Statistical_properties_of_stock_order_books_empirical results and models
a r X i v :c o n d -m a t /0203511v 1 25 M a r 2002Statistical properties of stock order books:empirical results and models Jean-Philippe Bouchaud ?,?,Marc M′e zard ??,?,Marc Potters ?November 6,2013?
Commissariat `a l’Energie Atomique,Orme des Merisiers 91191Gif-sur-Yvette cedex ,France ?Science &Finance,CFM,109-111rue Victor Hugo 92353Levallois cedex ,France ??Laboratoire de Physique Th′e orique et Mod`e les Statistiques Universit′e Paris Sud,Bat.100,91405Orsay cedex ,France November 6,2013Abstract We investigate several statistical properties of the order book of three liquid stocks of the Paris Bourse.The results are to a large degree inde-pendent of the stock studied.The most interesting features concern (i)the statistics of incoming limit order prices,which follows a power-law around the current price with a diverging mean;and (ii)the humped shape of the average order book,which can be quantitatively reproduced using a ‘zero
intelligence’numerical model,and qualitatively predicted using a simple approximation.Financial markets o?er an amazing source of detailed data on the collective behaviour of interacting agents.It is possible to ?nd many reproducible patterns and even to perform controled experiments,where the response of the price to agiven perturbation is studied.This brings this rather atypical subject into the realm of experimental science.The situation is simple and well de?ned,since many agents,with all the same goal,trade the very same asset.As such,the statistical analysis of ?nancial markets o?ers an interesting testing ground for more ambitious theories of human activities.One may wonder to what extent it is necessary to invoke human intelligence or rationality to explain the various universal statistical laws which have been recently unveiled by the systematic analysis of very large data sets.
Many statistical properties of?nancial markets have already been explored, and have revealed striking similarities between very di?erent markets(di?erent traded assets,di?erent geographical zones,di?erent epochs)[1,3,2].For exam-ple,the distribution of price changes exhibits a power-law tail with an apparently universal exponent[4,5,6,8].The volatility of most assets shows random vari-ations in time,with a correlation function which decays as a small power of the time lag,again quite universal across di?erent markets[7].
Here,we study the statistics of the order book,which is the ultimate‘mi-croscopic’level of description of?nancial markets.The order book is the list of all buy and sell‘limit’orders,with their corresponding price and volume,at a given instant of time.A limit order speci?es the maximum(resp.minimum) price at which an investor is willing to buy(resp.sell)a certain number of shares (volume).At a given instant of time,all limit buy orders are below the best buy o?er called the bid price,while all sell orders are above the best sell order called the ask price.When a new order appears(say a buy order),it either adds to the book if it is below the ask price,or generates a trade at the ask if it is above (or equal to)the ask price.1The price dynamics is therefore the result of the interplay between the order book and the order?ow.
Complete data on the order book of certain markets,such as the Paris Bourse (now Euronext),is now available,and contains a particularly abundant informa-tion.All orders on all stocks are stored,which makes possible the reconstitution of the full order book at any instant of time.On the example of‘France-Telecom’during the month of February2001,around280,000orders were placed,and the number of transactions was176,000.(Limit orders may never be executed,or may be cancelled at any time).The total number of market orders was around 30,000.Many questions can be studied;the systematic investigation of these data sets is only very recent[9,10,11]and has motivated a number of interest-ing theoretical work[12,13,10,14,15,16].Here,we mostly focus on‘static’properties of the order book,such as the distribution of incoming limit orders, the average shape of the order book in the moving reference frame of the price, or the distribution of volume at the bid/ask.Many other‘dynamical’properties can also be analyzed,such as the response of the price to order?ow,the full temporal correlation of the book,etc.
Our main results are as follows:(a)the price at which new limit orders are placed is,somewhat surprisingly,very broadly(power-law)distributed around the current bid/ask;(b)the average order book has a maximum away from the current bid/ask,and a tail re?ecting the statistics of the incoming orders;(c)the distribution of volume at the bid(or ask)is very broad,and can be?tted by a Gamma distribution.Notably,we?nd exactly the same statistical features,for the three liquid stocks studied.We then study numerically a‘zero intelligence’
model of order book which reproduces most of these empirical results.Finally,we show how the characteristic shape of the average order book can be analytically predicted,using a simple approximation.
The data provided by Paris Bourse gives the history of all transaction prices, with their price,volume and time stamp,of all quotes(bid and ask prices)with the corresponding volumes,and of all orders,with their price,volume and time stamp.Some limit orders may be cancelled before being executed;unfortunately the data base does not contain the time at which a given order is cancelled. We only know at the end of the day whether or not it was cancelled.When reconstructing the order book at a given instant of time,we have therefore made two extreme assumptions,which lead to nearly identical conclusions.Either we discard these orders altogether,or we keep them until the time where we can be sure that they have been cancelled,otherwise they would have been executed since the transaction price was observed to be below(for buy orders)or above(for sell orders)the corresponding limit price.We had available the data corresponding to all stocks during February2001,from which we have extracted three of the most liquid stocks:France-Telecom,Vivendi and Total.We expect that our?ndings will not depend on the particular month considered,but that some di?erences may appear when one studies stocks with smaller capitalisation:these questions will be investigated in the near future.Let us call a(t)the ask price at time t and b(t)the level of bid price at time t.The midpoint m(t)is the average between the bid and the ask:m(t)=[a(t)+b(t)]/2.We will measure distances in‘ticks’,which is the discretization unit of price changes.The tick is0.05 Euros for France-Telecom and Vivendi,and0.10Euros for Total.The price of the stocks was on the order of100Euros for the three cases.We denote by b(t)??the price of a new buy limit order,and a(t)+?the price of a new sell limit order.Notice that?can be negative(this is the only case where the gap g(t)=a(t)?b(t)is reduced),but is always larger than?g(t)(otherwise there would be a transaction).A?rst interesting question concerns the distribution density of?,i.e.the distance between the current price and the incoming limit order.We?nd that P(?)is identical for buy and sell orders(up to statistical ?uctuations);the shape of P(?)is found to be very well?tted(see Fig.1)by a single power-law:
?μ0
P(?)∝
110
1001000
?110
100
1000
10000
P (?) Power?law, μ=0.5
France?Telecom
Figure 1:Cumulative distribution of the position ?of incoming orders,as a function of 1+?(in ticks).The symbols correspond to buy orders on France-Telecom,but sell orders behave similarly.The same power-law behaviour also holds for Vivendi and Total.
of course the distribution is ultimately cut-o?for large ?’s).
It is quite surprising to observe such a broad distribution of limit order prices,which tells us that in the opinion of market participants,the price of the stock in a near future could be anything from its present value to 50%above or below this value,with all intermediate possibilities.This means that market participants believe that large jumps in the price of stocks are always possible,and place orders very far from the current price in order to take advantage of these large potential ?uctuations.If this was their reasoning,the probability to place an order at distance ?should be proportional to the probability that the price moves more than ?in order to meet the order.Since the tail of the distribution of price increments is a power-law with an exponent μδp ≈3[4],this would indeed lead to a power-law for P (?),but with a value μ=μδp ?1≈2larger than the observed one.Market participants seem to overestimate the probability of very large moves (but see below).
Limit orders strongly vary in volume.We ?nd that the unconditional limit order size,φ,is distributed uniformly in log-size,between 10and 50,000(both for buy or sell orders).One can study the distribution of incoming volume φas a function of the distance from the current price ?.We ?nd that the conditional averaged volume φ |?is roughly independent of ?between 1and 20ticks,but decays as a power law ??ν,with ν?1.5beyond ??≈20ticks.Not unexpectedly,
050010001500
Figure 2:Average order book for the three stocks,as a function of the distance ?from the current bid (or ask).Both axis have been rescaled in order to collapse the curves corresponding to the three stocks.The thick dots correspond to the numerical model explained below,with Γ=10?3and p m =0.25.
extremely far limit orders tend to be of smaller https://www.wendangku.net/doc/a11491354.html,ing the shape of P (?),the distribution of incoming volume should then decay as ??1?μfor ???,a feature that we have con?rmed directly.We note that μ+ν≈μδp ?1,suggesting that the argument above might concern volume ?ow rather than order ?ow.
We now turn to the characteristic humped shape of the order book.The order ?ow is maximum around the current price,but an order very near to the current price has a larger probability to be executed and disappear from the book.It is thus not a priori clear what will be the shape of the average order book.Quite interestingly,we ?nd that the (time-averaged)order book is symmetrical,and has a maximum away from the current bid (ask):see Fig. 2.This was also noted in [9].This shape furthermore appears to be universal ,since the three stocks studied lead to the same result,up to a rescaling of the both ?axis and the volume axis.The empirical determination of the average order book is the central result of this paper,and simple models that explain this shape will be discussed below.
The next question concerns the volume ?uctuations around this average shape.We ?rst study the distribution R (V )of volume at the bid (or ask).Again,the two are identical,and can be ?tted by a Gamma distribution for the volume (Fig.
012
345
Log 10(V)05000
10000
15000
R (V )
France Telecom (Asks)
Gamma distribution, γ=0.7
Figure 3:Distribution of the log-volume at the ask (same for bids),for France Telecom.The ?t corresponds to Gamma distribution,Eq.(2),with γ=0.7.
3):
R (V )=V γ?1exp(?V
V 2 |?? V |2?
one.Here we depart from[16],where all orders have unit size and the‘sprinkling’distribution is uniform between0and a certain?max.The size distribution was actually found to play a minor role,but the choosing the correct P(?)is crucial to reproduce the shape of the order book.We also launch a certain fraction p m of market orders(chosen to be typically1/6?1/4th of the total)in order to trigger trades.Finally,with a constant probabilityΓ(typically of order10?3)per unit time,independently of both size and position in the book,an order is cancelled. We have found that this simple model is able to reproduce quantitatively many of the features observed in the empirical data,such a the shape of the order book (see Fig.2),or the Gamma distribution of volume at the bid/ask R(V),with a similar value for the exponentγ.This model also reproduces other empirical properties,such as the short time dynamics of the‘midpoint’and the sublinear volume dependence of the response to an incoming market order[17,16,18].
Finally,we discuss a simple analytical approximation which allows us to com-pute the average order book from the ingredients of the numerical model.This was also attempted in[16].Although very few details were given in[16],our method appears to be quite di?erent from theirs.Consider sell orders.Those at distance?from the current ask at time t are those which were placed there at a time t′ ρ(?,t)= t?∞d t′ duP(?+u)P(u|C(t,t′))e?Γ(t?t′),(4) where P(u|C(t,t′))is the conditional probability that the time evolution of the price produces a given value of the ask di?erence u=a(t)?a(t′),given the condition that the path always satis?es?+a(t)?a(t′′)≥0at all intermediate times t′′∈[t′,t].The evaluation of P requires the knowledge of the statistics of the price process.Because of the exponential cut-o?e?Γ(t?t′),only the short time behaviour of the process is relevant where the con?nement e?ects of the order book on the price are particularly important[16].Nevertheless,to make progress,we will assume that the process is purely di?usive.In this case,P can be calculated using the method of images.One?nds: P(u|C(t,t′))=12πDτ exp ?u22Dτ ,(5) whereτ=t?t′and D is the di?usion constant of the price process. After a simple computation,one?nally?nds,up to a multiplicative constant which only a?ects the overall normalisation ofρst(?)=ρ(?,t→∞):ρst(?)=e?α? ?0d uP(u)sinh(αu)+sinh(α?) ∞?d uP(u)e?αu,(6) 024 6810 ?00.5 11.501 10 ?00.5 1 1.5 2 Figure 4:The average order book of the numerical model with various choices of parameters (μ=.6,p m ∈{1/4,1/6},and Γ∈{10?3,510?4}is compared to the approximate analytical prediction,(full curve),Eq.(7).After rescaling the axes,the various results roughly scale on the same curve,which is well reproduced by our simple analytic argument.where α?1= the competition between a power-law?ow of limit orders with a?nite lifetime, and the price dynamics that removes the orders close to the current price.These e?ects lead to a universal shape which will presumably hold for many di?er-ent markets,provided the lifetime of orders is su?ciently long compared to the typical time between trades2. In conclusion,we have investigated several‘static’properties of the order book.We have found that the results appear to be independent of the stock studied.The most interesting features concern(i)the statistics of incoming limit order prices,which follows a power-law around the current price with a diverging mean–suggesting that market participants believe that very large variations of the price are possible within a rather short time horizon;and(ii)the humped shape of the average order book,which can be quantitatively reproduced using a‘zero intelligence’numerical model,and qualitatively predicted using a simple approximation.One of the most interesting open problems is,in our view,to explain the clear power-law behaviour of the incoming orders and the value ofμthat we have found.The dynamical properties of the order book are also very interesting,and will be the subject of a further study[18]. Acknowledgements:We thank Jean-Pierre Aguilar,Jelle Boersma,Laurent Laloux, Andrew Matacz,Philip Seager and Denis Ullmo for stimulating and useful dis-cussions. References [1]for a recent review,see:R.Cont,Quantitative Finance,1,223(2001),and refs.therein. [2]J.-P.Bouchaud and M.Potters,Th′e orie des Risques Financiers,Al′e a-Saclay, 1997;Theory of Financial Risks,Cambridge University Press,2000. [3]R.Mantegna&H.E.Stanley,An Introduction to Econophysics,Cambridge University Press,1999. [4]V.Plerou,P.Gopikrishnan,L.A.Amaral,M.Meyer,H.E.Stanley,Phys. Rev.E606519(1999). [5]T.Lux,Applied Financial Economics,6,463,(1996). [6]D.M.Guillaume,M.M.Dacorogna,R.D.Dav′e,U.A.M¨u ller,R.B.Olsen and O.V.Pictet,Finance and Stochastics195(1997). [7]for a review,see e.g.J.-F.Muzy,J.Delour,E.Bacry,Eur.Phys.J.B17, 537-548(2000),and refs.therein. [8]F.Longin,Journal of Business,69383(1996) [9]B.Biais,P.Hilton,C.Spatt,An empirical analysis of the limit order book and the order?ow in the Paris Bourse,Journal of Finance,50,1655(1995) [10]S.Maslov,Simple model of a limit order-driven market,Physica A278, 571(2000);S.Maslov,https://www.wendangku.net/doc/a11491354.html,lis,Price?uctuations from the order book perspective–empirical facts and a simple model,Physica A299,234(2001) [11]D.Challet,R.Stinchcombe,Analyzing and modelling1+1d markets,preprint cond-mat/0106114 [12]P.Bak,M.Paczuski,and M.Shubik,Physica A246,430(1997) [13]David L.C.Chan,David Eliezer,Ian I.Kogan,Numerical analysis of the Minimal and Two-Liquid models of the Market Microstructure,preprint cond-mat/0101474 [14]H.Luckock,A statistical model of a limit order market,Sidney University preprint(September2001). [15]F.Slanina,Mean-?eld approximation for a limit order driven market model, preprint cond-mat/0104547 [16]Marcus G.Daniels,J.Doyne Farmer,Giulia Iori,Eric Smith,How storing supply and demand a?ects price di?usion,preprint cond-mat/0112422. [17]P.Gopikrishnan,V.Plerou,X.Gabaix,H.E.Stanley,Statistical Properties of Share volume traded in Financial Markets,Phys.Rev E62,R4493(2000). [18]J.P.Bouchaud,M.M′e zard,M.Potters,in preparation.