Presentation on theme: "Trading mechanisms, Roll model"— Presentation transcript:
1Trading mechanisms, Roll model Fabrizio LilloLecture 2
2Market microstructure Market microstructure “is devoted to theoretical, empirical, and experimental research on the economics of securities markets, including the role of information in the price discovery process, the definition, measurement, control, and determinants of liquidity and transactions costs, and their implications for the efficiency, welfare, and regulation of alternative trading mechanisms and market structures” (NBER Working Group)
3Garman (1976)“We depart from the usual approaches of the theory of exchange by (1) making the assumption of asynchronous, temporally discrete market activities on the part of market agents and (2) adopting a viewpoint which treats the temporal microstructure, i.e. moment-to-moment aggregate exchange behavior, as an important descriptive aspect of such markets.”
4CharacteristicsSecurities value comprises private and common componentsCommon: cash flowPrivate: Information, investment horizon, risk exposureHeterogeneous private valuesSpecific about the market mechanismMultiple characterization of prices (no Walrasian tatonnement)
5Liquidity Elasticity Cost of trading “Depth, breadth, and resiliency” Above the current market price there is a large incremental quantity for saleMany market participants, none with significant market powerPrice effects die out quicklyLiquidity suppliers (sell side: brokers, dealers, intermediaries)Liquidity demanders (buy side: individuals and institutional investors)Market (liquidity) consolidation vs fragmentation“Liquidity is a lot like pornography: we know it when we see it and we can measure it even if we do not know how to define it” (Lehman)
6The Questions (Hasbrouck) What are the optimal trading strategies for typical trading problems?Exactly how is information impounded in prices?How do we enhance the information aggregation process?How do we avoid market failures?What sort of trading arrangements maximize efficiency?How is market structure related to the valuation of securities?What can market/trading data tell us about the informational environment of the firm?What can market/trading data tell us about long term risk?
7Financial dataIn the last thirty years the degree of resolution of financial data has increasedDaily dataTick by tick dataOrder book data“Agent” resolved dataHigh frequency data are like a powerful microscope -> Change in the nature of social sciences ?High frequency data require sophisticated stochastic modeling and statistical techniques
8Daily dataDaily financial data are available at least since nineteenth centuryUsually these data contains opening, closing, high, and low price in the day together with the daily volumeStandard time series methods to investigate these data(from Gopikrishnan et al 1999)
9Tick by tick dataFinancial high frequency data usually refer to data sampled at a time horizon smaller than the trading dayThe usage of such data in finance dates back to the eighties of the last centuryBerkeley Option Data (CBOE)TORQ database (NYSE)HFDF93 by Olsen and Associates (FX)CFTC (Futures)
10Higher resolution means new problems data size: example of a year of a LSE stock12kB (daily data)15MB (tick by tick data)100MB (order book data)the total volume of data related to US large caps in 2007 was 57 million lines, approximately a gigabyte of stored dataA large cap stock in 2007 had on average 6 transactions per second, and on the order of 100 events per second affecting the order bookirregular temporal spacing of eventsthe discreteness of the financial variables under investigationproblems related to proper definition of financial variablesintraday patternsstrong temporal correlationsspecificity of the market structure and trading rules.
12More structured data require more sophisticated statistical tools data size:more computational powerbetter filtering proceduresirregular temporal spacing of eventspoint processes, ACD model, CTRW model…the discreteness of the financial variables under investigationdiscrete variable processesproblems related to proper definition of financial variablesintraday patternsstrong temporal correlationsmarket microstructurespecificity of the market structure and trading rules.better understanding of the trading process
13Trading mechanismsA trade is a preliminary agreement which sets in motion the clearing and settlement procedures that will result in a transfer of securities and fundsTrading often involves a broker which may simply provide a conduit to the market but may also act as the customer’s agent.Principal-agent relationshipThe broker’s duty is “best execution”: it provides credit, clearing services, information, analytics. Usually it is not a counterparty.
14Limit order bookMany stock exchanges (NYSE, LSE, Paris) works through a double auction mechanismDifferent levels oftransparency (recentat NYSE). Hidden andIceberg ordersMany order books vsa consolidated limit orderBookPrice-time priorityWalk the book
15Representation of limit order book dynamics (Ponzi, Lillo, Mantegna 2007)
17Floor market Old market structure Now used only in US commodity futures markets (Chicago Board of Trade, New York Mercantile Exchange, Chicago Mercantile Exchange)Little transparency on dataBund futures at London International Financial Futures Exchange (LIFFE) and Eurex (1997)
18Dealer marketA dealer is an intermediary who is willing to act as a counterparty for the trades of his customersForeign exchange, corporate bond, swap marketsCustomers cannot typically place limit ordersA large customer can have relationships with many competing dealersLow transparency: quotes in response to customer inquiries and not publicly availableInterdealer trading is also important (for dealer inventory management). Limit order book for FXDealers facilitate large (block) trades. Upstairs or off book market
19Crossing networksA buyer and a seller are paired anonymously for an agreed-on quantityThe trade is priced by reference to a price in another (typically the listed) marketITG’s POSIT, INSTINET, etc..
21Roll model (1984)“It illustrates a dichotomy fundamental to many microstructure models: the distinction between price components due to fundamental security value and those attributable to the market organization and trading process”It is also didactic in showing how to go from a structural model to a statistical model
22Roll model (1984) Random walk model of the efficient price where ut is an i.i.d. noise with variance s2uAll trades are conducted through a dealer who posts bid and ask pricesDealer incurs a noninformational cost c per trade (due to fixed costs: computers, telephone)The bid and the ask areand thus the spread is 2c
23The transaction price pt is where qt=+1 (-1) if the customer is buyer (seller).Moreover qt are assumed serially independent.The variance of transaction price increments isThe autocovariance of transaction price increments isand it is zero for lag larger than one
24Roll model was used to determine the spread from transaction data For example (Hasbrouck) the estimated first order covariance for PCO (Oct 2003) is g1= , which implies c=$0.017 and a spread 2c=$ The time weighted spread is $0.032Roll model was used to determine the spread from transaction dataAutocorrelation function of transaction price returns (solid circles), absolute value of transaction price returns (open circles), returns of midprice sampled before each transaction (filled triangles), and returns of midprice sampled at each market event (filled squares) for the AstraZeneca (AZN) stock traded at LSE in The lags on the horizontal line are measured in event time.
25Time series perspective The autocovariance structure of transaction price increments in the Roll model is the same as in a MA(1) processThe Roll model is a structural model, while its time series representation is a statistical modelBy assuming covariance stationarity we can use the Wold theorem: any zero-mean covariance stationary process can be represented aswhere et is a zero mean white noise process and kt is a linearly deterministic process (can be predicted arbitrarily well by a linear projection on past observation of xt)
27In the Roll model: everyone possesses the same information trades have no impactTrading is not a strategic problem because agents always face the same spreadCost is fixed and uninformational
28Asymmetric information models The security payoff is usually of a common value natureThe primary benefit derived from ownership of the security is the resale value or terminal liquidating dividend that is the same for all holdersBut for trade to exist, we also need private value components, that is, diversification or risk exposure needs that are idiosyncratic to each agent
29Public information initially consists of common knowledge concerning the probability structure of the economy, in particular the unconditional distribution of terminal security value and the distribution of types of agentsAs trading unfolds, the most important updates to the public information set are market data, such as bids, asks, and the prices and volumes of tradesThe majority of the asymmetric information models in microstructure examine market dynamics subject to a single source of uncertaintyThus, the trading process is an adjustment from one well-defined information set to another (no stationarity, ergodicity, time-homogeneity)
30Sequential trade models: randomly selected traders arrive at the market singly, sequentially, and independently. (Copeland and Galai (1983) and Glosten and Milgrom (1985)).When an individual trader only participates in the market once, there is no need for her to take into account the effect her actions might have on subsequent decisions of others.Strategic trader models: A single informed agent who can trade at multiple times.A trader who revisits the market, however, must make such calculations, and they involve considerations of strategyIn both models a trade reveals something about the agent’s private information
31Strategic model, Kyle model Fabrizio LilloLecture 4
32Strategic model: Kyle (1985) The model describes a case of information asymmetry and the way in which information is incorporated into price.It is an equilibrium modelThere are several variants: single period, multiple period, continuous timeThe model postulates three (types of) agents: an informed trader, a noise trader, and a market maker (MM)
33The terminal (liquidation) value of the asset is v, normally distributed with mean p0 and variance S0.The informed trader knows v and enters a demand xNoise traders submit a net order flow u, normally distributed with mean 0 and variance s2u.The MM observes the total demand y=x+u and then sets a price p. All the trades are cleared at p, any imbalance is exchanged by the MM.
34The informed trader wants to trade as much as possible to exploit her informational advantage However the MM knows that there is an informed trader and if the total demand is large (in absolute value) she is likely to incur in a loss. Thus the MM protects herself by setting a price that is increasing in the net order flow.The solution to the model is an expression of this trade-off
35p=(v-p)x =x[v-l(u+x)-m] Informed traderThe informed trader conjectures that the MM uses a linear price adjustment rule p=ly+m, where l is inversely related to liquidity.The informed trader’s profit isp=(v-p)x =x[v-l(u+x)-m]and the expected profit isE[p]=x(v-lx-m)The informed traders maximizes the expected profit, i.e.x=(v-m)/2lIn Kyle’s model the informed trader can loose money, but on average she makes a profit
36Market makerThe MM conjectures that the informed trader’s demand is linear in v, i.e. x=a+bvKnowing the optimization process of the informed trader, the MM solves(v-m)/2l=a+bva=-m/2l b=1/2lAs liquidity drops the informed agent trades lessThe MM observes y and setsp=E[v|y]
37E[Y|X=x]=mY+(sXY/sx2)(x-mX) SolutionIf X and Y are bivariate normal variables, it isE[Y|X=x]=mY+(sXY/sx2)(x-mX)This can be used to findE[v|y]=E[v|u+a+bv]The solution is
38Solution (II)The impact is linear and the liquidity increases with the amount of noise tradersThe informed agent trades more when she can hide her demand in the noise traders demandThe expected profit of the informed agent depends on the amount of noise tradersThe noise traders loose money and the MM breaks even (on average)
39Kyle model - summaryThe model can be extended to multiple periods in discrete or in continuous timeThe main predictions of the model areThe informed agent “slices and dices” her order flow in order to hide in the noise trader order flowLinear price impactUncorrelated total order flowPermanent and fixed impact
40Current microstructural paradigm There are two types of traders: informed and uninformedInformed traders have access to valuable information about the future price of the asset (fundamental value)Informed traders sell (buy) over- (under-)priced stocks making profit AND, through their own impact, drive quickly back the price toward its fundamental valueExamples: Kyle, Glosten-MilgromIn this way information is incorporated into prices, markets are efficient, and prices are unpredictable
42Is this the right explanation? Orders of magnitude InformationHow large is the relative uncertainty on the fundamental value? 10-3 or 1 (Black 1986)Financial experts are on the whole pretty bad in forecasting earnings and target pricesMany large price fluctuations not explained by public newsTimeTime scale for news: 1 hour-1day (?)Time scale for trading: 10-1s:100sTime scale for market events: 10-2:10-1sTime scale for “large” price fluctuations: 10 per dayVolumeDaily volume: 10-3:10-2 of the market capitalization of a stockVolume available in the book at a given time: 10-4:10-5 of the market capitalizationVolume investment funds want to buy: up to 1% of a company
43ConsequencesFinancial markets are in a state of latent liquidity, meaning that the displayed liquidity is a tiny fraction of the true (hidden) liquidity supplied/demandedDelayed market clearing: traders are forced to split their large orders in pieces traded incrementally as the liquidity becomes availableMarket participants forms a kind of ecology, where different species act on different sides of liquidity demand/supply and trade with very different time scales
45Mechanical price formation “Market”Price impactOrder flowPrice processPrice impact is the correlation between an incoming order and the subsequent price changeFor traders impact is a cost -> Controlling impactVolume vs temporal dependence of the impact
46Price (or market) impact Price impact is the correlation between an incoming order and the subsequent price changeFor traders impact is a cost -> Controlling impactVolume vs temporal dependence of the impact
47Why price impact?Given that in any transaction there is a buyer and a seller, why is there price impact?Agents successfully forecast short term price movements and trade accordingly (i.e. trade has no effect on price and noise trades have no impact)The impact of trades reveals some private information (but if trades are anonymous, how is it possible to distinguish informed trades?)Impact is a statistical effect due to order flow fluctuations (zero-intelligence models, self-fulfilling prophecy)“Orders do not impact prices. It is more accurate to say that orders forecast prices” (Hasbrouck 2007)
48Market impact Market impact is the price reaction to trades However it may indicate many different quantitiesPrice reaction to individual tradesPrice reaction to an aggregate number of tradesPrice reaction to a set of orders of the same sign placed consecutively by the same trader (hidden order)Price reaction in a market to a trade in another market (e.g. electronic market and block market)
49Volume and temporal component of market impact of individual trades Market impact is the expected price change due to a transaction of a given volume. The response function is the expected price change at a future time
50Master curve for individual impact NYSELondon Stock ExchangeImpact of individual transaction is NOT universalParis Bourse(Potters et al. 2003)Individual market impact is a concave function of the volume
51Master curve for individual impact (Lillo et al. Nature 2003)GROUP A -> least capitalized group GROUP T -> most capitalized groupNYSE
52Price impact from book shape Let indicate the cumulative number of shares (depth) up to price return rA market order of size V will produce a returnFor example ifthen the price impact is
53Real and virtual impact Is this explanation in terms of the relation between price impact and the limit order book shape correct ?The basic assumptions are:the traders placing market orders trade their desired volume irrespectively from what is present on the limit order bookthe limit order book is filled in a continuous way, i.e. all the price levels are filled with limit ordersWe test the first assumption by measuring the virtual price impact, i.e. the price shift that would have been observed in a given instant of time if a market order of size V arrived in the marketWe test the second assumption by considering the fluctuations of market impact
56Fluctuations of the impact Let us decompose the conditional probability of a return r conditioned to an order of volume V asand we investigate the cumulative probabilityfor several different value of V.This is the cumulative probability of a price return r conditioned to the volume and to the fact that price moves
57The fluctuations in market impact are important Different curves correspond to different trade volumeIndependent fromthe volume !!The role of the transaction volume is negligible. The volume is important in determining whether the price moves or notThe fluctuations in market impact are important
58How can small volume transactions create large price changes ? The impact function is NOT deterministic and the fluctuations of price impact are very large.These results show that the picture of the book as an approximately constant object is substantially wrongCentral role of fluctuations in the state of the bookHow can small volume transactions create large price changes ?
59Casestudyfirst gapLarge price changes are due to the granularity of supply and demandThe granularity is quantified by the size of gaps in the Limit Order Book
60Origin of large price returns First gap distribution (red) and return distribution (black)Large price returns are caused by the presence of large gaps in the order book
61Tail exponents (Farmer et al 2004) Low liquidity (red), medium liquidity (blue), high liquidity (green)A similar exponent describes also the probability density of thesuccessive gaps
62Walking up the book?The analysis of transactions in both large and small tick size LSE stocks reveal that the “walking up” of the book, i.e. a trades that involves more than one price level in the limit order book, is an extremely rare eventThis again strengthens the idea that market order traders strongly condition their order size to the best available volumeThus the use of the instantaneous shape of the limit order book for computing the market liquidity risk can be very misleading# of gaps12345>5AZN44%49%5.8%0.80%0.15%0.026%0.22%VOD64%34%1.7%0.094%0.010%0.0002%0.19%
63Financial markets are sometimes found in a state of temporary liquidity crisis, given by a sparse state of the book. Even small transactions can trigger large spread and market instabilities.Are these crises persistent?How does the market react to these crises?What is the permanent effect of the crises on prices?
64Persistence of spread and gap size DFA of spread (Plerou et al 2005)
65Market reaction to temporary liquidity crises We quantify the market reaction to large spread changes.The presence of large spread poses challenging questions to thetraders on the optimal way to trade.Liquidity takers have a strong disincentive for submitting marketorders given that the cost,the bid-ask spread, is largeLiquidity provider canprofit of a large spread byplacing limit orders andobtaining the best position.However the optimal orderplacement inside the spread isa nontrivial problem.Rapidly closing the spread -> priority but risk of informed tradersSlowly closing the spread -> “testing” the informed traders but risk of losing priority
66Traders’ behavior (order flow) Spread is an important determinant in order placement decision process(see also Mike and Farmer 2006)
67Limit order placement inside the spread (see also Mike and Farmer 2006) Waiting time between two spread variations
68We wish to answer the question: how does the spread s(t) return to a normal value after a spread variation?To this end we introduce the quantity
69Permanent impact completely permanent impact zero permanent impact Permanent impact is roughly proportional to immediate impact
70Limit order placement(from Zovko and Farmer 2002)Limit order price is power law distributed with a tail exponent in the rangeMoreover limit order price is correlated with volatility
71An utility maximization argument For a given limit price D, volatility s, and investment horizon T the investor is faced with the lotteryFrom first passage time of the price random walkSo the expected utility is where u(D) is the utility function.Agent optimizes her limit order placement by choosing the lottery (i.e. the value of D) which maximizes the expected utility.For several choices of u(D) it is possible to solve the problem analytically. For example for power utility function u(x)=C xaSome variable must be highly fluctuating: market (s) or agent (a or T)?
72Market member (broker) data The investigated markets are:Spanish Stock Exchange (BME)London Stock Exchange (LSE).Firms are credit entities and investment firms which are members of the stock exchange and are entitled to trade in the market.Roughly 200 Firms in the BME and LSE (350/250 in the NYSE)Investigation at the level of market members and not of the agents (individuals and institutionsThe dataset covers the whole marketThe resolution is at the level of individual trade (no temporal aggregation)
73It is possible to show that heterogeneity in volatility or utility function cannot explain the value of the exponent empirically observed.The only possibility is a strong heterogeneity in investment horizon THeterogeneity in volatilityHeterogeneity in traders preferencesCuriously, the same exponent of the time scale distribution is obtained by considering the time to fill of limit orders, metaorder splitting (see below), and a modified GARCH (Borland and Bouchaud 2005)
77Order flowOne of the key problem in the analysis of markets is the understanding of the relation between the order flow and the process of price formation.The order flow is strongly depending on- the decisions and strategies of traders- the exploiting of arbitrage opportunitiesThis problem is thus related to the balance between supply and demand and to the origin of the market efficiency.
78Market order flowWe investigate the London Stock Exchange in the periodWe consider market orders, i.e. orders to buy at the best available price triggering a tradeWe consider the symbolic time series obtained by replacing buy orders with +1 and sell orders with -1The order flow is studied mainly in event time....+1,+1,-1,-1,-1,+1,-1,+1,+1,+1,-1,-1,+1,...
79Order flow dynamics Why is the order flow a long-memory process ? Time series of signs of market orders is a long memory process (Lillo and Farmer 2004, Bouchaud et al 2004)Why is the order flow a long-memory process ?We show (empirically) that:It is likely due to splittingIt requires a huge heterogeneity in agent size
80The sign time series of the three types of orders Limit OrdersCancellationsThe sign time series of the three types of ordersis a long-memory processHurst exponent
81Long memory and efficiency How can the long memory of order flow be compatible with market efficiency?In the previous slides we have shown two empirical factsSingle transaction impact is on average non zero and given byThe sign time series is a long memory process
82Naïve model Consider a naïve random walk model of price dynamics It follows thatIf market order signs et are strongly correlated in time, price returns are also strongly correlated, prices are easily predictable, and the market inefficient.
83There are two possible modifications It is not possible to have an impact model where the impact is both fixed and permanentThere are two possible modificationsA fixed but transient impact model (Bouchaud et al. 2004)A permanent but variable (history dependent) impact model (Lillo and Farmer 2004, Gerig 2007, Farmer, Gerig, Lillo, Waelbroeck)
84Fixed but transient impact model (Bouchaud et al 2004) The model assumes that the price just after the (t-1)-th transaction isand return iswhere the propagator G0(k) is a decreasing function.The propagator can be chosen such as to make the market exactly efficient. This can be done by imposing that the volatility diffuses normally. The volatility at scale iswhere D is a correlation-induced contribution
85The correlation in the order flow decays as a power law with exponent g
86The model is able to make predictions on the response function defined as which can be re-expressed in terms of the propagator and of the order sign correlation Cj
88History dependent, permanent impact model We assume that agents can be divided in three classesDirectional traders (liquidity takers) which have large hidden orders to unload and create a correlated order flowLiquidity providers, who post bid and offer and attempt to earn the spreadNoise tradersThe strategies of the first two types of agents will adjust to remove the predictability of price changes
89Model for price diffusion We neglect volume fluctuations and we assume that the naïve model is modified aswhere W is the information set of the liquidity provider.Ex post there are two possibilities, either the predictor was right or wrongLet q+t (q-t) be the probability that the next order has the same (opposite) sign of the predictor and r+t (r-t) are the corresponding price change
90The efficiency condition Et-1[rt|W]=0 can be rewritten as The market maker has expectations on q+t and q-t given her information set W and adjusts r+t and r-t in order to make the market efficient-----> MARKET EFFICIENCYASYMMETRIC LIQUIDITY MODEL
92A linear model - the information set W of the liquidity provider The history dependent, permanent model is completely defined when one fixes- the information set W of the liquidity provider- the model used by the liquidity provider to build her forecastAs an important example we consider the case in which- the information set is made only by the past order flow- the liquidity provider uses a finite or infinite order autoregressivemodel to forecast order flow
93If the order flow is long memory, i. e If the order flow is long memory, i.e the optimal parameters of the autoregressive model are and the number of lags K in the model should be infinite.If, more realistically, K is finite the optimal parameters of the autoregressive model follows the same scaling behavior with kUnder these assumptions and if K is infinite the linear model becomes mathematically equivalent to the fixed-temporary model (or propagator) model by Bouchaud et al. with
94Asymmetric liquidity depends on the information set W. This model predicts the existence of two classes of traders that are natural counterparties in many transactionsLarge institutions creates predictable component of order flow by splitting their large hidden ordersHedge funds and high frequency traders removes this predictability by adjusting liquidity (and making profit)This ecology of market participants is empirically detectable?What is the interaction pattern between market participants?
101Hidden ordersIn financial markets large investors usually need to trade large quantities that can significantly affect prices. The associated cost is called market impactFor this reason large investors refrain from revealing their demand or supply and they typically trade their large orders incrementally over an extended period of time.These large orders are called packages or hidden orders and are split in smaller trades as the result of a complex optimization procedure which takes into account the investor’s preference, risk aversion, investment horizon, etc..We want to detect empirically the presence of hidden orders from the trading profile of the investors
102Model of order splitting There are N hidden orders (traders).An hidden order of size L is composed by L revealed ordersThe initial size L of each hidden order is taken by a given probability distribution P(L). The sign si (buy or sell) of the hidden order is initially set to +1 or -1 in a random way.At each time step an hidden order i is picked randomly and a revealed order of sign si is placed in the market. The size of the hidden order is decreased by one unit..When an hidden order is completely executed, a new hidden order is created with a new size and a new sign.
103We assume that the distribution of initial hidden order size is a Pareto distribution The rationale behind this assumption is thatIt is known that the market value of mutual funds is distributed as a Pareto distribution (Gabaix et al., 2003)It is likely that the size of an hidden order is proportional to the firm placing the order
104We prove that the time series of the signs of the revealed order has an autocorrelation function decaying asymptotically as
105Testing the modelsIt is very difficult to test the model because it is difficult to have information on the size and number of hidden orders present at a given time.We try to cope with this problem by taking advantage of the structure of financial markets such as London Stock Exchange (LSE).At LSE there are two alternative methods of trading- The on-book (or downstairs) market is public and execution is completely automated (Limit Order Book)- The off-book (or upstairs) market is based on personal bilateral exchange of information and trading.We assume that revealed orders are placed in the on-book market, whereas off-book orders are proxies of hidden orders
106Volume distributionThe volume of on-book and off-book trades have different statistical propertiesThe exponent a=1.5 for the hidden order size and the market order sign autocorrelation exponent g are consistent with the order splitting model which predicts the relation g=a-1.Is it possible to identify directly hidden orders?
107Our investigationThe investigated market is the Spanish Stock Exchange (BME).Firms are credit entities and investment firms which are members of the stock exchange and are entitled to trade in the market.200 Firms in the BME (350/250 in the NYSE)Investigation at the level of market members and not of the agents (individuals and institutionsThe dataset covers the whole marketThe resolution is at the level of individual trade (no temporal aggregation)
109Detecting hidden orders We developed a statistical method to identify periods of time when an investor was consistently (buying or selling) at a constant rate -> Hidden orders
110Distributional properties of hidden orders Investment horizonNumber of transactionsVolume of the orderCircles and squares are data taken from Chan and Lakonishok at NYSE (1995) and Gallagher and Looi at Australian Stock Exchange (2006)
111Large hidden ordersThe distributions of large hidden orders sizes are characterized by power law tails.Power law heterogeneity of investor typical (time or volume) scaleThese results are not consistent with the theory of Gabaix et al. Nature 2003)
112Allometric relations of hidden orders We measure the relation between the variables characterizing hidden orders by performing a Principal Component Analysis to the logarithm of variables.
113CommentsThe almost linear relation between N and V indicates that traders do not increase the transaction size above the available liquidity at the best (see also Farmer et al 2004)For the Nm-Vm and the T-Nm relations the fraction of variance explained by the first principal value is pretty highFor the T-Vm relation the fraction of variance explained by the first principal value is smaller, probably indicating an heterogeneity in the level of aggressiveness of the firm.Also in this case our exponents (1.9, 0.66, 1.1) are quite different from the one predicted by Gabaix et al theory (1/2, 1, 1/2)
114Role of agents heterogeneity We have obtained the distributional properties and the allometric relations of the variables characterizing hidden orders by pooling together all the investigated firmsAre these results an effect of the aggregation of firms or do they hold also at the level of individual firm?
115Heterogeneity and power law tails For each firm with at least 10 detected hidden orders we performed a Jarque-Bera test of the lognormality of the distribution of T, Nm, and VmFor the vast majority of the firms we cannot reject the hypothesis of lognormalityThe power law tails of hidden order distributions is mainly due to firms (size?) heterogeneityBBVA SAN TEF