategies. Our analysisconfirms that quantitative analysis of data is crucial to validate qualitativehypothesis on investors’ behavior in the regulated environment of orderplacement and to connect these micro-structural behaviors to the proper-ties of the collective dynamics of the system as a whole, such for instancemarket efficiency. Finally we discuss the relation between some of thedescribed properties and the theory of reflexivity proposing that in theprocess of price formation positive and negative feedback loops betweenthe cognitive and manipulative function of agents are present. ∗ Authors acknowledge partial support by the grant SNS13LILLB ”Systemic risk in financialmarkets across time scales”. ntroduction Understanding price movements, both their origin and their properties, is oneof the most important challenges in economics. In Finance this problem has along history which has seen two antagonist schools of thought confronting in thefirst half of the twentieth century. On one side there were the fundamentalistswho posited that the price of an asset is the discounted value of its ”intrinsic”or ”fundamental” value and equals the discounted cash flow which that securitygives title to. On the other side there were the econometricians, who, apply-ing statistical analyses to empirical time series of prices discovered that stockprices develop patterns that look like those of a random walk. The latter is thetime series that can be obtained, for example, by tossing a coin and movingup (down) the price by one unit when the outcome is head (tail). The erraticand random behavior of prices seemed to clash with the fundamentalist viewand seemed to give support to those believing that stock market is essentially acasino. By using the words of LeRoy ”If stock prices were patternless, was thereany point to fundamental analysis?”. It is well known that the solution to thisproblem was given by the seminal 1965 paper by Paul Samuelson ”Proof thatProperly Anticipated Prices Fluctuate Randomly” (even if Bachelier 1900 andCowles 1933 arrived to somewhat similar conclusions). Samuelson showed thatin an informationally efficient market, i.e. a market which fully incorporatesthe expectations and information of all market participants, price changes mustbe unforecastable. This is the celebrated Efficient Market Hypothesis (EMH), acornerstone of modern Finance, which reconciles the fundamentalist and econo-metrician view.However the details of how information is impounded into price are stilla matter of debate, as well as the question of whether markets are truly effi-cient. Market microstructure is devoted to the empirical and theoretical studyof how information (private or public) is incorporated into price and how priceis formed through the action of many interacting agents. Given the importanceof the problem (even outside Finance, think for example to the new markets foradvertisement on search engines in the Internet), the increasing availability ofhigh resolution data, and the significant changes observed in the organizationof markets, market microstructure has experienced a large development in thelast fifteen years. In this paper we review the problem of price formation froma microstructure point of view focusing on how strategies are slowly translatedinto orders in the market. We do not aim to be exhaustive, but to highlightthe main elements which have recently emerged in the field, trying to avoidas much as possible technicalities and the use of mathematical formalism. Inthe last part of the paper we will discuss about possible analogies and relationsbetween the newly discovered properties of price formation and the theory ofsocial reflexivity, proposed, among others, by George Soros.The contribution is divided in four sections. In the first section we presentthe efficient market hypothesis, the concept of market impact, and we describea widespread market mechanism, namely the limit order book. In the secondsection we present the order flow and describe one important properties, its long2emory, which allows us to understand the strategy of order splitting followedby large investors. In this section we also discuss how the long memory of orderflow can be reconciled with efficient market hypothesis. In the third section wereview Soros’ theory of social reflexivity and finally in the last section we outlineour interpretation of empirical results in the context of social reflexivity.
The market is where supply and demand meet, that is where buyers and sellersexchange goods. The efficient-market hypothesis (EMH) states [1, 2] that stockmarket efficiency implies existing share prices to always incorporate and reflectall relevant information. According to EMH, any information in the hand ofan investor on the future value of an assets should thus be instantaneouslyincorporated into the price. This occurs through the choice of the strategy theinvestor chooses to trade the asset. The trading strategy can be seen as themedium that translate information into price changes. Loosely speaking marketimpact refers to the correlation between an incoming order or strategy (to buyor to sell) and the subsequent price change. The strategy can be coded in theseries of orders to buy or to sell, which are sent to the market. The order flow is the aggregation of the orders of all the agents (or a subset of them).However, the causal relation between trading strategies (or order flow) andprice changes is far from trivial and at least three explanations for this correla-tion (i.e. the existence of market impact) can be given • Agents successfully forecast short term price movements and trade ac-cordingly. This does result in measurable correlation between trades andprice changes, even if the trades by themselves have absolutely no effecton prices at all. If an agent correctly forecasts price movements and ifthe price is about to rise, the agent is likely to buy in anticipation of it.According to this explanation, trades with no information content haveno price impact. • The impact of trades reveals some private information. The arrival of newprivate information causes trades, which cause other agents to update theirvaluations, leading to a price change. But if trades are anonymous andthere is no easy way to distinguish informed traders from non informedtraders, then all trades must impact the price since other agents believethat at least of fraction of these trades contains some private information,but cannot decide which ones. • Impact is a purely statistical effect. Imagine for example a completelyrandom order flow process, that leads to a certain order book dynamics.Conditional to an extra buy order, the price will on average move upif everything else is kept constant. Fluctuations in supply and demandmay be completely random, unrelated to information, but a well defined3igure 1.1: The typical structure of an order book. [3]notion of price impact still emerges. In this case impact is a completelymechanical - or better, statistical - phenomenon.The distinction between these three scenarios is not fully understood. Inorder to discriminate among these alternatives, it is useful to specialize into aconcrete case, since markets have different structures and rules. In other words,one way to address this problem in a quantitative fashion is through dynamicalmodels of market microstructure, which are based on a specific, ’internalist’,knowledge of what kind of orders can be placed by investors and what is theirdynamics. Here we will focus on the limit order book , a very common mechanismadopted in most electronic markets. In a limit order book market, an agent (oran intermediary acting on her behalf) can place two types of orders , namelylimit and market orders: • Limit orders are orders to buy or sell a given volume of shares at a specifiedprice or better. • Market orders are orders placed to buy or sell an investment immediatelyat the best available current price.The best buy and sell order on the market are called ask and bid respectively andtheir difference is the bid-ask spread. Buy (sell) limit orders with a price lower(higher) than the ask (bid) do not initiate a transaction and are stored, visibleto other market participants, waiting for the price moving in their direction. Anagent can decide to cancel a limit order at any time. Figure 1 shows a snapshotof a real order book.Orders are not typically placed directly by individual investors, mutual fundsor hedge funds: all these economic agents need to ask intermediaries to place Even though various financial markets may have more kinds of slightly different orders,these are the two main types- .Notice that even an uninformed trade moves mechanically the price and there isno relation between information of the trade and price movement (see the thirdexplanation in the bulleted list above). This is even reinforced by the fact thatelectronic markets are typically anonymous and there is no way of knowing theidentity of the counterpart.The second possible origin of market impact is due to the reaction of the restof the market to the trade. Even if the market order does not move mechanicallythe price, the other agents might react to the observation of the trade, revisingthe price of their limit orders and thus moving the price. As we will see below,the induced market impact plays an important role. In general, the more liquidis the market, i.e. the more limit orders are present in the order book then thelower is the market impact and the less a single investor can induce large pricechanges.A number of empirical studies [5, 12, 13] has established that market impact,both mechanical and induced, is statistically different from zero (despite beingquite noisy). This means that buy (sell) market orders move on average theprice upward (downward). Generically it is found that the larger the volume ofthe trade the larger on average the price change and the dependence betweenthese two quantities is strongly concave. In simple words this means that if thevolume of a market order is doubled, the price impact is significantly less thandoubled. The fact that even uninformed trades can change the price (again,also because it is almost impossible that the market understands quickly if thetrader is informed or not) raises several interesting questions on the relationbetween information and price changes and therefore on the Efficient MarketHypothesis. Anticipating the discussion we will do later in this paper, it suggeststhe presence of positive feedback loops, where an uninformed trade transientlyimpacts price, but the other market participants, being unable to discern if thetrade is informed or not, revise their valuation of the asset and trade concur-rently, creating more price change in the direction of the random initiator trade.This amplification mechanism resembles the reflexivity hypothesis (see below formore details). Note also that if market order volume is smaller than or equal to the volume at theopposite best, the order is executed at the best price; on the other hand if its volume exceedsthe volume of the opposite best in the order book, then it penetrates the book and reachesthe second best price or more. In this case the price is a weighted average over the variouslimit orders that are needed to execute the market order.
In the last ten years there have been major efforts to understand investorsbehaviors through detailed modeling and data-analysis. A significant amountof the literature has been devoted to analyze data on order books of real financialmarkets, [4, 5]. In particular we review here the empirical properties of orderflow and its consequences for the modeling and understanding of market impact.Understanding the inter-temporal properties of the order flow is quite chal-lenging. This is due to the intrinsic multi dimensional structure (each type oforder is a different variable) whose properties depend on the current state ofthe book, which in turn is the result of the past order flow. The order flowand limit order book modeling is still an open problem and new models andempirical results are continuously proposed.Here we focus on a small but important subset of the order flow. Specifi-cally we shall consider the flow of market orders (hence the orders triggering atransaction) and we discard the information on the volume of the order, focus-ing only on its sign. The sign ǫ ( t ) of the t -th trade is equal to +1 for a buyorder and − − ǫ ( t ) is theexpectation (or the mean) of the product of two transaction signs separated by τ transactions. For a totally random sequence (for example the one obtained bytossing a coin), this function is equal to zero for all τ s because the occurrence ofa head now does not give any information about the likelihood that a head or atail will occur τ tosses in the future. The autocorrelation function is thereforerelated to the predictability of trade signs.A series of papers in the last ten years have shown [6, 7, 8] that in the vastmajority of financial markets the autocorrelation function of the order flow is aslowly decaying function of the lag τ , well described by a form τ − γ , where γ isan exponent empirically found to be between 0 and 1. These kind of processesare called long memory processes because it can be shown that they lack atypical time scale beyond which the process appears to be random. In otherwords the present state of the system is determined by the whole past historyof the process. The exponent γ measures the memory of the process, which isslowly decaying for large τ . It is possible to quantify this slowness from theempirical order flow auto-covariance: the smaller the exponents, the slower the6ecay. This slowness is commonly interpreted and quantified through the Hurstexponent H . In the case of random diffusion, where no memory is present andorder signs are drawn randomly with equal probability at each time-step, theHurst exponent is 1 /
2. For long-memory processes H is larger than 1 /
2, as it isfor the the order flow so that the it can be regarded as a super-diffusive process,where fluctuations grows with time through an exponent larger than 1 / H vary across markets but they alway remain larger than 1 /
2. Long-memoryprocesses have been observed also in the dynamic behavior of other financialvariables: volatility of prices and trading volume have been recognized as long-memory processes [10].The observation of long memory of order flow raises two interesting ques-tions: (i) what is the behavioral origin for the sign persistence in the order flowand (ii) how is it possible to reconcile the predictable order flows with marketefficiency (i.e. unpredictability of price changes).The present explanations for long-memory fall into two classes: the firstis that the long memory of the order flow holds for each investor and it linksthe persistence in the order flow with the presence of large meta-orders thatare splitted in subsequent small orders of the same sign; the second class ofexplanations calls into question collective behaviors of investors imitating eachother [11]. Evidence gathered so far seems to favor the first class of explanations,in particular data show that large investors do split and execute their orders.Furthermore predictions concerning the relation between trade volumes, marketimpact and order flow are in agreement with the first type of explanations [9].The presence of meta-orders is indeed a simple and clear explanation forlong-memory in the order book. Let us consider an investor that has to executea large trade and who does not want to influence the price abruptly. Insteadof placing directly a big market order that would penetrate the order book andreach limit order at strongly unfavorable prices, the investor prefers to split themeta-order and both gain the chance not to influence the market and to getbetter prices if on average other investors are not following the same strategy.The other important question is how the long memory of order flow is con-sistent with EMH. Since a buy (sell) trade moves the price upward (downward)a persistent order flow sign time series would induce a persistent price changetime series. This means that by observing that recent past price changes were,for example, typically positive one could predict that in the near future theprice will move up. This (hypothetical) predictability would allow to make easyprofits and is inconsistent with EMH and with empirical observation.A possible solution of this apparent paradox is the asymmetric liquiditymechanism [9]. According to it the price impact of a trade depends on the pre-dictability of its sign. For example, if past trades were typically buys, implyingthat the next trade is more likely a buy than a sell, the asymmetric liquiditypostulates that if the more predictable event (the buy) actually occurs, it willimpact the price less than if the less likely event (the sell) occurs. In other wordsthe price change is not fixed, but it is history dependent and adapts itself to the7evel of predictability of its source (the order flow).The asymmetric liquidity mechanism has been verified empirically [9] andits microscopic origin has been explored and elucidated in [18]. In this lastpaper it has been shown that agents executing incrementally a large order bysplitting it in a large number of trades adjust the size of these trades in order tominimize their impact on price. This adjustment becomes stronger and strongeras the execution proceeds. In other words investors decide their strategy exactlybecause they are conscious of their impact on price. Finally, it has been shownthat this mechanism is able to reconcile the long memory of the order flow withthe uncorrelated price change time series.In conclusion, the splitting of metaorders and its detection from the rest ofthe market is critical to understand the dynamics of price in financial markets.From a strategic point of view, the origin of splitting has been explained andmotivated theoretically in the seminal work of Kyle [15]: the optimal strategyfor an investor with private information about the future price of an asset isto trade incrementally through time. This strategy allows earlier executions tobe made at better prices and minimizes execution cost. Moreover this strategyminimizes the information leakage, i.e. it allows the trader to hide her intention(and information). This strategy can be seen as a different form of reflexivity.In fact the purpose of splitting is to modify, with the action of trading and theimpact it generates, the beliefs of the other market participants. Differently fromthe impact case described in the previous section, here this form of interactionbetween action and beliefs creates a negative feedback. In the next two Sectionswe discuss in more detail the relations between these empirical facts and thetheory of reflexivity.
In social systems agents do not merely observe but also actively participatein the system themselves, this is the simple observation leading to reflexivitytheory in social sciences. Soros first exposed his theory in his book
The Alchemyof Finance in 1987 where he builds his conceptual framework on two principles.The first one is the principle of fallibility: in social systems the participants’views and consequently their perspectives are always biased, inconsistent, orboth.Human beings are utterly familiar with the principle of fallibility that af-fects all aspects of human life, from perception to action. Fallibility is strictlyconnected to the world complexity ; social facts are so complex that cannot beperceived and understood completely and thus they require a simplification thatintroduces biases and inconsistencies in our understanding. But fallibility is notjust about perception.If we interpret fallibility as the inability of a human being to elaborate anoptimal response in a given social situation then we can distinguish differentcauses of fallibility: (i) A subjective cognition fallibility: we are unable to act inthe best possible way because we perceive our situation in a subjective and in-8omplete fashion; (ii) a general cognition fallibility: we perceive the situation ofall the other social agents in a subjective and incomplete fashion; (iii) a manip-ulative fallibility: even assuming a perfect perception of the system situation,we can act in a wrong way.The second is the principle of reflexivity: the imperfect views of participantscan influence the situation to which they relate through their actions. In par-ticular Soros gives [16] this specific example: ”if investors believe that marketsare efficient then that belief will change the way they invest, which in turn willchange the nature of the markets in which they are participating (though notnecessarily making them more efficient).”The role of reflexivity can be described with a circle of social actions, allaffected by fallibility: the social agent perceives a given situation, formulatesan interpretation, decides a strategy, and finally acts but by acting she changesthe situation so that the act might no longer have the same effect that washypothesized at first.This circle can generate two kinds of feedback loops, negative or positive.Negative feedback loops of participants’ actions can stabilize the system, inthe sense that the situation of the system becomes more and more similar tothe participants’ perceptions thanks to their actions, thus helping to reach anequilibrium.Conversely positive feedback loops destabilize the system, since participants’perceptions and the real situation of the system differ bringing the system farfrom equilibrium towards an eventual sudden change, e.g. the case of bubbles and crashes in financial markets.Indeed financial markets are excellent and profitable laboratories to testreflexivity theory, and cases of positive feedback loops are investigated in [5].In the next section we will discuss some of the previously described propertiesof the price formation mechanism, market impact in the order book and ordersplitting, as examples of reflexivity where positive and negative feedback loopsmight play a major role.
In the first two sections we have given a general introduction and a specificdescription of the intriguing phenomenology of price formation in financial mar-kets, limit order books, market impact, and order flow. We have discussed thepossible origin of market impact, and the still open issue of its cause. We havealso presented the properties of order flow, i.e. the dynamics of supply and de-mand arriving into the market. We have presented the long-memory propertyof order flow and explained it as a consequence of the splitting of meta-ordersby investors. Furthermore we have shown how the dependence of market im-pact from trade predictability can explain the coexistence of long-memory in theorder flow and the fast decay of autocorrelation of price changes. Even if thedescription is by necessity short and synthetic, we hope that we convinced thereaders that the process of price formation is interesting and still not fully under-9tood. It is obviously at the heart of many economic (and not necessarily onlyfinancial) phenomena and has a significant number of practical consequences.In the text we have also proposed some analogies between the some elementsof price formation and the theory of social reflexivity (reviewed in Section 3).The decisions of investors in financial markets depend on their beliefs on thetraded assets and clearly price plays an important role of signal in this cognitiveactivity of agents. However price is affected, via market impact, by the decisionof the investors and this creates the simultaneous presence of the manipulativeand the cognitive function of humans, a key condition, according to Soros, forsocial reflexivity [16, 9]. We therefore believe that (financial) market microstruc-ture is a perfect playground for studying reflexivity and understanding feedbackloops between these two functions.More in detail, in this chapter we have sketched two possible mechanismsfor reflexivity in price formation. The first is at the core of microstructure,since it concerns the origin of market impact. In fact, price moves in responseto informed trades, but, at least on the short term, it moves also mechanicallyas a consequence of trading orders. Since other market participants cannotdiscern informed trades, also uninformed trades are able to move the price.This manipulative function modifies the cognitive activity of the other marketparticipants, who revise their valuation of the asset as a consequence of theimpact of a (possibly uninformed) trade. This process creates a positive feedbackloop where small price fluctuations, even when generated by uninformed trades,can be amplified by the activity of reflexive agents, and this process can createprice bubbles or short term large price fluctuations.The second mechanism for reflexivity is related to the activity of order split-ting. We have seen that, in consequence of the small liquidity present in financialmarkets and of the existence of market impact, investors who want to trade largequantities split their orders in small pieces and trade them incrementally overperiods of time ranging from few minutes to several days. Other agents con-tinuously monitor the flow of orders arriving to the market with the objectiveof identifying such splitting activities. This is because (i) the splitting activityof a large investor can signal her information on the future value of the priceand (ii) knowing that a large investor is unloading an order gives the opportu-nity of front loading the investor, i.e. trading quickly with the hope to be thecounterpart of the large investor in the future (and realizing a profit). Also inthis case there are several interactions between the cognitive and manipulativefunctions of the agents. The large investor has a belief on the future price ofan asset and through her trading moves the price in that direction. The otheragents monitor the price and the order flow, learning the presence of a largeinvestor and her belief on the price. Through their trading activity they modifythe price pattern which in turn can modify the beliefs of other agents (and evenof the splitting strategy of the large trader).The two mechanisms presented here are clearly not exhaustive of the possi-ble role of reflexivity in price formation and market microstructure. One of thelessons that can be learnt from this type of analysis is that the knowledge andmodeling of the detailed mechanism through which agents interact is critical to10nderstand some of the most important processes in economics and interactionof social agents. The second is that quantitative analysis of data is fundamen-tal to validate qualitative hypothesis on investors’ behavior in the market, toconnect these micro-structural behaviors to market efficiency, and to formulatenew hypotheses about the founding features of social systems.