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Price Impact

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Price Impact

Abstract

We define what "Price Impact" means, and how it is measured and modelled in the recent literature. Although this notion seems to convey the idea of a forceful and intuitive mechanism, we discuss why things might not be that simple. Empirical studies show that while the correlation between signed order flow and price changes is strong, the impact of trades on prices is neither linear in volume nor permanent. Impact allows private information to be reflected in prices, but by the same token, random fluctuations in order flow must also contribute to the volatility of markets.
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=Ntis:
pT=p0+
N1
X
n=0
pn=p0+λ
N1
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[(pTp0)·ǫ0]E[(pTp0)] 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=|vbvs|.
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[(pTp0)·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(pTp0)·PN1
n=0 ǫnvni
rE[(pTp0)]2Eh(PN1
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
N1
X
n=0
ǫnvn+
N1
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 ψ0.10.3 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) = rv
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)M0.3FM0.3v
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
N1
X
n=−∞
G(Nn)ǫnvψ
n,(11)
where G() describes the temporal behaviour of impact. One can show that
it is possible to choose a certain decaying shape for the impact function G()
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: vnv, n. One can
then show that if C() = E[ǫn·ǫn+] decays for large as γwith γ < 1
(typically γ0.5 for stocks), then G() should itself decay to zero as β
with β= (1 γ)/2 (Bouchaud et al. 2004). A permanent impact component
G(→ ∞)>0 is only compatible with the random nature of prices if C()
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()
and C() reads (Bouchaud et al. 2004):
R(T=t) = λvψ
G() + X
0<j<ℓ
G(j)C(j) + X
j>0
[G0(+j)G0(j)] C(j)
.
(12)
In other words, the impact G() 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() 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 ψ1 become acceptable. Gatheral has recently shown that if
β+ψ1, 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(ǫnE[ǫn|In1]),(13)
where In1is the information set available just before the nth interval of time.
If the ǫn’s are independent, then E[ǫn|In1] = 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 ǫnE[ǫn|In1] 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|In1], or the signs are opposite. It is easy to show that the more likely
outcome, i.e. ǫn= sign(E[ǫn|In1]), 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|In1] 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|In1] =
X
j=1
ajǫnj,(14)
where the coefficents ajcan be computed from the sign autocorrelation C()
using standard methods in filtering theory. Then the above transient impact
model is precisely recovered provided the following identification holds:
G() = 1
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=pn1+λvψ(1 E[ǫn|In1]) ; bn=pn1+λvψ(1E[ǫn|In1]) ,
(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=anbn= 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=σ1f.
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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11
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