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arXiv:0903.2428v1 [q-fin.TR] 13 Mar 2009

PRICE IMPACT

J. P. Bouchaud, Capital Fund Management

March 13, 2009

1 What is price impact?

Price impact refers to the correlation between an incoming order (to buy or

to sell) and the subsequent price change. That a buy trade should push the

price up seems at ﬁrst sight obvious and is easily demonstrated empirically,

as we will review below. It is also a dour reality for traders for whom price

impact is tantamount to a cost: their second buy trade is on average more

expensive than the ﬁrst because of their own impact (and vice-versa for

sells). Monitoring and controling impact is therefore one of the most active

and rapidly expanding domains of research within trading ﬁrms. The most

important questions are related to the volume dependence of impact (do

larger trades impact prices more?), and the temporal behaviour of impact

(is the impact of a trade immediate and permanent, or is there some lag

dependence of the impact?).

A moment of reﬂection suggests that the interpretation of price impact is

far from trivial, and may even lead to contradictions – isn’t a transaction a

fair deal between a buyer and a seller? So why is there price impact? Three

distinct possibilities come to mind:

1. Agents successfully forecast short term price movements and trade ac-

cordingly. This can result in measurable correlation between trades and

price changes, even if the trades by themselves have absolutely no eﬀect

on prices at all. If an agent correctly forecasts price movements and if

the price is about to rise, the agent is likely to buy in anticipation of it.

But in this framework ‘noise induced’ trades, based on no information

at all, should have no price impact.

1

2. The impact of trades reveal some private information. The arrival of

new private information causes trades, which cause other agents to

update their valuations, leading to a price change. But if trades are

anonymous and there is no easy way to distinguish informed traders

from non informed traders, then all trades will impact the price since

other agents will believe that a fraction of these trades might contain

some private information.

3. Impact is a statistical eﬀect due to order ﬂow ﬂuctuations. Imagine for

example a completely random order ﬂow process, that leads to a certain

order book dynamics (see, e.g. Farmer et al. 2005 for a description

of such models). Conditional to an extra buy order, the price will

on average move up if everything else is kept constant. Fluctuations

in supply and demand can thus be completely random, unrelated to

information, still a well deﬁned notion of price impact will emerge. In

this case impact is a completely mechanical – or better, statistical –

phenomenon.

All three of these mechanisms result in some “price impact”, i.e. a positive

correlation of trading volume and price movement, but they are conceptually

very diﬀerent. In the ﬁrst two pictures, as emphasized in Hasbrouck (2007),

“orders do not impact prices. It is more accurate to say that orders forecast

prices.” In the second picture, price impact is of paramount importance to

understand how information gets incorporated into prices. If some traders

really have an information on the “true” price at some time in the future, then

the observation of an excess of buy trades allows the market to guess that

the price will move up and to change the quotes accordingly (see Glosten-

Milgrom model, Kyle model for explicit implementations). Thus while

on one hand market impact is a friction, it should also be viewed as the

mechanism that allows prices to adjust to new information.

But if the third, mechanical, interpretation is correct, correlation between

price changes and order ﬂow is a tautology. If prices move only because

of trades, “information revelation” may merely be a self-fulﬁlling prophecy

which would occur even if the fraction of informed traders is zero. Possible

diﬀerences between these pictures may come about in the volume dependence

and temporal behavior of impact, which we will discuss below.

2

2 Linear and permanent impact: The Kyle

model

The simplest assumption is that impact is both linear in the traded volume

and permanent in time. These assumptions can be partly justiﬁed within the

Kyle model (Kyle, 1985), where an insider trader and noise traders submit

orders that are cleared by a Market Maker (MM) every time step ∆t. In this

model, the price adjustment rule ∆pof the MM must be linear in the total

signed volume ǫv, i.e:1

∆p=λǫv, (1)

where λis a measure of impact, and is inversely proportional to the liquidity

of the market. This price adjustement is furthermore permanent, i.e., the

price change between time t= 0 and t=T=N∆tis:

pT=p0+

N−1

X

n=0

∆pn=p0+λ

N−1

X

n=0

ǫnvn.(2)

This formula assumes that the impact λǫnvnof trades in the nth time interval

persists unabated up to time T. Huberman and Stanzl (2004), and Farmer

(2002) show that the linear price adjustement of the Kyle model is the only

speciﬁcation that does not allow price manipulations, provided impact is per-

manent, see below. From the above equation, it is clear that the sign of the

trades must be serially uncorrelated if the price is to follow an unpredictable

random walk. Within the setting of the Kyle model, the trading schedule of

the insider is precisely such that the ǫnare uncorrelated (Kyle, 1985). But

data from real markets reveal correlations of the sign of the traded volume

over long time scales; we will come back to this important point below.

3 Measures of price impact

An interesting measure of price impact is the correlation between the price

change between 0 and Tand the sign of the trade at time 0, deﬁned in general

as:

R(T) = E[(pT−p0)·ǫ0]−E[(pT−p0)] E[ǫ0].(3)

1ǫ= +1 if the volume of buys vbis larger than the volume of sells vsin the time interval

∆t, and ǫ=−1 in the opposite case; and v=|vb−vs|.

3

In the following, we will assume that drifts can be neglected so that only the

ﬁrst term in the RHS is not zero. This is often the case in practice if Tis

chosen to be suﬃciently small and/or if there is no strong buy/sell assymetry

(E[ǫ0] = 0).

Within the Kyle framework where the ǫs are uncorrelated, R(T) is easily

computed and is found to be:

R(T) = λ E[v] (4)

The impact function R(T) is therefore lag independent in this model. One

can of course also deﬁne a volume dependent impact function by conditioning

the above average on the incoming volume, i.e.:

R(T, v) = E[(pT−p0)·ǫ0|v0=v],(5)

which, in the Kyle model, is equal to λv, for all T. Note that both R(T) and

R(T, v) a priori depend on choice of the elementary time scale ∆t, but we

want to avoid heavy notations and skip this extra variable dependence.

Another commonly used measure of impact is the correlation ρ(T) be-

tween the price change from 0 to Tand the total signed volume in the same

interval, or more precisely:

ρ(T) = Eh(pT−p0)·PN−1

n=0 ǫnvni

rE[(pT−p0)]2Eh(PN−1

n=0 ǫnvn)2i(6)

In the Kyle model this correlation is trivially equal to unity, but this corre-

lation can decrease if liquidity ﬂuctuates (λbecomes time dependent), or if

other sources of volatility are taken into account. For example, the MM can

revise his quotes in the absence of any trades, say because of some incoming

news. This can be modelled by adding an extra term in Eq. (2) above:

pT=p0+λ

N−1

X

n=0

ǫnvn+

N−1

X

n=0

ηn,(7)

where the ηn’s are uncorrelated increments, representing causes of price vari-

ations not directly related to trading.

It is often useful to generalize the above deﬁnition of ρ(T) by replacing

the volumes vnby vψ

n, where ψis a certain exponent. As will be discussed

below, the measured correlations are found to be stronger when ψ < 1.

4

4 Empirical facts: impact cannot be so sim-

ple

Here we summarize brieﬂy the salient empirical facts that emerged in the last

twenty years, concerning the volume dependence and the temporal behaviour

of impact (see Bouchaud et al. 2009 for a recent review). Note that in order

to characterize impact empirically, one has to specify two time scales: a) the

“elementary” time scale ∆tover which trades are aggregated; b) the time

scale Tover which the impact of the initial trade is measured.

4.1 Concave volume dependence

The Kyle model assumes linear dependence of impact on the traded vol-

ume. One can measure the volume dependent instantaneous impact function

R(T= ∆t, v) for diﬀerent elementary time scales ∆t, ranging from the av-

erage transaction time to several hours or days. One typically ﬁnds that

the volume dependence of this impact is sublinear and well described by a

power-law:

R(T= ∆t, v)∝vψ(∆t);ψ(∆t)≤1.(8)

The exponent ψincreases with the elementary time scale, taking rather small

values ψ≃0.1−0.3 for individual trades, and increasing towards ψ= 1

when ∆tcorresponds to several thousands of trades, with some concavity

remaining for large volumes (Hasbrouck & Seppi (2001), Plerou et al. (2002)).

The correlation coeﬃcient ρ(T= ∆t) similarly increases as ∆tincrease, and

reaches rather large values >0.5 for daily returns of individual stocks, futures

or currencies (see Evans and Lyons, 2002).

Note that the small value of ψat the individual trade level means that

impact is only weakly dependent on volume, in line with many studies that

show a stronger correlation of price changes with the number of trades than

with the traded volume, see e.g. Jones et al. (1996). The small value of ψis

often interpreted in terms of discretionary trading: large market orders are

only submitted when there is a large prevailing volume at the best quote, a

conditioning that mitigates the impact of these large orders.

The above results are established using the total aggregated signed vol-

umes, independently of the origin of the trades. Large brokerage or trading

ﬁrms can also measure the price impact of their own trades; a concave impact

function is usually observed with a value of ψclose to 1/2, see e.g. Almgren

5

et al. (2005), and Execution Costs. For example, the BARRA price im-

pact model posits that, on a time interval ∆tneeded to complete a typical

trade,

R(∆t, v) = Aσrv

V,(9)

where σis the volatility and Vthe traded volume per unit time, and Aa

numerical coeﬃcient of order unity (see BARRA, 1997). Diﬀerent theoretical

justiﬁcations for this square-root impact law are given in BARRA (1997),

Grinold & Kahn (1999) and Gabaix et al. (2006).

In the case of stocks, one can also study empirically the inﬂuence of the

market capitalisation M. One ﬁnds that when ∆tcorresponds to a single

trade, the data can be approximately rescaled as (Lillo et al. 2003):

R(∆t, v)≈M−0.3FM0.3v

v,(10)

where vis the average volume per trade for a given stock, and F(.) a master

function that behaves as a power-law with exponent ψ.

4.2 Impact cannot be permanent

As we observed above, a permanent impact model leads to unpredictable

price changes only if the signed volume is uncorrelated. However, empirical

data shows that on a large variety of markets the autocorrelation of the signs

ǫndecays extremely slowly with time, over at least several days, representing

thousands of trades or more (see e.g. Bouchaud et al. 2009). The order

ﬂow is therefore found to be strongly persistent and predictable. This comes

from the fact that even “highly liquid” markets only oﬀer very small volumes

for immediate execution. The fact that the outstanding liquidity is so small

has an immediate consequence: trades must be fragmented, and need several

hours, days or even weeks to be completed. This clearly creates long memory

in the sign of the order ﬂow and shows that private information can only be

slowly incorporated into prices. This observation, however, is incompatible

with a permanent impact model such as Eq. (2), which would lead to trends,

i.e. strongly autocorrelated price changes.

4.3 A non-linear, transient impact model

In order to reconcile persistent order ﬂow with nearly unpredictable price

changes, one can postulate a non-linear, transient impact model that gener-

6

alizes Eq. (2) above:

pT=p−∞ +λ

N−1

X

n=−∞

G(N−n)ǫnvψ

n,(11)

where G(ℓ) describes the temporal behaviour of impact. One can show that

it is possible to choose a certain decaying shape for the impact function G(ℓ)

such as to oﬀset exactly the autocorrelation of the order ﬂow and ensure that

the price changes are white noise.

Assume for simplicity that volumes are all equal: vn≡v, ∀n. One can

then show that if C(ℓ) = E[ǫn·ǫn+ℓ] decays for large ℓas ℓ−γwith γ < 1

(typically γ≈0.5 for stocks), then G(ℓ) should itself decay to zero as ℓ−β

with β= (1 −γ)/2 (Bouchaud et al. 2004). A permanent impact component

G(ℓ→ ∞)>0 is only compatible with the random nature of prices if C(ℓ)

decays fast enough (γ > 1). Within this transient impact model with ﬁxed

volume of trades, the relation between the price impact function R(T), G(ℓ)

and C(ℓ) reads (Bouchaud et al. 2004):

R(T=ℓ∆t) = λvψ

G(ℓ) + X

0<j<ℓ

G(ℓ−j)C(j) + X

j>0

[G0(ℓ+j)−G0(j)] C(j)

.

(12)

In other words, the impact G(ℓ) of a single trade in isolation is diﬀerent from

the directly measurable impact R(T), which picks up contributions from the

fact that trades tend to repeat themselves in the same direction.

Note that even if the impact G(ℓ) of an individual trade decays with

time, one can show that both the total impact R(T) and the correlation

ρ(T) deﬁned by Eq. (6) tend to a non-zero limit when Tis large whenever

the relation β= (1 −γ)/2 holds.

Finally, we mentioned above that the linear impact model, corresponding

to ψ= 1, is the only choice consistent with the absence of price manipulation

strategies if impact is permanent (β= 0). But if impact is transient, other

values of ψ≤1 become acceptable. Gatheral has recently shown that if

β+ψ≥1, price manipulation is not possible (Gatheral, 2009).

4.4 Another point of view: surprise in the order ﬂow

If one insists a priori that prices must follow a strict random walk, then

only the surprise in the order ﬂow can impact the price. In other words, the

7

impact of the trades in the nth interval of time ∆tshould read (neglecting

volume ﬂuctuations):

∆pn=λvψ(ǫn−E[ǫn|In−1]),(13)

where In−1is the information set available just before the nth interval of time.

If the ǫn’s are independent, then E[ǫn|In−1] = 0 and one recovers the spec-

iﬁcation of the Kyle model. If on the other hand the ǫn’s are correlated, one

can form a prediction for the next value of ǫnbased on the past realisations

ǫm<n, such that the surprise component ǫn−E[ǫn|In−1] is by construction

uncorrelated for diﬀerent times.

Within this simpliﬁed framework, there are only two possible outcomes:

either the sign of the nth transaction matches the sign of the predictor

E[ǫn|In−1], or the signs are opposite. It is easy to show that the more likely

outcome, i.e. ǫn= sign(E[ǫn|In−1]), has the smaller impact (Gerig 2007).

Because of the positive correlation in order ﬂow, the impact of a buy follow-

ing a buy should be less than the impact of a sell following a buy – otherwise

trends would appear. The detailed microstructural mechanism for such an

history dependent asymmetric impact is a topic of research, see Bouchaud et

al. (2009) and references therein.

Using a linear autoregression model for E[ǫn|In−1] allows one to identify

the above surprise model, Eq. (13), with the transient impact model of the

previous section. Following Hasbrouck’s VAR model (Hasbrouck, 1991), one

may assume that:

E[ǫn|In−1] =

∞

X

j=1

ajǫn−j,(14)

where the coeﬃcents ajcan be computed from the sign autocorrelation C(ℓ)

using standard methods in ﬁltering theory. Then the above transient impact

model is precisely recovered provided the following identiﬁcation holds:

G(ℓ) = 1 −

ℓ−1

X

j=1

aj.(15)

8

5 Spread and impact are two sides of the

same coin

Since market makers (or liquidity providers) cannot guess the surprise of the

next trade, they post a bid price bnand an ask price angiven by:

an=pn−1+λvψ(1 −E[ǫn|In−1]) ; bn=pn−1+λvψ(−1−E[ǫn|In−1]) ,

(16)

The above rule ensures no ex-post regrets for the market maker: whatever

the sign of the trade, the traded price is always the ‘right’ one (see Madhavan

et al. (1997)). The bid-ask spread Sis then given by:

S=an−bn= 2λvψ,(17)

showing that spread and impact are two sides of the same coin: liquidity

providers must be compensated for the adverse impact of market order trades

(see Adverse Selection, Limit Order Markets). Conversely, one sees

that the impact of trades is expected to be proportional to the bid-ask spread,

suggesting that the volatility per trade σ1is also proportional to the bid-ask

spread. Such a correlation is well supported by empirical data, see Wyart

et al. (2008). The relation with the volatility per unit time σinvolves the

trading frequency f, through σ=σ1√f.

6 Conclusion

Although “price impact” seems to convey the idea of a forceful and intuitive

mechanism, the story behind it might not be that simple. Empirical stud-

ies show that the correlation between signed order ﬂow and price changes

is indeed strong, but the impact of trades is neither linear in volume nor

permanent, as assumed in several models, such as the Kyle model. Impact is

rather found to be strongly concave in volume and transient, the latter prop-

erty being a necessary consequence of the long-memory nature of the order

ﬂow. Only after averaging on a long time scale (on the order of days) may

an eﬀective linear and permanent model make sense. This is an important

observation for execution costs monitoring and control, but also for building

agent-based models of market activity that often posit linear and permanent

impact.

9

Coming back to Hasbrouck’s comment (Hasbrouck 2007), do trades im-

pact prices or do they forecast future price changes? Since trading on modern

electronic markets is anonymous, there cannot be any obvious diﬀerence be-

tween “informed” trades and “uninformed” trades if the strategies used for

their execution are similar. Hence, the impact of any trade must statistically

be the same, whether informed or not informed. Impact indeed allows pri-

vate information to be reﬂected in prices, but by the same token, random

ﬂuctuations in order ﬂow (induced by noise traders) must also contribute to

the volatility of markets.

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