8/13/2019 Farmer Single Curve Collapse of the Price Impact Function for the NYSE

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arXiv:cond-mat/0207428v1

17Jul2002

Single Curve Collapse of the Price Impact Function for the New York Stock Exchange

Fabrizio Lillo

Istituto Nazionale per la Fisica della Materia, Unita di Palermo, Viale delle Scienze, I-90128, Palermo, Italy

J. Doyne Farmer

McKinsey Professor, Santa Fe Institute, 1399 Hyde Park Road, Santa Fe, NM 87501

Rosario N. Mantegna

Istituto Nazionale per la Fisica della Materia, Unita di Palermo, Viale delle Scienze,I-90128, Palermo, Italy and Dipartimento di Fisica e Tecnologie Relative,

Universita di Palermo, Viale delle Scienze, I-90128, Palermo, Italy

We study the average price impact of a singletrade executed in the NYSE. After appropriateaveraging and rescaling, the data for the 1000 most highly capitalized stocks collapse onto a singlefunction, giving average price shift as a function of trade size. This function increases as a p owerthat is the order of 1/2 for small volumes, but then increases more slowly for large volumes. Weobtain similar results in each year from the period 1995 - 1998. We also find that small volumeliquidity scales as a power of the stock capitalization.

Although supply and demand are perhaps the mostfundamental concepts in economics, finding any generalform for their behavior has proved to be elusive. Herewe build on earlier studies of how trading affects prices[1, 2, 3, 4, 5, 6, 7, 8, 9, 10]. Our study adds to theseprevious efforts by using huge amounts of data, by look-ing at the short term response to a single trade, and bymeasuring the market activity in units of transactionsrather than seconds, so that we can more naturally ag-

gregate data for many different stocks. This allows usto find regularities in the response of prices to new or-ders that have previously been hidden by the extremelyhigh noise levels that dominate financial prices. We studythe 1000 largest stocks of the New York Stock Exchange,from (1995-1998), and find that, by appropriate aver-aging and rescaling, it is possible to collapse the priceshift caused by a transaction onto a single curve. Theprice shift grows slowly with transaction size, growingvery roughly as the 1/2 power in the small volume limit,and much more slowly in the large volume limit. The factthat such consistent results are seen across many stocksand for four different years suggests regularities in sup-ply and demand. Orders can be viewed as expressions

of changes in supply and demand, and the existence of amaster price impact curve reflects the fact that fluctua-tions from the supply and demand equilibrium for a largenumber of financial assets, differing in economic sectorsof activity and market capitalization, are governed by thesame statistical rule.

The response of prices to orders is a key property of amarket. If an attempt to buy or sell results in a smallchange in price, then the market is considered liquid; oth-erwise it is considered illiquid. One expects liquidity to

lillo@lagash.dft.unipa.itjdf@santafe.edumantegna@unipa.it

depend on properties of the asset, such as trading vol-ume, or for stocks, the market capitalization (the totalworth of the company, i.e. the total number of sharestimes their price). The data collapse that we observehere gives a clearer understanding of how liquidity de-pends on volume and market capitalization.

The study is based on the Trades And Quotes (TAQ)database, which contains the prices for all transactionsas well as price quotations (the best offers to buy and

sell at a given price at any given time) for the US equitymarkets. We analyze data for the period 1995-1998 forthe 1000 stocks with the largest market capitalizationtraded in the New York Stock Exchange. The analysis isbased on roughly 113 million transactions and 173 millionquotes.

Our goal is to understand how much the price changeson average in response to an order to buy or sell of a givensize. Of course, in each trade there is both a buyer anda seller. Nonetheless, one often loosely refers to a tradeas a buy or a sell depending on whether the initiatorof the trade was buying or selling. By initiatorwe meanthe agent who placed the more recent order. Buy orders

tend to drive the price up, and sell orders tend to driveit down. It is this price impactthat we are interested in.

Based on only transactions and quotes it is not possi-ble to know with certainty whether trades are initiated bybuyers or sellers. However, we can infer this indirectly us-ing an algorithm developed by Lee and Ready [11]. Thisalgorithm identifies the correct sign of trades by com-paring the prices of transactions with recent quotes. TheLee and Ready algorithm is able to classify the sign of ap-proximately 85% of the trades of our database. An orderby a single party may trigger transactions with multiplecounterparts; from the TAQ database we can only seetransactions. To cope with this, we lump together all

transactions with the same timestamp and treat them asa single trade.

We study the shift in the midquote price caused by the

mailto:jdf@santafe.edumailto:lillo@lagash.dft.unipa.itmailto:mantegna@unipa.itmailto:mantegna@unipa.itmailto:mantegna@unipa.itmailto:jdf@santafe.edumailto:lillo@lagash.dft.unipa.it

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normalized volume

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FIG. 1: Price shift vs. normalized transaction size for buyerinitiated order for two representative stocks, General ElectricCo. (squares) and International Rectifier Corp. (circles) in1995.)

most recent transaction. For each transaction of volume occurring at time t we observe two cases: (i) Whenthe next event is a quote revision, we compare the nextquote to the previous (prevailing) quote, and computethe difference in the logarithm of the midquote price.Letting the logarithm of the midquote price be p(t), wecompute the price shift p(ti+1) =p(ti+1)p(ti), where

ti is the time of the prevailing (previous) quote andti+1is the time of the next quote following the transaction;(ii) When the next event is a new transaction we set theprice shift p(ti) to zero [12]. We then investigate theaverage price shift as a function of the transaction size measured in dollars, doing this separately for buys andsells.

To investigate the average behavior we bin the databased on transaction size and compute the average priceshift for the data in each bin. To put all stocks on roughlythe same footing, we normalize the transaction size by di-viding by its average value for each stock in each year.The results of doing this for two representative stocks are

shown in figure 1. For one of the highest cap stocks (Gen-eral Electric) the average price impact increases roughlyas 0.6 throughout almost the entire volume range . Incontrast, for a mid-cap stock such as International Rec-tifier Corp. (IRF), for small (

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mean market capitalization

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liquidity

FIG. 3: Liquidity as a function of mean market capital-ization of each group of stocks for 1995 (black), 1996 (green),1997 (blue) and 1998 (red). The black dashed line is the powerlaw best fit on all points. The best fitting exponent is 0.39.

andyaxes of each group according to the transformations

x x/C y y C (1)

We then search for the values of and that do thebest job of placing all the points on a single curve. Todo this we divide the x axis into bins, and find values

that minimize the mean of the two dimensional variance= (y/y)2 + (x/x)2, where denotes the standarddeviation and denotes the mean, and y is the renor-malized return and x is the renormalized transactionsize. In all investigated years there is a clear minimumfor 0.3 (to be precise = 0.3 0.05 for allyears and= 0.30.05 for 1995, 1997 and 1998 whereas= 0.4 0.05 for 1996). The resulting rescaled price im-pact curves for buys in the investigated years are shownin Figure (4).

In all cases the collapse is quite good. The resultingmaster function spans three decades. It increases slowerthan a power law, and decreases more slowly in 1998than in 1995. The data from 1996 and 1997 show similarbehavior, with the slopes decreasing steadily from yearto year.

This slow rate of increase of the price impact functionshown here is surprising. Naive arguments predict that itshould increase at least exponentially for positive . Incontrast, many previous empirical studies of price impactsuggest concave behavior [2, 5,6,7,9,10]. However, thisresult has not been observed universally [14], and noneof these studies have given a clear indication as to func-tional form. We have solved the problems by focusing onthe most elementary response, which is the price impactfollowing a singletrade, by analyzing a huge amount ofdata, aggregating across different stocks and by scaling

the data based on market capitalization.

The traditional approach in economics to deriving de-mand curves is to assume that agents maximize their

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(a) (b)

(c) (d)

FIG. 4: The price shift vs. transaction size, for buy ordersin 1995 (a), 1996 (b), 1997 (c) and 1998 (d), renormalizedas described in the text in order to make the data collapseroughly onto a single curve. The parameter = 0.3 for allyears and the parameter= 0.3 for 1995, 1997, and 1998 and= 0.4 for 1996. Results for sell orders are very similar.

utility under assumptions about cognitive ability and ac-cess to information. The standard interpretation of ourresults would be that the size dependence of price im-pact is due to differences in the information content oftrades. In other words, some trades are based on moreinformation than others, and this is known by marketparticipants and factored into the price setting process.This hypothesis suffers from the problem that the infor-mation content of trades is difficult to assess a priori,making the hypothesis unfalsifiable. In contrast, an al-ternative approach is to study the mechanism for makingtransactions in detail, under the hypothesis that orderplacement and cancellation are largely random. This re-sults in predictions of price impact that are qualitativelyconsistent with those seen here [15, 16]. If these pre-dictions are born out quantitatively it will be significant

in demonstrating that it is important to model financialinstitutions in detail, and that for some purposes it ismay be more useful to model human behavior as randomrather than rational.

In summary, we have demonstrated a remarkable regu-larity in the immediate response of stock prices to fluctu-ations in supply or demand. In each year we are able toget a good data collapse with similar parameters. Thisscaling holds for stocks with trading volumes and marketcapitalizations that differ by 6 and 4 orders of magni-tude respectively. The resulting data collapse is usefulbecause it tells us how the liquidity of stocks varies withtheir market cap, increasing as powers of market cap, in

a way that is not obvious a priori.

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[1] J. Hasbrouck, Handbook of Statistics14, 647 (1996)[2] J.A. Hausman and A.W. Lo, Journal of Financial Eco-

nomics31, 319 (1992)[3] L.K.C. Chan and J. Lakonishok, Journal of Finance50,

1147 (1995)[4] A. Dufour and R.F. Engle, Journal of Finance 55, 2467

(2000)[5] J.D. Farmer, Slippage 1996, Prediction

Company internal technical report (1996),www.predict.com/jdf/slippage.pdf. This study wasbased on roughly 500 trades of crude oil futures. Asubsequent (proprietary) analysis based on more thanhalf a million American stocks by William Finnoff in1998 also supports