Market Microstructure and Algorithmic Tradingcfrei/PIMS/Almgren3.pdf · "micro" impact: individual...

Robert Almgren

Market Microstructureand

Algorithmic Trading

PIMS Summer School 2016University of Alberta, Edmonton

Lecture 3: July 5, 2016

Edmonton mini-course, July 2016

Outline

1. Micro impactindividual child orderspaid by liquidity demander to supplier

2. Macro impactneed parent orders (brokers only)incorporate time or no time

3. Models for trade optimization

2


Market impact

How your trades affect the marketHow much it costs you to trade"micro" impact: individual trades or events

execute trade with market orderor place/cancel limit order

"macro" impact: larger scale orders"buy 1000 lots across next 2 hours"how does price change during and after trading

Models for trade trajectory optimizationdependence of cost on scheduling decisions

3


Two conundrums of market impact

1. Buyer vs sellerwho pays impact to whom?

2. Impact vs alphatrade decision is not exogenousdepends on previous price changesand on anticipated price changes

4


Conundrum #1: Buyer vs seller

Every trade has two sidesWhich one pays market impact?

Answers"People like me" pay to "the market"More aggressive pays to less aggressive

5

Sell

Buy

Price goes up following trade

Positive impact

Negative impact


Data sources

Public market dataimpact of aggressive (market) ordersproblem: algo executions can be 50% passive

Broker or internal data setclient orders paying impact to marketproblem: in closed system, sum to zero

CME LDB data set (to 2012)trade volume tagged by "CTI code" (local/external)can demonstrate transfer to locals

6


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This report is for information purposes only, subject to change without notice and not to be constructed as a solicitation or off er to buy or sell any fi nancial instruments or securities. Newedge makes no representation or warranty that the information contained herein is accurate, complete, fair or correct or that any transaction is appropriate for any person and it should not be relied on as such. Subject to the nature and contents of this report, the investments described are subject to fl uctuations in price and/or value and investors may get back less than originally invested. Certain high volatility investments can be subject to sudden and large declines in value that could equal or exceed the amount invested. Futures and options, as well as certain other fi nancial instruments, are speculative products and the risk of loss can be substantial. Consequently only risk capital should be used to trade futures and options and other speculative products. Investors should, before act-ing on any information herein, fully understand the risks and potential losses and seek their own independent investment and trading advice having regard to their objectives, fi nancial situation and needs. This report and the information included are not intended to be construed as investment advice. Any forecasts are for illustrative purposes only and are not to be relied upon as advice or interpreted as a recommendation. Newedge accepts no liability for any direct, indirect, incidental or consequential damages or losses arising from the use of this report or its content. This report is not to be construed as providing investment services in any jurisdiction where the provision of such services would be illegal.

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Exhibit 1 also shows the average price at which each group bought and at which each group sold during this particular 15-minute interval. As it hap-pened, for the two insider groups – CTI 1+3 and CTI 2, the average buy price was lower than the average sell price, while for the one outsider group – CTI 4 – the average sell price was lower than the average buy price. This is consistent with our intuition that the first two groups are more interested in providing liquidity and getting paid for it, while the last group is more interested in access to liquidity and is willing to pay for it.

The longer viewIt would be a bit of a stretch to suggest that the dif-ference in average prices is a measure of market slippage for these three groups, but the evidence pro-vided in Exhibit 2 is at least consistent with the idea that outsiders are paying for access, while those who are closer to the market are in the business of provid-ing market liquidity.

In the upper panel, we have plotted the average values of these differences between sell and buy prices calculated for 15 minute intervals. The price differences are fairly stable and fairly small. On av-erage, traders in the CTI 4 group pay approximately 0.1/32nd, which is about 1/5th of the tick in this mar-ket and is worth about $3.00. This is much smaller than and a far cry from what one would have found in the open outcry days when someone in this group

could have counted on paying the full bid/ask spread – or very close to it – for access to the market.

On the other hand, the risks inherent in providing liquidity are evident in the lower panel where we have shown the differences in sell and buy prices when we calculate these averages over entire trading days instead of the narrow 15-minute intervals used to produce the values in the upper panel. The business of making markets – or of being a liquidity provider – is one of selling options. The most you can make is the bid/ask spread and the market can get away from you. In the lower panel, we see a clear quarterly cycle in the price difference, probably associated with the futures expiration. We can also see two extended episodes, in June 2009 and December 2010, when those in the CTI 4 group – the so-called liquidity demanders – were consistently winners across the day while the liquidity providers were the losers. We do not yet have an explanation for these episodes.

Robert Almgren is co-founder and Head of Research at Quantitative Brokers LLC, a broker dedicated to algorithmic execution and transaction cost measurement for interest rate products. He has an extensive re-search and publishing record in optimal trade execution and cost measurement.

Galen Burghardt is Director of Research at Newedge Group. He is a co-author of The Treasury Bond Ba-sis and Managed Futures: A Guide for Institutional Investors and author of The Eurodollar Futures and Options Handbook.

-0.25

-0.2

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2

0.25

CTI 1 + 3 CTI 2 CTI 4

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

The bid-ask spread for 10-year futures is 0.5/32

CTI 1 + 3 CTI 2 CTI 4 The bid-ask spread for 10-year futures is 0.5/32

Jan-2009

DateJan-2011Jul-2010 Oct-2010Apr-2010Jan-2010Oct-2009Jul-2009Apr-2009

Pric

e di

ffer

ence

(32n

ds)

Pric

e di

ffer

ence

(32n

ds)

Jan-2009

DateJan-2011Jul-2010 Oct-2010Apr-2010Jan-2010Oct-2009Jul-2009Apr-2009

Source: Quantitative Brokers based on CME LDB data

Source: Quantitative Brokers based on CME LDB data

Exhibit 2Difference between average sell price and average buy price Averages calculated over 15 minute intervals

Averages calculated over full trading days

A WINDOW INTO THE WORLD OF FUTURES MARKET LIQUIDITY 2

NEWEDGE PRIME BROKERAGE

7

The purpose of this snapshot is to call attention to an interesting data set maintained by the Chicago Mercantile Exchange (CME) that affords a unique insight into futures trading costs. As brokers, we use this data to help understand transactions costs and to keep them as low as possible for our clients.

The CME microstructure data allows us to conclude two things. First, those traders whom we tra-ditionally think of as liquidity takers do in fact pay for access to the pool of liquidity afforded by the exchange. Second, the net price paid for liquidity is remarkably small given the size of the bid/ask spread. In this example, which highlights trading in 10-year Treasury note futures, we find that the average price paid by “liquidity takers” is about $3 per contract per round turn, while the value of the bid/ask spread is just over $15.

A window into the world of futures market liquidity

NEWEDGE PRIME BROKERAGE | INVESTOR RESEARCH

Research Snapshot

Robert [email protected]

Galen [email protected]

March 30, 2011

CTI 1 – An individual member trading for his or her own account, a “local,”

CTI 2 – A member firm trading for its own proprietary account,

CTI 3 – A member firm trading for another member (very little volume), and

CTI 4 – All other participants.

CME LDB data(no longer available)


Conundrum #2: impact vs alpha

Decision to trade is never exogenousTrader buys because expects price rise

subsequent rise is impact or alpha?Ideal study: send random ordersMust calibrate impact model for each trade style

short-term alpha vs long termExample: cross-impact

correlation due to cross impact or correlated trading?Example: serial correlation of trade sign

buys followed by buys, sells by sells

8


Micro impact

Price change following market order executionOnly study you can do with public data

9

Ask

Bid

Trade is a "buy"because at ask price:

buy market order with sell limit order

Δt

Δp

Market impact model:Δp as function of Δt and trade size


0 200 400 600 800

-15

-10

-5

0

5

10

15

Trade size in lots

Pric

e ch

ange

in ti

cks

ESU5 on 03 Aug / 60 sec

dp = 0.378 +0.00199 size

10

Typical market impact data:many small trades,

lots of noise

Clustering oninteger multiples

of tick size


1 5 10 50 100 500 1000

-15

-10

-5

0

5

10

15

Trade size in lots

Pric

e ch

ange

in ti

cks

ESU5 on 03 Aug / 60 sec

dp = 0.378 +0.00199 size

11

Same with log scale for trade size


Challenges in micro impact

Buy/sell classification is arbitrarylegitimate study, but may not be what you want

Market order may depend on quote sizesmicroprice (quote imbalance) is common signal

Market orders are serially correlatedimpact may be due to earlier and later orderscause may be slicing of larger orders, ortrade decisions reacting to trade activity

12


2,000 lots2,000 lots

130.98

130.99

131.00

131.01

131.02

131.03

131.04

131.05

13:50:30 13:51:30 13:52:30 13:53:30 13:54:30 13:55:30 13:56:30

CET on Thu 10 Dec 2015

BUY 61 FGBMH6 BOLT

13:51:17

Done at 13:56:08

Exec 131.02VWAP 131.02

Strike 131.015

Sweep 131.020

FGBM

H6

Aggressive fillsPassive fillsCumulative execMarket tradesLimit ordersCumulative VWAPMicropriceBid-ask

Exec = 131.02 Cost to strike = 0.48 tick = €4.84 per lot

13:50:30 13:51:30 13:52:30 13:53:30 13:54:30 13:55:30 13:56:30

0

10

20

30

40

50

60

Exec

uted

and

wor

king

qua

ntity

1k

2k

3k

4k

5kC

umul

ativ

e m

arke

t vol

ume

13:51:17

Done at 13:56:08

15 @ 131.02

11 @ 131.02

17 @ 131.02

9 @ 131.02

4 @ 131.022 @ 131.02

1 @ 131.02

1.3% mkt

1 passv 60 aggrAggressive fillsPassive fillsWorking quantityCumulative Market VolumeFilled quantityAggressive quantity

13

Aggressive orderswhen microprice is away (opposite quote is small)

because anticipate price motion


Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

PRICE DYNAMICS IN A MARKOVIAN LIMIT ORDER MARKET 13

0

5

10

15

20

25

30

05

1015

2025

30

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

n

p

Figure 5. Conditional probability of a price increase, as a function of the bid and ask queue sizes, comparedwith empirical transition frequencies for Citigroup stock price tick-by-tick data on June 26th, 2008.

Noting that e−r(t) = 2− cos(t)−!

(2− cos(t))2 − 1 we obtain the result.Note that the conditional probabilities (7) are, in the case of a balanced order book,

independent of the parameters describing the order flow.The expression (7) is easily computed numerically: Figure 5 displays the shape of the

function pup1 . Comparison with empirical data for Citigroup stock (June, 2008) shows goodagreement between the theoretical value (7) and the empirical transition frequencies of theprice conditional on the state of the consolidated order book.

3.3. Probability of price increase: Asymmetric order flow. In this section we relax thesymmetry assumptions above and allow the intensity of limit and market orders at the bidand the ask to be different; more precisely we assume the following:

• Limit orders at the ask arrive at independent, exponential times with parameter λa.• Market orders and cancellations at the ask arrive at independent, exponential times

with parameter µa + θa.• Limit orders at the bid arrive at independent, exponential times with parameter λb.• Market orders and cancellations at the bid arrive at independent, exponential times,

with parameter µb + θb.

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Copyright © by SIAM. Unauthorized reproduction of this article is prohibited.

SIAM J. FINANCIAL MATH. c⃝ 2013 Society for Industrial and Applied MathematicsVol. 4, pp. 1–25

Price Dynamics in a Markovian Limit Order Market∗

Rama Cont† and Adrien de Larrard†

Abstract. We propose a simple stochastic model for the dynamics of a limit order book, in which arrivals ofmarket orders, limit orders, and order cancellations are described in terms of a Markovian queueingsystem. Price dynamics are endogenous and result from the execution of market orders againstoutstanding limit orders. Through its analytical tractability, the model allows us to obtain analyticalexpressions for various quantities of interest, such as the distribution of the duration between pricechanges, the distribution and autocorrelation of price changes, and the probability of an upwardmove in the price, conditional on the state of the order book. We study the diffusion limit of theprice process and express the volatility of price changes in terms of parameters describing the arrivalrates of buy and sell orders and cancellations. These analytical results provide some insight into therelation between order flow and price dynamics in limit order markets.

Key words. limit order book, market microstructure, queueing, diffusion limit, high-frequency data, liquidity,duration analysis, point process

AMS subject classifications. 60J28, 60J70, 60K25, 90B22

DOI. 10.1137/110856605

1. Introduction. An increasing number of financial instruments are traded in electronic,order-driven markets, in which orders to buy and sell are centralized in a limit order book avail-able to market participants and market orders are executed against the best available offers inthe limit order book. The dynamics of prices in such markets are interesting not only from theviewpoint of market participants—for trading and order execution (Alfonsi, Schied, and Schulz(2010), Predoiu, Shaikhet, and Shreve (2011))—but also from a fundamental perspective sincethey provide a rare glimpse into the dynamics of supply and demand and their role in priceformation (Cont (2011)).

Equilibrium models of price formation in limit order markets (see Parlour (1998), Rosu(2009)) have shown that the evolution of the price in such markets is rather complex and de-pends on the state of the order book. On the other hand, empirical studies on limit order books(see Bouchaud, Farmer, and Lillo (2008), Farmer et al. (2004), Gourieroux, Jasiak, and Le Fol(1999), Hollifield, Miller, and Sandas (2004), Smith et al. (2003)) provide an extensive list ofstatistical features of order book dynamics that are challenging to incorporate in a singlemodel. While most of these studies have focused on unconditional/steady-state distributionsof various features of the order book, empirical studies (Harris and Panchapagesan (2005),Cont, Kukanov, and Stoikov (2010a)) show that the state of the order book contains infor-mation on short-term price movements, so it is of interest to provide forecasts of various

∗Received by the editors November 28, 2011; accepted for publication (in revised form) October 10, 2012;published electronically January 29, 2013.

http://www.siam.org/journals/sifin/4/85660.html†Laboratoire de Probabilites et Modeles Aleatoires, CNRS-Universite Pierre et Marie Curie, 75252 Paris, France

([email protected], [email protected]).

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Ask smallBid large

Ask largeBid small

Conditional probability of a price increase

Edmonton mini-course, July 2016 15

RE S E A R C H PA P E R Q UANTITATIVE F I N A N C E V O L U M E 4 (2004) 176–190quant.iop.org IN S T I T U T E O F P H Y S I C S P U B L I S H I N G

Fluctuations and response in financialmarkets: the subtle nature of‘random’ price changes

Jean-Philippe Bouchaud1,2, Yuval Gefen3, Marc Potters2 andMatthieu Wyart1

1 Commissariat a l’Energie Atomique, Orme des Merisiers,91191 Gif-sur-Yvette Cedex, France2 Science and Finance, Capital Fund Management, 109–111 rue Victor Hugo,92 532 Levallois Cedex, France3 Condensed Matter Physics Department, Weizmann Institute of Science,76 100 Rehovot, Israel

Received 28 August 2003, in final form 21 November 2003Published 19 December 2003Online at stacks.iop.org/Quant/4/176 (DOI: 10.1088/1469-7688/4/2/007)

AbstractUsing trades and quotes data from the Paris stock market, we show that therandom walk nature of traded prices results from a very delicate interplaybetween two opposite tendencies: long-range correlated market orders thatlead to super-diffusion (or persistence), and mean reverting limit orders thatlead to sub-diffusion (or anti-persistence). We define and study a modelwhere the price, at any instant, is the result of the impact of all past trades,mediated by a non-constant ‘propagator’ in time that describes the responseof the market to a single trade. Within this model, the market is shown to be,in a precise sense, at a critical point, where the price is purely diffusive andthe average response function almost constant. We find empirically, anddiscuss theoretically, a fluctuation–response relation. We also discuss thefraction of truly informed market orders, that correctly anticipate short-termmoves, and find that it is quite small.

1. IntroductionThe efficient market hypothesis (EMH) posits that all availableinformation is included in prices, which emerge at all timesfrom the consensus between fully rational agents, that wouldotherwise immediately arbitrage away any deviation from thefair price [1, 2]. Price changes can then only be the result ofunanticipated news and are by definition totally unpredictable.The price is at any instant of time the best predictor of futureprices. One of the central predictions of EMH is thus thatprices should be random walks in time, which (to a goodapproximation) they indeed are. This was interpreted early onas a success of EMH. However, as pointed out by Shiller, theobserved volatility of markets is far too high to be compatible

with the idea of fully rational pricing [3]. The frantic activityobserved in financial markets is another problem: on liquidstocks, there is typically one trade every 5 s, whereas thetime lag between relevant news is certainly much larger.More fundamentally, the assumption of rational, perfectlyinformed agents seems intuitively much too strong, and hasbeen criticized by many [4–6]. Even the very concept of thefair price of a company appears to be somewhat dubious.

There is a model at the other extreme of the spectrumwhere prices also follow a pure random walk, but for a totallydifferent reason. Assume that agents, instead of being fullyrational, have zero intelligence and take random decisions tobuy or to sell, but that their action is interpreted by all the othersagents as potentially containing some information. Then, the

176 1469-7688/04/020176+15$30.00 © 2004 IOP Publishing Ltd PII: S1469-7688(04)68104-8

J-P Bouchaud et al QUANTITATIVE FI N A N C E

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Figure 1. Plot of√

D(ℓ)/ℓ as a function of ℓ for France-Telecom,during three different periods. The variation of D(ℓ)/ℓ with ℓ isvery small, in particular in the small-tick (0.01 Euros) period (July2001–December 2002). For the large-tick-size period (0.05 Euros;January 2001–June 2001), there is a systematic downward trend:see also figure 2.

systematically discarded the first ten and the last ten minutesof trading in a given day, to remove any artifacts due to theopening and closing of the market. Many quantities of interestin the following are two-time observables, that is, comparetwo observables at (trade) time n and n + ℓ. In order to avoidovernight effects, we have restricted our analysis mostly tointraday data, i.e. both n and n + ℓ belong to the same tradingday. We have also assumed that our observables only dependon the time lag ℓ.

In the example of France-Telecom, on which we willfocus mostly, there are on the order of 10 000 trades/day.For example, the total number of trades on France-Telecomduring 2002 was close to 2 × 106; this allows quite accuratestatistical estimates of various quantities. The volume of eachtrade was found to be roughly log-normally distributed, with⟨ln V ⟩ ≃ 5.5 and a root mean square of " ln V ≃ 1.8. Therange of observed values of ln V is between 1 and 11.

2.2. Price fluctuation and diffusionThe simplest quantity to study is the average mean squarefluctuation of the price between (trade) time n and n+ℓ. Here,the price pn is defined as the mid-point just before the nth trade:pn ≡ mn− . In this paper, we always consider detrended prices,such that the empirical drift is zero. We thus define D(ℓ) as

D(ℓ) = ⟨(pn+ℓ − pn)2⟩. (1)

As is well known, in the absence of any linear correlationsbetween successive price changes, D(ℓ) has a strictly diffusivebehaviour, i.e.

D(ℓ) = Dℓ, (2)

where D is a constant. In the presence of short-rangecorrelations, one expects deviations from this behaviour at

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Figure 2. Plot of√

D(ℓ)/ℓ as a function of ℓ for other stocksduring the year 2002, except Barclays (May–June 2002). The y-axishas been rescaled arbitrarily for clarity. We note that stocks withlarger tick size tend to reveal a stronger mean-reverting effect.

short times. However, on liquid stocks with relatively smalltick sizes such as France-Telecom (FT), one finds a remarkablylinear behaviour for D(ℓ), even for small ℓ. The absenceof linear correlations in price changes is compatible with theidea that (statistical) arbitrage opportunities are absent, evenfor high-frequency trading. In fact, in order to emphasizethe differences from a strictly diffusive behaviour, we havestudied the quantity

√D(ℓ)/ℓ (which has the dimension of

Euros). We show this quantity in figure 1 for FT, averagedover three different periods: the first semester of 2001 (wherethe tick size was 0.05 Euros), the second semester of 2001and the whole of 2002 (where the tick size was 0.01 Euros).One sees that D(ℓ)/ℓ is indeed nearly constant, with a small‘oscillation’ on which we will comment later. Similar plotscan be observed for other stocks (see figure 2). We have notedthat for stocks with larger ticks a slow decrease of D(ℓ)/ℓ isobserved, corresponding to a slight anti-persistence (or sub-diffusion) effect. Note that since we study the mid-point, theanti-correlations reported here are not related to the trivial bid–ask bounce.

The conclusion is that the random walk (diffusive)behaviour of stock prices appears even at the trade by tradelevel, with a diffusion constant D which is of the order ofthe typical bid–ask squared. From figure 1, one indeed seesthat

√D(1) ∼ 0.01 Euros, which is precisely the tick size,

whereas FT has an average bid–ask spread equal to two ticks.Hence, each transaction typically moves the mid-point by halfthe bid–ask.

2.3. Response function and market impact

In order to better understand the impact of trading on pricechanges, one can study the following response function R(ℓ),defined as

R(ℓ) = ⟨(pn+ℓ − pn)εn⟩, (3)

178

QUANTITATIVE FI N A N C E Fluctuations and response in financial markets

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Figure 3. Average response function R(ℓ) for FT, during threedifferent periods (full symbols). We have given error bars for the2002 data. For the 2001 data, the y-axis has been rescaled to bestcollapse onto the 2002 data. Using the same rescaling factor, wehave also shown the data of figure 1. The fact that the samerescaling works approximately for D(ℓ) as well will be consideredfurther in section 2.5 below.

where εn is the sign of the nth trade, introduced in section 2.1.The quantity R(ℓ) measures how much, on average, the pricemoves up conditioned to a buy order at time zero (or a sellorder moves the price down) a time ℓ later. As will be clearbelow, this quantity is however not the market response to asingle trade, a quantity that will later be denoted by G0. Amore detailed object can in fact be defined by conditioning theaverage to a certain volume V of the nth trade:

R(ℓ, V ) = ⟨(pn+ℓ − pn)εn⟩|Vn=V . (4)

Previous empirical studies have mostly focused on the volumedependence of R(ℓ, V ), and established that this functionis strongly concave as a function of the volume [7, 27–30].In [31], a thorough analysis of US stocks was performedin terms of a piecewise power-law dependence for R(ℓ =1, V ) ∝ V α , with an exponent α ≃ 0.4 for small volumes,and a smaller value (α ≃ 0.2) for larger volumes (seealso [32]). In a previous publication [33], two of us haveproposed that this dependence might in fact be logarithmic(see also a footnote in [30]): R(ℓ = 1, V ) = R1 ln V

(where R1 is a stock dependent constant), a law that seems tosatisfactorily account for all the data that we have analysed.The empirical determination of the temporal structure ofR(ℓ, V ) has been much less investigated (although one can findin [30] somewhat related results on a coarse-grained versionof R(ℓ, V )). Preliminary empirical results, published in [33],reported that R(ℓ, V ) could be written in a factorized form(first suggested on theoretical grounds in [12]):

R(ℓ, V ) ≈ R(ℓ)f (V ); f (V ) ∝ ln V, (5)

where R(ℓ) is a slowly varying function that initially increasesup to ℓ ∼ 100–1000 and then is seen to decrease back, with a

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!

Figure 4. Average response function R(ℓ) for a restricted selectionof stocks, during the year 2002.

rather small overall range of variation. The initial increase ofR(ℓ) was reported in [27] and has also recently been noticedby Lillo and Farmer [17]. Here, we provide much better datathat supports both the above assertions. We show for examplein figure 3 the temporal structure of R(ℓ) for France-Telecom,for different periods. Note that R(ℓ) increases by a factor∼2 between ℓ = 1 and ℓ∗ ≈ 1000, before decreasing again.Including overnights allows one to probe larger values of ℓ

and confirm that R(ℓ) decreases, and even becomes negativebeyond ℓ ≃ 5000 (why this may be so will be explainedby our model). Similar results have been obtained for manydifferent stocks as well: figure 4 shows a small selection ofother stocks, where the non-monotonic behaviour of R(ℓ)

is shown. However, in some cases (such as Pechiney), themaximum is not observed. One possible reason is that thenumber of daily trades is in this case much smaller (∼1000),and that ℓ∗ is beyond the maximum intraday time lag. Onthe other hand, the model discussed below does also allow formonotonic response functions.

The existence of a timescale ℓ∗ beyond which R(ℓ)

decreases is thus both statistically significant, and to a largedegree independent of the considered stock. On the otherhand, the amplitude of the change of R(ℓ) seems to be stockdependent. As will be clear later, the fact that R(ℓ) slowlyincreases before decreasing back to negative values is a non-trivial result that requires a specific interpretation.

Turning now to the factorization property of R(ℓ, V ),equation (5), we illustrate its validity in figure 5, whereR(ℓ, V )/f (V ) is plotted as a function of ℓ for different valuesof V . The function f (V ) was chosen for best visual rescaling,and is found to be close to f (V ) = ln V , as expected. Note thatfor the smallest volume (open circles) the long-time behaviourof R(ℓ, V ) seems to be different, which is probably due tothe fact that small volumes are in fact more likely to be largevolumes chopped up into small pieces.

179

J-P Bouchaud et al QUANTITATIVE FI N A N C E

!!

"

"#""$

"#"%"

"#"%$

"#"!"

"

"

&'(&!""!&'(&!""%

!#$&&&&%")$ $&&&&%")$

Figure 8. Average diffusion constant D = D(ℓ)/ℓ, computed forℓ = 128, and conditioned to a certain value of R2(ℓ), also computedfor ℓ = 128 (FT). The open symbols correspond to 2002, whereasthe black symbols are computed using the first semester of 2001,where the tick size was five times larger. Correspondingly, thex-axis was rescaled down by a factor of 25 and the y-axis by a factorof five for this data set.

mentioned above, the statistics of price changes reveals verylittle temporal correlation, the correlation function of the signεn of the trades, on the other hand, reveals very slowly decayingcorrelations as a function of trade time. This correlation hasbeen mentioned in some papers before; see e.g. [7, 27]. Here,we propose that these correlations decay as a power law of thetime lag, at least up to ℓ ≈ 15 000 (two trading days) beyondwhich we do not have sufficiently accurate data.

More precisely, one can consider the following correlationfunction:

C0(ℓ) = ⟨εn+ℓεn⟩ − ⟨εn⟩2. (6)

If trades were random, one should observe that C0(ℓ) decaysto zero beyond a few trades. Surprisingly, this is not whathappens: on the contrary, C0(ℓ) is strong and decays veryslowly toward zero, as an inverse power law of ℓ (see figure 9):

C0(ℓ) ≃ C0

ℓγ, (ℓ ! 1). (7)

The value of γ seems to be somewhat stock dependent. Forexample, for FT, one finds γ ≈ 1/5, whereas for Totalγ ≈ 2/3. In their study, Lillo and Farmer found a somewhatlarger value of γ ≈ 0.39 for Vodafone [17]. In any case,the value of γ is found to be smaller than one, which is veryimportant because the integral of C0(ℓ) is then divergent. Thisis in fact the precise definition of ‘long-term’ correlations.Now, as will be shown more precisely in the next section, theintegral of C0(ℓ) can intuitively be thought of as the effectivenumber Ne of correlated successive trades. Hence, out of—say—1000 trades, one should group together

Ne ≃ 1 +1000!

ℓ=1

C0(ℓ) ≈ 1 +C0

1 − γ10001−γ (8)

(*+,&-./01,23

%"%

%""

%")%

%")!

%")4

%"" %"! %"5%"% %"4

Figure 9. Volume-weighted sign autocorrelation functions as afunction of time lag: C0, C1 and C2 (see the text for definitions). Thestraight line corresponds to ℓ−γ with γ = 1/5. The dotted curvescorrespond to the simple approximation given by equations (11).Using data across different days allows one to extend the power-lawdecay at least up to 15 000 trades.

‘coherent’ trades. For FT, γ ≈ 1/5 and C0 ≈ 0.2, whichmeans that the effect of one trade should be amplified, throughthe correlations, by a factor Ne ≈ 50! In other words, both theresponse function R and the diffusion constant should increaseby a factor of 50 between ℓ = 1 and 1000, in stark contrastwith the observed empirical data. This is the main puzzle thatone should try to elucidate: how can one reconcile the strong,slowly decaying correlations in the sign of the trades with thenearly diffusive nature of the price fluctuations, and the non-monotonic response function?

Before presenting a mathematical transcription of theabove question and proposing a possible resolution, letus comment on two related correlation functions that willnaturally appear in the following, namely

C1(ℓ) = ⟨εn+ℓεn ln Vn⟩, (9)

andC2(ℓ) = ⟨εn+ℓ ln Vn+ℓεn ln Vn⟩. (10)

We have found empirically that these two ‘mixed’ correlationfunctions are proportional to C0(ℓ) (see figure 8):

C1(ℓ) ≈ ⟨ln V ⟩C0(ℓ); C2(ℓ) ≈ ⟨ln V ⟩2C0(ℓ). (11)

There are however small systematic deviations, which indicatethat (i) small volumes contribute more to the long-rangecorrelations than larger volumes and (ii) ln V − ⟨ln V ⟩ is aquantity exhibiting long-range correlations as well.

3. A micro-model of price fluctuations3.1. Set up of the modelIn order to understand the above results, we will postulate thefollowing trade superposition model, where the price at time

182

Price motion has no serial correlation, even though is response to correlated order flow. Other traders anticipate future orders.

Modern models: "propagators"


Macro impact

Need to know "parent order"Plot slippage vs sizeFit linear or nonlinear model

16


Cost model

Inputs:X = executed order sizeB = benchmark price, bid-ask midpoint at startP = average executed priceC = P - B = trade cost or slippage (for buy order) = -(P - B) for sell orderModel C as function of X: C = f(X)

17

This structure takes no account of how the order is executed or over what time horizon.

No use for optimizing execution!


Nondimensionalization

18

V = daily volume (actual, average, or forecast) σ = daily volatility

Idea: Measure your order relative to what the market is doing anyway

Lets you compare different assets and different days(with widely varying volume and volatility) in same model

"Trading 1% daily volume costs 5% of daily volatility"


Structure of f(x)

19

Minimize sum of squares error to order data

Linear

Nonlinear

j indexes ordersa, b, k are universal


0 1000 2000 3000 4000 5000

-2

-1

0

1

2

3

4

Executed size in lots

Slip

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min

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incr

weighted mean = 1.02

501000.3370.383

ES from Fri 02 Jan 2015 to Thu 10 Dec 20154210 data pointsConfidence bands at 1,2 standard deviationsKernel smoothers at 0.1,0.2,0.5 decadesWeighted mean = 1.02Fit with exponent 0.906

SP500 (ES) 2015 (unscaled)


1 5 10 50 100 500 1000 5000

-2

-1

0

1

2

3

4


Slip

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incr


50 1000.336 0.382

ES from Fri 02 Jan 2015 to Thu 10 Dec 20154199 data pointsConfidence bands at 1,2 standard deviationsKernel smoothers at 0.1,0.2,0.5 decadesWeighted mean = 1.02Fit with exponent 0.908

SP500 (ES) 2015 (unscaled)


0 200 400 600 800

-5

0

5

10


Slip

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incr


50

1001.74

2.62

CL from Fri 02 Jan 2015 to Thu 10 Dec 20151866 data pointsConfidence bands at 1,2 standard deviationsKernel smoothers at 0.1,0.2,0.5 decadesWeighted mean = 3.76Fit with exponent 0.649

Crude Oil (CL) 2015 (unscaled)


1 5 10 50 100 500 1000

-5

0

5

10


Slip

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incr


50

1001.74

2.63

CL from Fri 02 Jan 2015 to Thu 10 Dec 20151865 data pointsConfidence bands at 1,2 standard deviationsKernel smoothers at 0.1,0.2,0.5 decadesWeighted mean = 3.75Fit with exponent 0.642

Crude Oil (CL) 2015 (unscaled)


0.000 0.001 0.002 0.003 0.004 0.005

-0.05

0.00

0.05

0.10

Executed size as fraction of daily volume

Slip

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ly v

olat

ility


CL,HO,NG,RB from Fri 02 Jan 2015 to Thu 10 Dec 20157605 data pointsConfidence bands at 1,2 standard deviationsKernel smoothers at 0.1,0.2,0.5 decadesWeighted mean = 0.0114Fit with exponent 0.57

Scaled fit for several energy products


Advantages of this model (parent order level):simple to dosimple to interpretgives immediate useful results for cost estimation

Disadvantages of this modelnot useful for order scheduling or optimizationno microscopic description of mechanism

Caveatssome orders may be cancelled based on market moves

solution: restrict sample to fully executed ordersdifferent strategies have different short term alpha

solution: results are client-specific and strategy-specific

25


Robert F. Almgren 2

EDUCATION

1989 Ph.D. in Applied and Computational Mathematics, Princeton University1984 M.S. in Applied Mathematics, Harvard University1983 B.S. in Physics and B.S. in Mathematics, Massachusetts Institute of Technology

PUBLICATIONS

Finance and optimization

1. T. M. Li and RFA, “Option hedging with smooth market impact”, Market Microstructure and Liquidity,to appear 2016.

2. RFA and A. Tourin, “Optimal soaring via Hamilton-Jacobi-Bellman equations”,Optim. Control Appl. Meth, 36 (2015) 475–495.

3. “Execution strategies in fixed-income markets”, in Maureen O’Hara, Marcos Lopez de Prado and DavidEasley, editors, High Frequency Trading, Risk Books 2013.

4. “Optimal trading with stochastic liquidity and volatility”, SIAM J. Financial Math., 3 (2012) 163–181.5. J. Lorenz and RFA, “Mean-variance optimal adaptive execution”, Appl. Math. Fin., 18 (2011) 395–422.6. “Execution Costs”, in Rama Cont, editor-in-chief, Encyclopedia of Quantitative Finance, Wiley 2009.7. RFA and J. Lorenz, “Adaptive arrival price”, in Algorithmic Trading III: Precision, Control, Execution,

Brian R. Bruce, editor, Institutional Investor Journals 2007.8. RFA and J. Lorenz, “Bayesian adaptive trading with a daily cycle”, J. Trading Fall 2006.9. RFA and N. Chriss, “Optimal portfolios from ordering information”, J. Risk Fall 2006.

10. RFA, C. Thum, E. Hauptmann, and H. Li, “Equity market impact”, Risk, July 2005.11. RFA and N. Chriss, “Bidding principles”, Risk, June 2003.12. “Optimal execution with nonlinear impact functions and trading-enhanced risk”,

Appl. Math. Fin., 10 (2003) 1–18.13. “Financial derivatives and partial differential equations”, Amer. Math. Mon, Jan. 2002.14. RFA and N. Chriss, “Optimal execution of portfolio transactions”, J. Risk 3 (2000–01) 5–39.15. RFA and N. Chriss, “Value under liquidation”, Risk, Dec. 1999.

Free boundary problems

16. K. Glasner and RFA, “Dual fronts in a phase field model,” Physica D 146 (2000) 328–340.17. B. Johnson, R. Sekerka, and RFA, “Thermodynamic basis for a variational model for crystal growth,”

Phys. Rev. E 60 (1999) 705–714.18. “Second order phase field asymptotics with unequal conductivities,”

SIAM J. Appl. Math. 59 (1999) 2086–210719. N. Provatas, N. Goldenfeld, J. Dantzig, J. LaCombe, A. Lupulescu, M. Koss, M. Glicksman, and RFA,

“Crossover scaling in dendritic evolution at low undercooling”, Phys. Rev. Lett. 82 (1999) 4496–4499.

Direct Estimation ofEquity Market Impact⇤

Robert Almgren†, Chee Thum‡,Emmanuel Hauptmann‡, and Hong Li‡

May 10, 2005§

Abstract

The impact of large trades on market prices is a widely discussedbut rarely measured phenomenon, of essential importance to sell-and buy-side participants. We analyse a large data set from theCitigroup US equity trading desks, using a simple but realistic the-oretical framework. We fit the model across a wide range of stocks,determining the dependence of the coefficients on parameters suchas volatility, average daily volume, and turnover. We reject the com-mon square-root model for temporary impact as function of traderate, in favor of a 3/5 power law across the range of order sizesconsidered. Our results can be directly incorporated into optimaltrade scheduling algorithms and pre- and post-trade cost estima-tion.

⇤We are grateful to Stavros Siokos of Citigroup Equity Trading Strategy and to NeilChriss of SAC Capital Management for helpful feedback and perspective.

†University of Toronto Departments of Mathematics and Computer Science, andCitigroup Global Quantitative Research; [email protected].

‡Citigroup Global Quantitative Research, New York and London.§Original version December 20, 2004

1

Direct Estimation ofEquity Market Impact⇤

Robert Almgren†, Chee Thum‡,Emmanuel Hauptmann‡, and Hong Li‡

May 10, 2005§

Abstract

The impact of large trades on market prices is a widely discussedbut rarely measured phenomenon, of essential importance to sell-and buy-side participants. We analyse a large data set from theCitigroup US equity trading desks, using a simple but realistic the-oretical framework. We fit the model across a wide range of stocks,determining the dependence of the coefficients on parameters suchas volatility, average daily volume, and turnover. We reject the com-mon square-root model for temporary impact as function of traderate, in favor of a 3/5 power law across the range of order sizesconsidered. Our results can be directly incorporated into optimaltrade scheduling algorithms and pre- and post-trade cost estima-tion.

⇤We are grateful to Stavros Siokos of Citigroup Equity Trading Strategy and to NeilChriss of SAC Capital Management for helpful feedback and perspective.

†University of Toronto Departments of Mathematics and Computer Science, andCitigroup Global Quantitative Research; [email protected].

‡Citigroup Global Quantitative Research, New York and London.§Original version December 20, 2004

1

Equity Market Impact May 10, 2005 10

distorting the results by including excessive overnight volatility, but wehave found this to be give more consistent results than the alternative oftruncating at the close.

Based on these prices, we define the following impact variables:

Permanent impact: I =

Spost � S0

S0

Realized impact: J =

S � S0

S0.

The “effective” impact J is the quantity of most interest, since it deter-mines the actual cash received or spent on the trade. In the model below,we will define temporary impact to be J minus a suitable fraction of I,and this temporary impact will be the quantity described by our theory.We cannot sensibly define temporary impact until we write this model.

On any individual order, the impacts I, J may be either positive ornegative. In fact, since volatility is a very large contributor to their values,they are almost equally likely to have either sign. They are defined so thatpositive cost is experienced if I, J have the same sign as the total orderX: for a buy order with X > 0, positive cost means that the price S(t)moves upward. We expect the average values of I, J, taken across manyorders, to have the same sign as X.

Volume time The level of market activity is known to vary substantiallyand consistently between different periods of the trading day; this intra-day variation affects both the volume profile and the variance of prices.To capture this effect, we perform all our computations in volume time⌧ , which represents the fraction of an average day’s volume that has exe-cuted up to clock time t. Thus a constant-rate trajectory in the ⌧ variablecorresponds to a VWAP execution in real time, as shown in Figure 1. Therelationship between t and ⌧ is independent of the total daily volume;we scale it so that ⌧ = 0 at market open and ⌧ = 1 at market close.

We map each of the clock times t0, . . . , tn in the data set to a cor-responding volume time ⌧0, . . . ,⌧n. Since the stocks in our sample areheavily traded, in this paper we use a nonparametric estimator that di-rectly measures differences in ⌧ : the shares traded during the periodcorresponding to the exection of each order. Figure 2 illustrates the em-pirical profiles. The fluctuations in these graphs are the approximatesize of statistical error in the volume calculation for a 15-minute trade;these errors are typically 5% or less, and are smaller for longer trades.


izing each individual stock (in addition to daily volume and volatility).There are several candidates for the inputs to L:

• Shares outstanding: We constrain the form of L to be

L =

✓⇥V

◆�

.

where ⇥ is the total number of shares outstanding, and the expo-nent � is to be determined. The dimensionless ratio ⇥/V is theinverse of “turnover,” the fraction of the company’s value tradedeach day. This is a natural explanatory variable, and has been usedin empirical studies such as Breen, Hodrick, and Korajczyk (2002).

• Bid-ask spread: We have found no consistent dependence on thebid ask spread across our sample, so we do not include it in L.

• Market capitalization: Market cap differs from shares outstandingby the price per share, so including this factor is equivalent to in-cluding a “price effect.” Our empirical studies suggest that there is apersistent price effect, as also found by Lillo, Farmer, and Mantegna(2003), but that the dependence is weak enough that we neglect itin favor of the conceptually simpler quantity ⇥/V .

Temporary In extensive preliminary exploration, we have found thatthe temporary cost function h(v) does not require any stock-specificmodification: liquidity cost as a fraction of volatility depends only onshares traded as a fraction of average daily volume.

Determination of exponent After assuming the functional form ex-plained above, we confirm the model and determine the exponent � byperforming a nonlinear regression of the form

I

�

= � T sgn(X)��X

VT

��↵

✓⇥V

◆�

+ hnoisei (7)

1�

✓J �

I

2

◆= ⌘ sgn(X)

��X

VT

��

+ hnoisei (8)

where “noise” is again the heteroskedastic error term from (1), and sgnis the sign function. We use a modified Gauss-Newton optimization algo-


izing each individual stock (in addition to daily volume and volatility).There are several candidates for the inputs to L:

• Shares outstanding: We constrain the form of L to be

L =

✓⇥V

◆�

.

where ⇥ is the total number of shares outstanding, and the expo-nent � is to be determined. The dimensionless ratio ⇥/V is theinverse of “turnover,” the fraction of the company’s value tradedeach day. This is a natural explanatory variable, and has been usedin empirical studies such as Breen, Hodrick, and Korajczyk (2002).

• Bid-ask spread: We have found no consistent dependence on thebid ask spread across our sample, so we do not include it in L.

• Market capitalization: Market cap differs from shares outstandingby the price per share, so including this factor is equivalent to in-cluding a “price effect.” Our empirical studies suggest that there is apersistent price effect, as also found by Lillo, Farmer, and Mantegna(2003), but that the dependence is weak enough that we neglect itin favor of the conceptually simpler quantity ⇥/V .

Temporary In extensive preliminary exploration, we have found thatthe temporary cost function h(v) does not require any stock-specificmodification: liquidity cost as a fraction of volatility depends only onshares traded as a fraction of average daily volume.

Determination of exponent After assuming the functional form ex-plained above, we confirm the model and determine the exponent � byperforming a nonlinear regression of the form

I

�

= � T sgn(X)��X

VT

��↵

✓⇥V

◆�

+ hnoisei (7)

1�

✓J �

I

2

◆= ⌘ sgn(X)

��X

VT

��

+ hnoisei (8)

where “noise” is again the heteroskedastic error term from (1), and sgnis the sign function. We use a modified Gauss-Newton optimization algo-


rithm to determine the values of �, ↵, and � that minimize the normal-ized residuals. The results are

↵ = 0.891 ± 0.10

� = 0.267 ± 0.22

� = 0.600 ± 0.038.

Here, as throughout this paper, the error bars expressed with ± are onestandard deviation, assuming a Gaussian error model. Thus the “true”value can be expected to be within this range with 67% probability, andwithin a range twice as large with 95% probability.

From these values we draw the following conclusions:

• The value ↵ = 1, for linear permanent impact, cannot reliably berejected. In view of the enormous practical simplification of linearpermanent impact, we choose to use ↵ = 1.

• The liquidity factor is very approximately � = 1/4.

• For temporary impact, our analysis confirms the concavity of thefunction with � strictly inferior to 1. This confirms the fact thatthe bigger the trades made by fund managers on the market, theless additional cost they experience per share traded. At the 95%confidence level, the square-root model � = 1/2 is rejected. We willtherefore fix on the temporary cost exponent � = 3/5. In compari-son with the square-root model, this gives slightly smaller costs forsmall trades, and slightly larger costs for large trades.

Note that because � > 0, for fixed values of the number X of shares in theorder and the average daily volume V , the cost increases with ⇥, the totalnumber of shares outstanding. In effect, a larger number of outstandingshares means that a smaller fraction of the company is traded each day,so a given fraction of that flow has greater impact.

Therefore these results confirm empirically the theoretical argumentsof Huberman and Stanzl (2004) for permanent impact that is linear inblock size, and the concavity of temporary impact as has been widelydescribed in the literature for both theoretical and empirical reasons.

4.3 Determination of coefficients

After fixing the exponent values, we determine the values of � and ⌘

by linear regression of the models (7,8), using the heteroskedastic error


Market impact models for trading

Two types of market impact(both active, both important):

• Permanentdue to information transmissionaffects public market price

• Temporarydue to finite instantaneous liquidity“private” execution price not reflected in market

Many richer structures are possible

27


Temporary vs. permanent market impact

28

price

time

Pre-trade

Post-trade

Execution(sell)

Temporary impact(liquidity cost)

depends on rateof execution

Permanent impact(information)

independent ofexecution strategy

Instantaneous relaxationfrom temporary impact

to permanent level

Jim Gatheral: richer time structures for decay


Permanent impact

29

✓t = instantaneous rate of trading

Xt = X0 +Z T

0✓s ds

Linear to avoid round-trip arbitrage (Huberman & Stanzl, Gatheral) (Schönbucher & Wilmott 2000: knock-out option--also need temporary impact)

Pt = P0 + � Wt + ⌫(Xt �X0)

G(✓) = ⌫ ✓

dPt = � dWt + G(✓t) dt

Cost to execute net X shares = 12⌫ X2

(independent of path)

t

X(t)


Temporary impact

30

We trade at Pt 6= Pt

Pt depends on instantaneous trade rate ✓t

Require finite instantaneous trade rate⇒ imperfect hedging

Pt = Pt + H(✓t)


Example: bid-ask spread

31

✓t

buy at ask

sell at bid

Pt

Pt s

Pt = Pt +12s sgn(✓t)

12s sgn(✓t) · ✓t �t =

12s|✓t|�t

“Linear” model: cost to trade shares✓t �t


Critique of linear cost model

independent of trade sizenot suitable for large traders

in practice, effective execution near midpointspread cost not consistent with modern cost modelsliquidity takers act as liquidity providers

32


Proportional temporary cost model

33

✓tH(0) = 0

concave(empirical)

H(✓t)

Special case: linear for simplicity

⟹ Quadratic total cost:

H(✓) = 12�✓

H(✓) · ✓�t = 12�✓2�t

Pt = Pt + H(✓t)


Conclusions

Market impact not easy to define or measuretrading and price changes are relatedwho pays trading cost to whom

Micro models from public dataincluding trade sizeexcluding trade size

Macro models from private trade dataexcluding timeincluding time

34

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